Unit 14 — End of Topic Tests (reviewer copy: all keys and model answers visible)

[
 {
  "value": 4,
  "misconception": "you read the mass number as the neutron count — the mass number 4 counts protons AND neutrons together",
  "feedback": "This answer is wrong — you read the mass number as the neutron count; the mass number 4 counts protons AND neutrons together, so subtract the 2 protons."
 },
 {
  "value": 6,
  "misconception": "you added the two numbers — the neutron count is the mass number MINUS the proton count",
  "feedback": "This answer is wrong — you added the two numbers; the neutron count is the mass number MINUS the proton count: 4 − 2 = 2."
 },
 {
  "value": 0,
  "misconception": "you gave the particle no neutrons — 2 of its 4 nuclear particles are neutrons",
  "feedback": "This answer is wrong — you gave the particle no neutrons; 2 of its 4 nuclear particles are neutrons."
 }
]

{
 "keyExplanation": "Neutron count = mass number − proton count = 4 − 2 = 2. The alpha particle's 4 nuclear particles are 2 protons and 2 neutrons.",
 "anticipatedWrong": [
  {
   "value": 4,
   "misconception": "you read the mass number as the neutron count — the mass number 4 counts protons AND neutrons together",
   "feedback": "This answer is wrong — you read the mass number as the neutron count; the mass number 4 counts protons AND neutrons together, so subtract the 2 protons."
  },
  {
   "value": 6,
   "misconception": "you added the two numbers — the neutron count is the mass number MINUS the proton count",
   "feedback": "This answer is wrong — you added the two numbers; the neutron count is the mass number MINUS the proton count: 4 − 2 = 2."
  },
  {
   "value": 0,
   "misconception": "you gave the particle no neutrons — 2 of its 4 nuclear particles are neutrons",
   "feedback": "This answer is wrong — you gave the particle no neutrons; 2 of its 4 nuclear particles are neutrons."
  }
 ]
}

[
 {
  "value": "²²³₈₅At",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²²¹₈₉Ac",
  "misconception": "you kept the atomic number — losing 2 protons changes the element",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
 },
 {
  "value": "²²⁹₉₁Pa",
  "misconception": "you added the alpha particle's numbers instead of subtracting them — the nucleus LOSES 2 protons and 2 neutrons",
  "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them; the nucleus LOSES 2 protons and 2 neutrons."
 }
]

{
 "keyExplanation": "Mass number: 225 − 4 = 221. Atomic number: 89 − 2 = 87. The element with atomic number 87 is francium, so alpha decay leaves ²²¹₈₇Fr behind.",
 "anticipatedWrong": [
  {
   "value": "²²³₈₅At",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²²¹₈₉Ac",
   "misconception": "you kept the atomic number — losing 2 protons changes the element",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
  },
  {
   "value": "²²⁹₉₁Pa",
   "misconception": "you added the alpha particle's numbers instead of subtracting them — the nucleus LOSES 2 protons and 2 neutrons",
   "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them; the nucleus LOSES 2 protons and 2 neutrons."
  }
 ]
}

[
 {
  "value": "¹⁴¹₅₇La",
  "misconception": "you dropped the atomic number instead of raising it — a neutron becomes a proton, so the count rises",
  "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; a neutron becomes a proton, so the count rises."
 },
 {
  "value": "¹⁴¹₅₈Ce",
  "misconception": "you kept the atomic number — one more proton means the atom is no longer cerium",
  "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer cerium."
 },
 {
  "value": "¹⁴⁰₅₉Pr",
  "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
 }
]

{
 "keyExplanation": "Beta decay keeps the mass number and raises the atomic number by 1: the mass number stays 141, and 58 + 1 = 59. The element with atomic number 59 is praseodymium, so the nuclide left behind is ¹⁴¹₅₉Pr.",
 "anticipatedWrong": [
  {
   "value": "¹⁴¹₅₇La",
   "misconception": "you dropped the atomic number instead of raising it — a neutron becomes a proton, so the count rises",
   "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; a neutron becomes a proton, so the count rises."
  },
  {
   "value": "¹⁴¹₅₈Ce",
   "misconception": "you kept the atomic number — one more proton means the atom is no longer cerium",
   "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer cerium."
  },
  {
   "value": "¹⁴⁰₅₉Pr",
   "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
  }
 ]
}

[
 {
  "value": 88,
  "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
  "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 88 + 2 = 90."
 },
 {
  "value": 92,
  "misconception": "you added the alpha particle's mass number instead of its atomic number — the atomic-number total uses the bottom numbers",
  "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers: 88 + 2."
 },
 {
  "value": 86,
  "misconception": "you subtracted the alpha particle's 2 instead of adding it — the right side's atomic numbers add together",
  "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
 }
]

{
 "keyExplanation": "The right side holds ²²³₈₈Ra and ⁴₂He. Their atomic numbers add: 88 + 2 = 90, matching the left side's 90 — the atomic numbers balance, because electric charge never appears or disappears.",
 "anticipatedWrong": [
  {
   "value": 88,
   "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
   "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 88 + 2 = 90."
  },
  {
   "value": 92,
   "misconception": "you added the alpha particle's mass number instead of its atomic number — the atomic-number total uses the bottom numbers",
   "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers: 88 + 2."
  },
  {
   "value": 86,
   "misconception": "you subtracted the alpha particle's 2 instead of adding it — the right side's atomic numbers add together",
   "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
  }
 ]
}

[
 {
  "value": "⁵⁹₂₆Fe → ⁵⁹₂₅Mn + ⁰₋₁e",
  "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
  "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
 },
 {
  "value": "⁵⁹₂₆Fe → ⁵⁸₂₇Co + ⁰₋₁e",
  "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
 },
 {
  "value": "⁵⁹₂₆Fe → ⁵⁹₂₇Co",
  "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
  "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
 }
]

{
 "keyExplanation": "The mass number stays 59; the atomic number rises to 26 + 1 = 27, and the element with atomic number 27 is cobalt. The complete equation is ⁵⁹₂₆Fe → ⁵⁹₂₇Co + ⁰₋₁e; check: 59 = 59 + 0 and 26 = 27 + (−1).",
 "anticipatedWrong": [
  {
   "value": "⁵⁹₂₆Fe → ⁵⁹₂₅Mn + ⁰₋₁e",
   "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
   "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
  },
  {
   "value": "⁵⁹₂₆Fe → ⁵⁸₂₇Co + ⁰₋₁e",
   "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
  },
  {
   "value": "⁵⁹₂₆Fe → ⁵⁹₂₇Co",
   "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
   "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
  }
 ]
}

[
 {
  "value": "²⁵⁷₁₀₀Fm → ²⁵⁵₉₆Cm + ⁴₂He",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²⁵⁷₁₀₀Fm → ²⁵³₁₀₀Fm + ⁴₂He",
  "misconception": "you kept the atomic number — losing 2 protons changes the element",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
 },
 {
  "value": "²⁵⁷₁₀₀Fm → ²⁵³₉₈Cf",
  "misconception": "you left out the alpha particle — without it, neither total balances",
  "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
 }
]

{
 "keyExplanation": "New mass number: 257 − 4 = 253. New atomic number: 100 − 2 = 98, and the element with atomic number 98 is californium. The complete equation is ²⁵⁷₁₀₀Fm → ²⁵³₉₈Cf + ⁴₂He; check: 257 = 253 + 4 and 100 = 98 + 2.",
 "anticipatedWrong": [
  {
   "value": "²⁵⁷₁₀₀Fm → ²⁵⁵₉₆Cm + ⁴₂He",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²⁵⁷₁₀₀Fm → ²⁵³₁₀₀Fm + ⁴₂He",
   "misconception": "you kept the atomic number — losing 2 protons changes the element",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
  },
  {
   "value": "²⁵⁷₁₀₀Fm → ²⁵³₉₈Cf",
   "misconception": "you left out the alpha particle — without it, neither total balances",
   "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
  }
 ]
}

[
 {
  "value": "¹⁴⁰₅₉Pr",
  "misconception": "you added 1 to the atomic number instead of adding the beta particle's −1 — the left side must total 58 + (−1) = 57",
  "feedback": "This answer is wrong — you added 1 instead of adding the beta particle's −1; the left side must total 58 + (−1) = 57."
 },
 {
  "value": "¹⁴⁰₅₈Ce",
  "misconception": "you copied the shown nucleus — the missing species must make BOTH totals balance, so its atomic number is 57",
  "feedback": "This answer is wrong — you copied the shown nucleus; the missing species must make BOTH totals balance, so its atomic number is 57."
 },
 {
  "value": "¹³⁹₅₇La",
  "misconception": "you dropped the mass number — the beta particle's mass number is 0, so the totals keep the mass at 140",
  "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so the totals keep the mass at 140."
 }
]

{
 "keyExplanation": "The missing species sits on the left, so its numbers are the right side's totals. Mass number: 140 + 0 = 140. Atomic number: 58 + (−1) = 57 — lanthanum. The starting nucleus is ¹⁴⁰₅₇La, and the equation reads ¹⁴⁰₅₇La → ¹⁴⁰₅₈Ce + ⁰₋₁e.",
 "anticipatedWrong": [
  {
   "value": "¹⁴⁰₅₉Pr",
   "misconception": "you added 1 to the atomic number instead of adding the beta particle's −1 — the left side must total 58 + (−1) = 57",
   "feedback": "This answer is wrong — you added 1 instead of adding the beta particle's −1; the left side must total 58 + (−1) = 57."
  },
  {
   "value": "¹⁴⁰₅₈Ce",
   "misconception": "you copied the shown nucleus — the missing species must make BOTH totals balance, so its atomic number is 57",
   "feedback": "This answer is wrong — you copied the shown nucleus; the missing species must make BOTH totals balance, so its atomic number is 57."
  },
  {
   "value": "¹³⁹₅₇La",
   "misconception": "you dropped the mass number — the beta particle's mass number is 0, so the totals keep the mass at 140",
   "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so the totals keep the mass at 140."
  }
 ]
}

[
 {
  "value": 4,
  "misconception": "you read the mass number as the neutron count — the 4 counts protons AND neutrons together",
  "feedback": "This answer is wrong — you read the mass number as the neutron count; the 4 counts protons AND neutrons together."
 },
 {
  "value": 6,
  "misconception": "you added the symbol's two numbers — subtract them instead: 4 − 2",
  "feedback": "This answer is wrong — you added the symbol's two numbers; the neutron count is 4 − 2 = 2."
 },
 {
  "value": 0,
  "misconception": "you gave the particle no neutrons — half of its 4 nuclear particles are neutrons",
  "feedback": "This answer is wrong — you gave the particle no neutrons; half of its 4 nuclear particles are neutrons."
 }
]

{
 "keyExplanation": "The top number counts protons plus neutrons and the bottom number counts protons, so the neutrons are 4 − 2 = 2 — the alpha particle is 2 protons and 2 neutrons bound together.",
 "anticipatedWrong": [
  {
   "value": 4,
   "misconception": "you read the mass number as the neutron count — the 4 counts protons AND neutrons together",
   "feedback": "This answer is wrong — you read the mass number as the neutron count; the 4 counts protons AND neutrons together."
  },
  {
   "value": 6,
   "misconception": "you added the symbol's two numbers — subtract them instead: 4 − 2",
   "feedback": "This answer is wrong — you added the symbol's two numbers; the neutron count is 4 − 2 = 2."
  },
  {
   "value": 0,
   "misconception": "you gave the particle no neutrons — half of its 4 nuclear particles are neutrons",
   "feedback": "This answer is wrong — you gave the particle no neutrons; half of its 4 nuclear particles are neutrons."
  }
 ]
}

[
 {
  "value": "²¹⁹₈₃Bi",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²¹⁷₈₇Fr",
  "misconception": "you kept the atomic number — the nucleus loses 2 protons, so the atom is no longer francium",
  "feedback": "This answer is wrong — you kept the atomic number; the nucleus loses 2 protons, so the atom is no longer francium."
 },
 {
  "value": "²²⁵₈₉Ac",
  "misconception": "you added the alpha particle's numbers — the nucleus loses them; it does not gain them",
  "feedback": "This answer is wrong — you added the alpha particle's numbers; the nucleus loses them — it does not gain them."
 }
]

{
 "keyExplanation": "Mass number: 221 − 4 = 217. Atomic number: 87 − 2 = 85. The element with atomic number 85 is astatine, so the nuclide left behind is ²¹⁷₈₅At.",
 "anticipatedWrong": [
  {
   "value": "²¹⁹₈₃Bi",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²¹⁷₈₇Fr",
   "misconception": "you kept the atomic number — the nucleus loses 2 protons, so the atom is no longer francium",
   "feedback": "This answer is wrong — you kept the atomic number; the nucleus loses 2 protons, so the atom is no longer francium."
  },
  {
   "value": "²²⁵₈₉Ac",
   "misconception": "you added the alpha particle's numbers — the nucleus loses them; it does not gain them",
   "feedback": "This answer is wrong — you added the alpha particle's numbers; the nucleus loses them — it does not gain them."
  }
 ]
}

[
 {
  "value": "¹⁶⁹₆₇Ho",
  "misconception": "you dropped the atomic number instead of raising it — the neutron-to-proton change adds a proton",
  "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; the neutron-to-proton change adds a proton."
 },
 {
  "value": "¹⁶⁹₆₈Er",
  "misconception": "you kept the atomic number — one more proton means the atom is no longer erbium",
  "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer erbium."
 },
 {
  "value": "¹⁶⁸₆₉Tm",
  "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron, so the mass number is unchanged",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron, so the mass number is unchanged."
 }
]

{
 "keyExplanation": "The mass number stays 169, and the atomic number rises: 68 + 1 = 69. The element with atomic number 69 is thulium, so the nuclide left behind is ¹⁶⁹₆₉Tm.",
 "anticipatedWrong": [
  {
   "value": "¹⁶⁹₆₇Ho",
   "misconception": "you dropped the atomic number instead of raising it — the neutron-to-proton change adds a proton",
   "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; the neutron-to-proton change adds a proton."
  },
  {
   "value": "¹⁶⁹₆₈Er",
   "misconception": "you kept the atomic number — one more proton means the atom is no longer erbium",
   "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer erbium."
  },
  {
   "value": "¹⁶⁸₆₉Tm",
   "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron, so the mass number is unchanged",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron, so the mass number is unchanged."
  }
 ]
}

[
 {
  "value": 82,
  "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
  "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 82 + 2 = 84."
 },
 {
  "value": 86,
  "misconception": "you added the alpha particle's mass number instead of its atomic number",
  "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; use the bottom numbers: 82 + 2."
 },
 {
  "value": 80,
  "misconception": "you subtracted the alpha particle's 2 instead of adding it",
  "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
 }
]

{
 "keyExplanation": "Add the right side's atomic numbers: 82 + 2 = 84. That matches the 84 on the left — the atomic numbers balance, because electric charge never appears or disappears.",
 "anticipatedWrong": [
  {
   "value": 82,
   "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
   "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 82 + 2 = 84."
  },
  {
   "value": 86,
   "misconception": "you added the alpha particle's mass number instead of its atomic number",
   "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; use the bottom numbers: 82 + 2."
  },
  {
   "value": 80,
   "misconception": "you subtracted the alpha particle's 2 instead of adding it",
   "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
  }
 ]
}

[
 {
  "value": "¹⁶⁶₆₇Ho → ¹⁶⁶₆₆Dy + ⁰₋₁e",
  "misconception": "you subtracted 1 from the atomic number — the neutron-to-proton change RAISES it by 1",
  "feedback": "This answer is wrong — you subtracted 1 from the atomic number; the neutron-to-proton change RAISES it by 1."
 },
 {
  "value": "¹⁶⁵₆₈Er + ⁰₋₁e as the right side",
  "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
 },
 {
  "value": "¹⁶⁶₆₇Ho → ¹⁶⁶₆₈Er",
  "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
  "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
 }
]

{
 "keyExplanation": "The mass number stays 166; the atomic number rises to 67 + 1 = 68, and the element with atomic number 68 is erbium. The complete equation is ¹⁶⁶₆₇Ho → ¹⁶⁶₆₈Er + ⁰₋₁e; check: 166 = 166 + 0 and 67 = 68 + (−1).",
 "anticipatedWrong": [
  {
   "value": "¹⁶⁶₆₇Ho → ¹⁶⁶₆₆Dy + ⁰₋₁e",
   "misconception": "you subtracted 1 from the atomic number — the neutron-to-proton change RAISES it by 1",
   "feedback": "This answer is wrong — you subtracted 1 from the atomic number; the neutron-to-proton change RAISES it by 1."
  },
  {
   "value": "¹⁶⁵₆₈Er + ⁰₋₁e as the right side",
   "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
  },
  {
   "value": "¹⁶⁶₆₇Ho → ¹⁶⁶₆₈Er",
   "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
   "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
  }
 ]
}

[
 {
  "value": "¹⁵⁴₆₆Dy → ¹⁵²₆₂Sm + ⁴₂He",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "¹⁵⁴₆₆Dy → ¹⁵⁰₆₅Tb + ⁴₂He",
  "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
  "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
 },
 {
  "value": "¹⁵⁴₆₆Dy + ⁴₂He → ¹⁵⁰₆₄Gd",
  "misconception": "you put the alpha particle on the left — it is produced by the decay and belongs on the right",
  "feedback": "This answer is wrong — you put the alpha particle on the left; it is produced by the decay and belongs on the right."
 }
]

{
 "keyExplanation": "New mass number: 154 − 4 = 150. New atomic number: 66 − 2 = 64, and the element with atomic number 64 is gadolinium. The complete equation is ¹⁵⁴₆₆Dy → ¹⁵⁰₆₄Gd + ⁴₂He; check: 154 = 150 + 4 and 66 = 64 + 2.",
 "anticipatedWrong": [
  {
   "value": "¹⁵⁴₆₆Dy → ¹⁵²₆₂Sm + ⁴₂He",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "¹⁵⁴₆₆Dy → ¹⁵⁰₆₅Tb + ⁴₂He",
   "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
   "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
  },
  {
   "value": "¹⁵⁴₆₆Dy + ⁴₂He → ¹⁵⁰₆₄Gd",
   "misconception": "you put the alpha particle on the left — it is produced by the decay and belongs on the right",
   "feedback": "This answer is wrong — you put the alpha particle on the left; it is produced by the decay and belongs on the right."
  }
 ]
}

[
 {
  "value": "¹⁵⁶₆₂Sm",
  "misconception": "you subtracted the −1 as if it were +1 — 63 = ? + (−1) forces the missing atomic number UP to 64",
  "feedback": "This answer is wrong — you subtracted the −1 as if it were +1; 63 = ? + (−1) forces the missing atomic number UP to 64."
 },
 {
  "value": "¹⁵⁶₆₃Eu",
  "misconception": "you kept the starting atomic number — the beta particle's −1 raises the new nucleus's atomic number to 64",
  "feedback": "This answer is wrong — you kept the starting atomic number; the beta particle's −1 raises the new nucleus's atomic number to 64."
 },
 {
  "value": "¹⁵⁵₆₄Gd",
  "misconception": "you dropped the mass number — the beta particle carries mass number 0, so 156 stays",
  "feedback": "This answer is wrong — you dropped the mass number; the beta particle carries mass number 0, so 156 stays."
 }
]

{
 "keyExplanation": "Missing mass number: 156 − 0 = 156. Missing atomic number: 63 − (−1) = 64 — gadolinium. The missing species is ¹⁵⁶₆₄Gd, and the equation reads ¹⁵⁶₆₃Eu → ¹⁵⁶₆₄Gd + ⁰₋₁e.",
 "anticipatedWrong": [
  {
   "value": "¹⁵⁶₆₂Sm",
   "misconception": "you subtracted the −1 as if it were +1 — 63 = ? + (−1) forces the missing atomic number UP to 64",
   "feedback": "This answer is wrong — you subtracted the −1 as if it were +1; 63 = ? + (−1) forces the missing atomic number UP to 64."
  },
  {
   "value": "¹⁵⁶₆₃Eu",
   "misconception": "you kept the starting atomic number — the beta particle's −1 raises the new nucleus's atomic number to 64",
   "feedback": "This answer is wrong — you kept the starting atomic number; the beta particle's −1 raises the new nucleus's atomic number to 64."
  },
  {
   "value": "¹⁵⁵₆₄Gd",
   "misconception": "you dropped the mass number — the beta particle carries mass number 0, so 156 stays",
   "feedback": "This answer is wrong — you dropped the mass number; the beta particle carries mass number 0, so 156 stays."
  }
 ]
}

[
 {
  "value": 4,
  "misconception": "you took the top number as the neutron count — it is the mass number, protons plus neutrons",
  "feedback": "This answer is wrong — you took the top number as the neutron count; it is the mass number, protons plus neutrons."
 },
 {
  "value": 6,
  "misconception": "you added 4 and 2 — the neutron count comes from subtracting the proton count",
  "feedback": "This answer is wrong — you added 4 and 2; the neutron count comes from subtracting: 4 − 2 = 2."
 },
 {
  "value": 0,
  "misconception": "you counted no neutrons — the particle carries 2 protons and 2 neutrons",
  "feedback": "This answer is wrong — you counted no neutrons; the particle carries 2 protons and 2 neutrons."
 }
]

{
 "keyExplanation": "Neutrons = mass number − proton count = 4 − 2 = 2. Two protons and two neutrons is exactly the alpha particle's makeup.",
 "anticipatedWrong": [
  {
   "value": 4,
   "misconception": "you took the top number as the neutron count — it is the mass number, protons plus neutrons",
   "feedback": "This answer is wrong — you took the top number as the neutron count; it is the mass number, protons plus neutrons."
  },
  {
   "value": 6,
   "misconception": "you added 4 and 2 — the neutron count comes from subtracting the proton count",
   "feedback": "This answer is wrong — you added 4 and 2; the neutron count comes from subtracting: 4 − 2 = 2."
  },
  {
   "value": 0,
   "misconception": "you counted no neutrons — the particle carries 2 protons and 2 neutrons",
   "feedback": "This answer is wrong — you counted no neutrons; the particle carries 2 protons and 2 neutrons."
  }
 ]
}

[
 {
  "value": "¹⁷²₆₈Er",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "¹⁷⁰₇₂Hf",
  "misconception": "you kept the atomic number — losing 2 protons means the atom is no longer hafnium",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons means the atom is no longer hafnium."
 },
 {
  "value": "¹⁷⁸₇₄W",
  "misconception": "you added the alpha particle's numbers instead of subtracting them",
  "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them."
 }
]

{
 "keyExplanation": "Mass number: 174 − 4 = 170. Atomic number: 72 − 2 = 70. The element with atomic number 70 is ytterbium, so alpha decay leaves ¹⁷⁰₇₀Yb behind.",
 "anticipatedWrong": [
  {
   "value": "¹⁷²₆₈Er",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "¹⁷⁰₇₂Hf",
   "misconception": "you kept the atomic number — losing 2 protons means the atom is no longer hafnium",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons means the atom is no longer hafnium."
  },
  {
   "value": "¹⁷⁸₇₄W",
   "misconception": "you added the alpha particle's numbers instead of subtracting them",
   "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them."
  }
 ]
}

[
 {
  "value": "¹⁷⁵₆₉Tm",
  "misconception": "you dropped the atomic number instead of raising it — the proton count RISES by 1 in beta decay",
  "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; the proton count RISES by 1 in beta decay."
 },
 {
  "value": "¹⁷⁵₇₀Yb",
  "misconception": "you kept the atomic number — the neutron-to-proton change means the atom is no longer ytterbium",
  "feedback": "This answer is wrong — you kept the atomic number; the neutron-to-proton change means the atom is no longer ytterbium."
 },
 {
  "value": "¹⁷⁴₇₁Lu",
  "misconception": "you dropped the mass number by 1 — the total count of protons plus neutrons is unchanged",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the total count of protons plus neutrons is unchanged."
 }
]

{
 "keyExplanation": "Beta decay keeps the mass number at 175 and raises the atomic number: 70 + 1 = 71. The element with atomic number 71 is lutetium, so the nuclide left behind is ¹⁷⁵₇₁Lu.",
 "anticipatedWrong": [
  {
   "value": "¹⁷⁵₆₉Tm",
   "misconception": "you dropped the atomic number instead of raising it — the proton count RISES by 1 in beta decay",
   "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; the proton count RISES by 1 in beta decay."
  },
  {
   "value": "¹⁷⁵₇₀Yb",
   "misconception": "you kept the atomic number — the neutron-to-proton change means the atom is no longer ytterbium",
   "feedback": "This answer is wrong — you kept the atomic number; the neutron-to-proton change means the atom is no longer ytterbium."
  },
  {
   "value": "¹⁷⁴₇₁Lu",
   "misconception": "you dropped the mass number by 1 — the total count of protons plus neutrons is unchanged",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the total count of protons plus neutrons is unchanged."
  }
 ]
}

[
 {
  "value": 84,
  "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
  "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 84 + 2 = 86."
 },
 {
  "value": 88,
  "misconception": "you added the alpha particle's mass number instead of its atomic number",
  "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers."
 },
 {
  "value": 82,
  "misconception": "you subtracted the alpha particle's 2 instead of adding it",
  "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
 }
]

{
 "keyExplanation": "The right side holds ²¹⁵₈₄Po and ⁴₂He, so the atomic numbers give 84 + 2 = 86 — matching the left side's 86. The atomic numbers balance, because electric charge never appears or disappears.",
 "anticipatedWrong": [
  {
   "value": 84,
   "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
   "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 84 + 2 = 86."
  },
  {
   "value": 88,
   "misconception": "you added the alpha particle's mass number instead of its atomic number",
   "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers."
  },
  {
   "value": 82,
   "misconception": "you subtracted the alpha particle's 2 instead of adding it",
   "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
  }
 ]
}

[
 {
  "value": "¹⁸⁷₇₄W → ¹⁸⁷₇₃Ta + ⁰₋₁e",
  "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
  "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
 },
 {
  "value": "¹⁸⁷₇₄W → ¹⁸⁶₇₅Re + ⁰₋₁e",
  "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
 },
 {
  "value": "¹⁸⁷₇₄W + ⁰₋₁e → ¹⁸⁷₇₅Re",
  "misconception": "you put the beta particle on the left — it is produced by the decay and belongs on the right",
  "feedback": "This answer is wrong — you put the beta particle on the left; it is produced by the decay and belongs on the right."
 }
]

{
 "keyExplanation": "The mass number stays 187; the atomic number rises to 74 + 1 = 75, and the element with atomic number 75 is rhenium. The complete equation is ¹⁸⁷₇₄W → ¹⁸⁷₇₅Re + ⁰₋₁e; check: 187 = 187 + 0 and 74 = 75 + (−1).",
 "anticipatedWrong": [
  {
   "value": "¹⁸⁷₇₄W → ¹⁸⁷₇₃Ta + ⁰₋₁e",
   "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
   "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
  },
  {
   "value": "¹⁸⁷₇₄W → ¹⁸⁶₇₅Re + ⁰₋₁e",
   "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
  },
  {
   "value": "¹⁸⁷₇₄W + ⁰₋₁e → ¹⁸⁷₇₅Re",
   "misconception": "you put the beta particle on the left — it is produced by the decay and belongs on the right",
   "feedback": "This answer is wrong — you put the beta particle on the left; it is produced by the decay and belongs on the right."
  }
 ]
}

[
 {
  "value": "²¹⁷₈₆Rn → ²¹⁵₈₂Pb + ⁴₂He",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²¹⁷₈₆Rn → ²¹³₈₆Rn + ⁴₂He",
  "misconception": "you kept the atomic number — losing 2 protons changes the element",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
 },
 {
  "value": "²¹⁷₈₆Rn → ²¹³₈₄Po",
  "misconception": "you left out the alpha particle — without it, neither total balances",
  "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
 }
]

{
 "keyExplanation": "New mass number: 217 − 4 = 213. New atomic number: 86 − 2 = 84, and the element with atomic number 84 is polonium. The complete equation is ²¹⁷₈₆Rn → ²¹³₈₄Po + ⁴₂He; check: 217 = 213 + 4 and 86 = 84 + 2.",
 "anticipatedWrong": [
  {
   "value": "²¹⁷₈₆Rn → ²¹⁵₈₂Pb + ⁴₂He",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²¹⁷₈₆Rn → ²¹³₈₆Rn + ⁴₂He",
   "misconception": "you kept the atomic number — losing 2 protons changes the element",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
  },
  {
   "value": "²¹⁷₈₆Rn → ²¹³₈₄Po",
   "misconception": "you left out the alpha particle — without it, neither total balances",
   "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
  }
 ]
}

[
 {
  "value": "²⁰¹₇₉Au",
  "misconception": "you subtracted instead of adding — the missing species is on the LEFT, so its numbers are the right side's totals: 205 + 4 and 81 + 2",
  "feedback": "This answer is wrong — you subtracted instead of adding; the missing species is on the LEFT, so its numbers are the right side's totals: 205 + 4 and 81 + 2."
 },
 {
  "value": "²⁰⁹₈₁Tl",
  "misconception": "you added only the mass number — the atomic total needs 81 + 2 = 83 as well",
  "feedback": "This answer is wrong — you added only the mass number; the atomic total needs 81 + 2 = 83 as well."
 },
 {
  "value": "²⁰⁷₈₅At",
  "misconception": "you swapped the two additions — mass 205 + 4 = 209 and atomic 81 + 2 = 83, not the reverse",
  "feedback": "This answer is wrong — you swapped the two additions; mass 205 + 4 = 209 and atomic 81 + 2 = 83, not the reverse."
 }
]

{
 "keyExplanation": "The missing species sits on the left, so its numbers are the right side's totals. Mass number: 205 + 4 = 209. Atomic number: 81 + 2 = 83 — bismuth. The starting nucleus is ²⁰⁹₈₃Bi, and the equation reads ²⁰⁹₈₃Bi → ²⁰⁵₈₁Tl + ⁴₂He.",
 "anticipatedWrong": [
  {
   "value": "²⁰¹₇₉Au",
   "misconception": "you subtracted instead of adding — the missing species is on the LEFT, so its numbers are the right side's totals: 205 + 4 and 81 + 2",
   "feedback": "This answer is wrong — you subtracted instead of adding; the missing species is on the LEFT, so its numbers are the right side's totals: 205 + 4 and 81 + 2."
  },
  {
   "value": "²⁰⁹₈₁Tl",
   "misconception": "you added only the mass number — the atomic total needs 81 + 2 = 83 as well",
   "feedback": "This answer is wrong — you added only the mass number; the atomic total needs 81 + 2 = 83 as well."
  },
  {
   "value": "²⁰⁷₈₅At",
   "misconception": "you swapped the two additions — mass 205 + 4 = 209 and atomic 81 + 2 = 83, not the reverse",
   "feedback": "This answer is wrong — you swapped the two additions; mass 205 + 4 = 209 and atomic 81 + 2 = 83, not the reverse."
  }
 ]
}

[
 {
  "value": 4,
  "misconception": "you answered with the mass number — that counts protons and neutrons together",
  "feedback": "This answer is wrong — you answered with the mass number; that counts protons and neutrons together."
 },
 {
  "value": 6,
  "misconception": "you combined the two numbers by adding — the neutron count is their difference",
  "feedback": "This answer is wrong — you combined the two numbers by adding; the neutron count is their difference, 4 − 2 = 2."
 },
 {
  "value": 0,
  "misconception": "you decided the particle holds only protons — 2 of its 4 particles are neutrons",
  "feedback": "This answer is wrong — you decided the particle holds only protons; 2 of its 4 particles are neutrons."
 }
]

{
 "keyExplanation": "Subtract the proton count from the mass number: 4 − 2 = 2 neutrons. That matches the alpha particle's makeup of 2 protons and 2 neutrons.",
 "anticipatedWrong": [
  {
   "value": 4,
   "misconception": "you answered with the mass number — that counts protons and neutrons together",
   "feedback": "This answer is wrong — you answered with the mass number; that counts protons and neutrons together."
  },
  {
   "value": 6,
   "misconception": "you combined the two numbers by adding — the neutron count is their difference",
   "feedback": "This answer is wrong — you combined the two numbers by adding; the neutron count is their difference, 4 − 2 = 2."
  },
  {
   "value": 0,
   "misconception": "you decided the particle holds only protons — 2 of its 4 particles are neutrons",
   "feedback": "This answer is wrong — you decided the particle holds only protons; 2 of its 4 particles are neutrons."
  }
 ]
}

[
 {
  "value": "²¹⁵₈₁Tl",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²¹³₈₅At",
  "misconception": "you kept the atomic number — the nucleus loses 2 protons, so the atom is no longer astatine",
  "feedback": "This answer is wrong — you kept the atomic number; the nucleus loses 2 protons, so the atom is no longer astatine."
 },
 {
  "value": "²²¹₈₇Fr",
  "misconception": "you added the alpha particle's numbers — the nucleus loses 2 protons and 2 neutrons",
  "feedback": "This answer is wrong — you added the alpha particle's numbers; the nucleus loses 2 protons and 2 neutrons."
 }
]

{
 "keyExplanation": "Mass number: 217 − 4 = 213. Atomic number: 85 − 2 = 83. The element with atomic number 83 is bismuth, so the nuclide left behind is ²¹³₈₃Bi.",
 "anticipatedWrong": [
  {
   "value": "²¹⁵₈₁Tl",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²¹³₈₅At",
   "misconception": "you kept the atomic number — the nucleus loses 2 protons, so the atom is no longer astatine",
   "feedback": "This answer is wrong — you kept the atomic number; the nucleus loses 2 protons, so the atom is no longer astatine."
  },
  {
   "value": "²²¹₈₇Fr",
   "misconception": "you added the alpha particle's numbers — the nucleus loses 2 protons and 2 neutrons",
   "feedback": "This answer is wrong — you added the alpha particle's numbers; the nucleus loses 2 protons and 2 neutrons."
  }
 ]
}

[
 {
  "value": "¹⁸⁸₇₄W",
  "misconception": "you dropped the atomic number instead of raising it — a neutron becomes a proton, so the count rises",
  "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; a neutron becomes a proton, so the count rises."
 },
 {
  "value": "¹⁸⁸₇₅Re",
  "misconception": "you kept the atomic number — one more proton means the atom is no longer rhenium",
  "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer rhenium."
 },
 {
  "value": "¹⁸⁷₇₆Os",
  "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
 }
]

{
 "keyExplanation": "The mass number stays 188; the atomic number rises to 75 + 1 = 76. The element with atomic number 76 is osmium, so the nuclide left behind is ¹⁸⁸₇₆Os.",
 "anticipatedWrong": [
  {
   "value": "¹⁸⁸₇₄W",
   "misconception": "you dropped the atomic number instead of raising it — a neutron becomes a proton, so the count rises",
   "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; a neutron becomes a proton, so the count rises."
  },
  {
   "value": "¹⁸⁸₇₅Re",
   "misconception": "you kept the atomic number — one more proton means the atom is no longer rhenium",
   "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer rhenium."
  },
  {
   "value": "¹⁸⁷₇₆Os",
   "misconception": "you dropped the mass number by 1 — the electron that leaves is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the electron that leaves is not a proton or neutron."
  }
 ]
}

[
 {
  "value": 88,
  "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
  "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in."
 },
 {
  "value": 92,
  "misconception": "you added the alpha particle's mass number instead of its atomic number",
  "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; use the bottom numbers: 88 + 2."
 },
 {
  "value": 86,
  "misconception": "you subtracted the alpha particle's 2 instead of adding it",
  "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
 }
]

{
 "keyExplanation": "Add the right side's atomic numbers: 88 + 2 = 90, matching the left side's 90 — the atomic numbers balance, because electric charge never appears or disappears.",
 "anticipatedWrong": [
  {
   "value": 88,
   "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
   "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in."
  },
  {
   "value": 92,
   "misconception": "you added the alpha particle's mass number instead of its atomic number",
   "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; use the bottom numbers: 88 + 2."
  },
  {
   "value": 86,
   "misconception": "you subtracted the alpha particle's 2 instead of adding it",
   "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
  }
 ]
}

[
 {
  "value": "¹⁶¹₆₅Tb → ¹⁶¹₆₄Gd + ⁰₋₁e",
  "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
  "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
 },
 {
  "value": "¹⁶⁰₆₆Dy + ⁰₋₁e as the right side",
  "misconception": "you dropped the mass number by 1 — the total count of protons plus neutrons is unchanged",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the total count of protons plus neutrons is unchanged."
 },
 {
  "value": "¹⁶¹₆₅Tb → ¹⁶¹₆₆Dy",
  "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
  "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
 }
]

{
 "keyExplanation": "The mass number stays 161; the atomic number rises to 65 + 1 = 66, and the element with atomic number 66 is dysprosium. The complete equation is ¹⁶¹₆₅Tb → ¹⁶¹₆₆Dy + ⁰₋₁e; check: 161 = 161 + 0 and 65 = 66 + (−1).",
 "anticipatedWrong": [
  {
   "value": "¹⁶¹₆₅Tb → ¹⁶¹₆₄Gd + ⁰₋₁e",
   "misconception": "you subtracted 1 from the atomic number — beta decay raises it by 1",
   "feedback": "This answer is wrong — you subtracted 1 from the atomic number; beta decay raises it by 1."
  },
  {
   "value": "¹⁶⁰₆₆Dy + ⁰₋₁e as the right side",
   "misconception": "you dropped the mass number by 1 — the total count of protons plus neutrons is unchanged",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the total count of protons plus neutrons is unchanged."
  },
  {
   "value": "¹⁶¹₆₅Tb → ¹⁶¹₆₆Dy",
   "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
   "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
  }
 ]
}

[
 {
  "value": "²²¹₈₈Ra → ²¹⁹₈₄Po + ⁴₂He",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "²²¹₈₈Ra → ²¹⁷₈₇Fr + ⁴₂He",
  "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
  "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
 },
 {
  "value": "²²¹₈₈Ra → ²¹⁷₈₆Rn",
  "misconception": "you left out the alpha particle — without it, neither total balances",
  "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
 }
]

{
 "keyExplanation": "New mass number: 221 − 4 = 217. New atomic number: 88 − 2 = 86, and the element with atomic number 86 is radon. The complete equation is ²²¹₈₈Ra → ²¹⁷₈₆Rn + ⁴₂He; check: 221 = 217 + 4 and 88 = 86 + 2.",
 "anticipatedWrong": [
  {
   "value": "²²¹₈₈Ra → ²¹⁹₈₄Po + ⁴₂He",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "²²¹₈₈Ra → ²¹⁷₈₇Fr + ⁴₂He",
   "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
   "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
  },
  {
   "value": "²²¹₈₈Ra → ²¹⁷₈₆Rn",
   "misconception": "you left out the alpha particle — without it, neither total balances",
   "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
  }
 ]
}

[
 {
  "value": "¹²⁷₅₁Sb",
  "misconception": "you subtracted the −1 as if it were +1 — 52 = ? + (−1) forces the missing atomic number UP to 53",
  "feedback": "This answer is wrong — you subtracted the −1 as if it were +1; 52 = ? + (−1) forces the missing atomic number UP to 53."
 },
 {
  "value": "¹²⁷₅₂Te",
  "misconception": "you kept the starting atomic number — the beta particle's −1 raises the new nucleus's atomic number to 53",
  "feedback": "This answer is wrong — you kept the starting atomic number; the beta particle's −1 raises the new nucleus's atomic number to 53."
 },
 {
  "value": "¹²⁶₅₃I",
  "misconception": "you dropped the mass number — the beta particle's mass number is 0, so 127 stays",
  "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so 127 stays."
 }
]

{
 "keyExplanation": "Missing mass number: 127 − 0 = 127. Missing atomic number: 52 − (−1) = 53 — iodine. The missing species is ¹²⁷₅₃I, and the equation reads ¹²⁷₅₂Te → ¹²⁷₅₃I + ⁰₋₁e.",
 "anticipatedWrong": [
  {
   "value": "¹²⁷₅₁Sb",
   "misconception": "you subtracted the −1 as if it were +1 — 52 = ? + (−1) forces the missing atomic number UP to 53",
   "feedback": "This answer is wrong — you subtracted the −1 as if it were +1; 52 = ? + (−1) forces the missing atomic number UP to 53."
  },
  {
   "value": "¹²⁷₅₂Te",
   "misconception": "you kept the starting atomic number — the beta particle's −1 raises the new nucleus's atomic number to 53",
   "feedback": "This answer is wrong — you kept the starting atomic number; the beta particle's −1 raises the new nucleus's atomic number to 53."
  },
  {
   "value": "¹²⁶₅₃I",
   "misconception": "you dropped the mass number — the beta particle's mass number is 0, so 127 stays",
   "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so 127 stays."
  }
 ]
}

[
 {
  "value": 4,
  "misconception": "you used the mass number as if it counted only neutrons — it counts protons too",
  "feedback": "This answer is wrong — you used the mass number as if it counted only neutrons; it counts protons too, so subtract the 2."
 },
 {
  "value": 6,
  "misconception": "you added the two numbers instead of subtracting them",
  "feedback": "This answer is wrong — you added the two numbers instead of subtracting them: 4 − 2 = 2."
 },
 {
  "value": 0,
  "misconception": "you left the particle with no neutrons — its 4 particles are 2 protons AND 2 neutrons",
  "feedback": "This answer is wrong — you left the particle with no neutrons; its 4 particles are 2 protons AND 2 neutrons."
 }
]

{
 "keyExplanation": "The mass number 4 counts protons plus neutrons and the 2 counts protons, so 4 − 2 = 2 neutrons — the alpha particle is 2 protons and 2 neutrons bound together.",
 "anticipatedWrong": [
  {
   "value": 4,
   "misconception": "you used the mass number as if it counted only neutrons — it counts protons too",
   "feedback": "This answer is wrong — you used the mass number as if it counted only neutrons; it counts protons too, so subtract the 2."
  },
  {
   "value": 6,
   "misconception": "you added the two numbers instead of subtracting them",
   "feedback": "This answer is wrong — you added the two numbers instead of subtracting them: 4 − 2 = 2."
  },
  {
   "value": 0,
   "misconception": "you left the particle with no neutrons — its 4 particles are 2 protons AND 2 neutrons",
   "feedback": "This answer is wrong — you left the particle with no neutrons; its 4 particles are 2 protons AND 2 neutrons."
  }
 ]
}

[
 {
  "value": "¹⁷⁸₇₀Yb",
  "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
  "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
 },
 {
  "value": "¹⁷⁶₇₄W",
  "misconception": "you kept the atomic number — losing 2 protons means the atom is no longer tungsten",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons means the atom is no longer tungsten."
 },
 {
  "value": "¹⁸⁴₇₆Os",
  "misconception": "you added the alpha particle's numbers instead of subtracting them",
  "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them."
 }
]

{
 "keyExplanation": "Mass number: 180 − 4 = 176. Atomic number: 74 − 2 = 72. The element with atomic number 72 is hafnium, so alpha decay leaves ¹⁷⁶₇₂Hf behind.",
 "anticipatedWrong": [
  {
   "value": "¹⁷⁸₇₀Yb",
   "misconception": "you swapped the drops — the mass number falls by 4 and the atomic number falls by 2",
   "feedback": "This answer is wrong — you swapped the drops; the mass number falls by 4 and the atomic number falls by 2."
  },
  {
   "value": "¹⁷⁶₇₄W",
   "misconception": "you kept the atomic number — losing 2 protons means the atom is no longer tungsten",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons means the atom is no longer tungsten."
  },
  {
   "value": "¹⁸⁴₇₆Os",
   "misconception": "you added the alpha particle's numbers instead of subtracting them",
   "feedback": "This answer is wrong — you added the alpha particle's numbers instead of subtracting them."
  }
 ]
}

[
 {
  "value": "¹⁹⁹₇₈Pt",
  "misconception": "you dropped the atomic number instead of raising it — beta decay raises it by 1",
  "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; beta decay raises it by 1."
 },
 {
  "value": "¹⁹⁹₇₉Au",
  "misconception": "you kept the atomic number — one more proton means the atom is no longer gold",
  "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer gold."
 },
 {
  "value": "¹⁹⁸₈₀Hg",
  "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
 }
]

{
 "keyExplanation": "Beta decay keeps the mass number at 199 and raises the atomic number: 79 + 1 = 80. The element with atomic number 80 is mercury, so the nuclide left behind is ¹⁹⁹₈₀Hg.",
 "anticipatedWrong": [
  {
   "value": "¹⁹⁹₇₈Pt",
   "misconception": "you dropped the atomic number instead of raising it — beta decay raises it by 1",
   "feedback": "This answer is wrong — you dropped the atomic number instead of raising it; beta decay raises it by 1."
  },
  {
   "value": "¹⁹⁹₇₉Au",
   "misconception": "you kept the atomic number — one more proton means the atom is no longer gold",
   "feedback": "This answer is wrong — you kept the atomic number; one more proton means the atom is no longer gold."
  },
  {
   "value": "¹⁹⁸₈₀Hg",
   "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
  }
 ]
}

[
 {
  "value": 93,
  "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
  "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 93 + 2 = 95."
 },
 {
  "value": 97,
  "misconception": "you added the alpha particle's mass number instead of its atomic number",
  "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers."
 },
 {
  "value": 91,
  "misconception": "you subtracted the alpha particle's 2 instead of adding it",
  "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
 }
]

{
 "keyExplanation": "The right side holds ²³⁹₉₃Np and ⁴₂He. Their atomic numbers add: 93 + 2 = 95, matching the left side's 95 — the atomic numbers balance, because electric charge never appears or disappears.",
 "anticipatedWrong": [
  {
   "value": 93,
   "misconception": "you took the new nucleus's atomic number alone — the alpha particle's 2 must be added in",
   "feedback": "This answer is wrong — you took the new nucleus's atomic number alone; the alpha particle's 2 must be added in: 93 + 2 = 95."
  },
  {
   "value": 97,
   "misconception": "you added the alpha particle's mass number instead of its atomic number",
   "feedback": "This answer is wrong — you added the alpha particle's mass number instead of its atomic number; the atomic-number total uses the bottom numbers."
  },
  {
   "value": 91,
   "misconception": "you subtracted the alpha particle's 2 instead of adding it",
   "feedback": "This answer is wrong — you subtracted the alpha particle's 2 instead of adding it; the right side's atomic numbers add together."
  }
 ]
}

[
 {
  "value": "¹³²₅₂Te → ¹³²₅₁Sb + ⁰₋₁e",
  "misconception": "you subtracted 1 from the atomic number — the neutron-to-proton change raises it by 1",
  "feedback": "This answer is wrong — you subtracted 1 from the atomic number; the neutron-to-proton change raises it by 1."
 },
 {
  "value": "¹³¹₅₃I + ⁰₋₁e as the right side",
  "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
  "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
 },
 {
  "value": "¹³²₅₂Te → ¹³²₅₃I",
  "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
  "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
 }
]

{
 "keyExplanation": "The mass number stays 132; the atomic number rises to 52 + 1 = 53, and the element with atomic number 53 is iodine. The complete equation is ¹³²₅₂Te → ¹³²₅₃I + ⁰₋₁e; check: 132 = 132 + 0 and 52 = 53 + (−1).",
 "anticipatedWrong": [
  {
   "value": "¹³²₅₂Te → ¹³²₅₁Sb + ⁰₋₁e",
   "misconception": "you subtracted 1 from the atomic number — the neutron-to-proton change raises it by 1",
   "feedback": "This answer is wrong — you subtracted 1 from the atomic number; the neutron-to-proton change raises it by 1."
  },
  {
   "value": "¹³¹₅₃I + ⁰₋₁e as the right side",
   "misconception": "you dropped the mass number by 1 — the departing electron is not a proton or neutron",
   "feedback": "This answer is wrong — you dropped the mass number by 1; the departing electron is not a proton or neutron."
  },
  {
   "value": "¹³²₅₂Te → ¹³²₅₃I",
   "misconception": "you left out the beta particle — without its −1, the atomic numbers cannot balance",
   "feedback": "This answer is wrong — you left out the beta particle; without its −1, the atomic numbers cannot balance."
  }
 ]
}

[
 {
  "value": "²⁴⁶₉₈Cf → ²⁴²₉₈Cf + ⁴₂He",
  "misconception": "you kept the atomic number — losing 2 protons changes the element",
  "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
 },
 {
  "value": "²⁴⁶₉₈Cf → ²⁴²₉₇Bk + ⁴₂He",
  "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
  "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
 },
 {
  "value": "²⁴⁶₉₈Cf → ²⁴²₉₆Cm",
  "misconception": "you left out the alpha particle — without it, neither total balances",
  "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
 }
]

{
 "keyExplanation": "New mass number: 246 − 4 = 242. New atomic number: 98 − 2 = 96, and the element with atomic number 96 is curium. The complete equation is ²⁴⁶₉₈Cf → ²⁴²₉₆Cm + ⁴₂He; check: 246 = 242 + 4 and 98 = 96 + 2.",
 "anticipatedWrong": [
  {
   "value": "²⁴⁶₉₈Cf → ²⁴²₉₈Cf + ⁴₂He",
   "misconception": "you kept the atomic number — losing 2 protons changes the element",
   "feedback": "This answer is wrong — you kept the atomic number; losing 2 protons changes the element."
  },
  {
   "value": "²⁴⁶₉₈Cf → ²⁴²₉₇Bk + ⁴₂He",
   "misconception": "you dropped the atomic number by 1 instead of 2 — the alpha particle carries away 2 protons",
   "feedback": "This answer is wrong — you dropped the atomic number by 1 instead of 2; the alpha particle carries away 2 protons."
  },
  {
   "value": "²⁴⁶₉₈Cf → ²⁴²₉₆Cm",
   "misconception": "you left out the alpha particle — without it, neither total balances",
   "feedback": "This answer is wrong — you left out the alpha particle; without it, neither total balances."
  }
 ]
}

[
 {
  "value": "¹⁰⁹₄₈Cd",
  "misconception": "you added 1 instead of adding the beta particle's −1 — the left side must total 47 + (−1) = 46",
  "feedback": "This answer is wrong — you added 1 instead of adding the beta particle's −1; the left side must total 47 + (−1) = 46."
 },
 {
  "value": "¹⁰⁹₄₇Ag",
  "misconception": "you copied the shown nucleus — the missing species must make both totals balance, so its atomic number is 46",
  "feedback": "This answer is wrong — you copied the shown nucleus; the missing species must make both totals balance, so its atomic number is 46."
 },
 {
  "value": "¹⁰⁸₄₆Pd",
  "misconception": "you dropped the mass number — the beta particle's mass number is 0, so the mass stays 109",
  "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so the mass stays 109."
 }
]

{
 "keyExplanation": "The missing species sits on the left, so its numbers are the right side's totals. Mass number: 109 + 0 = 109. Atomic number: 47 + (−1) = 46 — palladium. The starting nucleus is ¹⁰⁹₄₆Pd, and the equation reads ¹⁰⁹₄₆Pd → ¹⁰⁹₄₇Ag + ⁰₋₁e.",
 "anticipatedWrong": [
  {
   "value": "¹⁰⁹₄₈Cd",
   "misconception": "you added 1 instead of adding the beta particle's −1 — the left side must total 47 + (−1) = 46",
   "feedback": "This answer is wrong — you added 1 instead of adding the beta particle's −1; the left side must total 47 + (−1) = 46."
  },
  {
   "value": "¹⁰⁹₄₇Ag",
   "misconception": "you copied the shown nucleus — the missing species must make both totals balance, so its atomic number is 46",
   "feedback": "This answer is wrong — you copied the shown nucleus; the missing species must make both totals balance, so its atomic number is 46."
  },
  {
   "value": "¹⁰⁸₄₆Pd",
   "misconception": "you dropped the mass number — the beta particle's mass number is 0, so the mass stays 109",
   "feedback": "This answer is wrong — you dropped the mass number; the beta particle's mass number is 0, so the mass stays 109."
  }
 ]
}

[
 {
  "value": "1/7",
  "misconception": "you divided the original amount by the number of half-lives — seven half-lives means seven halvings",
  "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Seven half-lives means seven halvings: 1/2 down to 1/128."
 },
 {
  "value": "1/64",
  "misconception": "you halved one time too few — the seventh halving takes 1/64 to 1/128",
  "feedback": "This answer is wrong — you halved one time too few. The seventh halving takes 1/64 to 1/128."
 },
 {
  "value": "0",
  "misconception": "you emptied the sample — each half-life removes only half of what is left",
  "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so some always remains: 1/128."
 }
]

{
 "keyExplanation": "Halve once per half-life: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 — because after each half-life, half of the nuclei that remain decay. After 7 half-lives, 1/128 of the original cerium-141 nuclei remain.",
 "anticipatedWrong": [
  {
   "value": "1/7",
   "misconception": "you divided the original amount by the number of half-lives — seven half-lives means seven halvings",
   "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Seven half-lives means seven halvings: 1/2 down to 1/128."
  },
  {
   "value": "1/64",
   "misconception": "you halved one time too few — the seventh halving takes 1/64 to 1/128",
   "feedback": "This answer is wrong — you halved one time too few. The seventh halving takes 1/64 to 1/128."
  },
  {
   "value": "0",
   "misconception": "you emptied the sample — each half-life removes only half of what is left",
   "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so some always remains: 1/128."
  }
 ]
}

[
 {
  "value": 25,
  "misconception": "you divided 100% by the number of half-lives — four half-lives means four halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Four half-lives means four halvings: 100 → 50 → 25 → 12.5 → 6.25."
 },
 {
  "value": 12.5,
  "misconception": "you halved one time too few — the fourth halving takes 12.5% down to 6.25%",
  "feedback": "This answer is wrong — you halved one time too few. The fourth halving takes 12.5% down to 6.25%."
 },
 {
  "value": 0.0625,
  "misconception": "you gave the fraction instead of the percentage — 1/16 of the original is 6.25%",
  "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/16 of the original is 6.25%."
 }
]

{
 "keyExplanation": "Halve 100% once per half-life: 100 → 50 → 25 → 12.5 → 6.25. Because after each half-life, half of the nuclei that remain decay. 6.25% of the original nuclei remain.",
 "anticipatedWrong": [
  {
   "value": 25,
   "misconception": "you divided 100% by the number of half-lives — four half-lives means four halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Four half-lives means four halvings: 100 → 50 → 25 → 12.5 → 6.25."
  },
  {
   "value": 12.5,
   "misconception": "you halved one time too few — the fourth halving takes 12.5% down to 6.25%",
   "feedback": "This answer is wrong — you halved one time too few. The fourth halving takes 12.5% down to 6.25%."
  },
  {
   "value": 0.0625,
   "misconception": "you gave the fraction instead of the percentage — 1/16 of the original is 6.25%",
   "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/16 of the original is 6.25%."
  }
 ]
}

[
 {
  "value": 68,
  "misconception": "you halved only once — 83.1 days holds THREE 27.7-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 83.1 days holds THREE 27.7-day half-lives: 136 → 68 → 34 → 17."
 },
 {
  "value": 45.33,
  "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
  "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 136 → 68 → 34 → 17."
 },
 {
  "value": 34,
  "misconception": "you halved one time too few — the third halving takes 34 µg down to 17 µg",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 34 µg down to 17 µg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 136 µg, half-life = 27.7 days, time passed = 83.1 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 83.1 ÷ 27.7 = 3\nHalve the starting amount three times: 136 → 68 → 34 → 17\n17 µg of chromium-51 remains.",
 "anticipatedWrong": [
  {
   "value": 68,
   "misconception": "you halved only once — 83.1 days holds THREE 27.7-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 83.1 days holds THREE 27.7-day half-lives: 136 → 68 → 34 → 17."
  },
  {
   "value": 45.33,
   "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
   "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 136 → 68 → 34 → 17."
  },
  {
   "value": 34,
   "misconception": "you halved one time too few — the third halving takes 34 µg down to 17 µg",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 34 µg down to 17 µg."
  }
 ]
}

[
 {
  "value": 174,
  "misconception": "you halved only once — 146 hours holds TWO 73-hour half-lives",
  "feedback": "This answer is wrong — you halved only once. 146 hours holds TWO 73-hour half-lives: 348 → 174 → 87."
 },
 {
  "value": 43.5,
  "misconception": "you halved one time too many — two half-lives means exactly two halvings",
  "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 348 → 174 → 87."
 },
 {
  "value": 202,
  "misconception": "you subtracted the hours from the micrograms — amounts and times never subtract; count the half-lives, then halve",
  "feedback": "This answer is wrong — you subtracted the hours from the micrograms. Amounts and times never subtract; count the half-lives, then halve: 87 µg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 348 µg, half-life = 73 hours, time passed = 146 hours\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 146 ÷ 73 = 2\nHalve the starting amount twice: 348 → 174 → 87\n87 µg of thallium-201 remains.",
 "anticipatedWrong": [
  {
   "value": 174,
   "misconception": "you halved only once — 146 hours holds TWO 73-hour half-lives",
   "feedback": "This answer is wrong — you halved only once. 146 hours holds TWO 73-hour half-lives: 348 → 174 → 87."
  },
  {
   "value": 43.5,
   "misconception": "you halved one time too many — two half-lives means exactly two halvings",
   "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 348 → 174 → 87."
  },
  {
   "value": 202,
   "misconception": "you subtracted the hours from the micrograms — amounts and times never subtract; count the half-lives, then halve",
   "feedback": "This answer is wrong — you subtracted the hours from the micrograms. Amounts and times never subtract; count the half-lives, then halve: 87 µg."
  }
 ]
}

[
 {
  "value": 356,
  "misconception": "you used the ratio 72 ÷ 9 = 8 as the number of halvings — 72 g → 36 g → 18 g → 9 g is 3 halvings",
  "feedback": "This answer is wrong — you used the ratio 72 ÷ 9 = 8 as the number of halvings. 72 g → 36 g → 18 g → 9 g is 3 halvings: 3 × 44.5 = 133.5 days."
 },
 {
  "value": 89,
  "misconception": "you stopped counting one halving early — 72 g → 36 g → 18 g → 9 g is 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting one halving early. 72 g → 36 g → 18 g → 9 g is 3 halvings, not 2: 133.5 days."
 }
]

{
 "keyExplanation": "Count the halvings: 72 g → 36 g → 18 g → 9 g.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 44.5 days\ntime passed = 133.5 days",
 "anticipatedWrong": [
  {
   "value": 356,
   "misconception": "you used the ratio 72 ÷ 9 = 8 as the number of halvings — 72 g → 36 g → 18 g → 9 g is 3 halvings",
   "feedback": "This answer is wrong — you used the ratio 72 ÷ 9 = 8 as the number of halvings. 72 g → 36 g → 18 g → 9 g is 3 halvings: 3 × 44.5 = 133.5 days."
  },
  {
   "value": 89,
   "misconception": "you stopped counting one halving early — 72 g → 36 g → 18 g → 9 g is 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting one halving early. 72 g → 36 g → 18 g → 9 g is 3 halvings, not 2: 133.5 days."
  }
 ]
}

[
 {
  "value": 148,
  "misconception": "you read the time for the mass to fall to a quarter of the start — that is two half-lives, not one",
  "feedback": "This answer is wrong — you read the time for the mass to fall to a quarter of the start. That is two half-lives; one half-life is 74 days."
 },
 {
  "value": 80,
  "misconception": "you took the halfway point of the time axis instead of the point where the MASS has halved",
  "feedback": "This answer is wrong — you took the halfway point of the time axis. Read where the MASS has halved: 120 g at 74 days."
 }
]

{
 "keyExplanation": "Half of the starting amount: half of 240 g is 120 g. The curve reaches 120 g at 74 days. The half-life of iridium-192 is 74 days.",
 "anticipatedWrong": [
  {
   "value": 148,
   "misconception": "you read the time for the mass to fall to a quarter of the start — that is two half-lives, not one",
   "feedback": "This answer is wrong — you read the time for the mass to fall to a quarter of the start. That is two half-lives; one half-life is 74 days."
  },
  {
   "value": 80,
   "misconception": "you took the halfway point of the time axis instead of the point where the MASS has halved",
   "feedback": "This answer is wrong — you took the halfway point of the time axis. Read where the MASS has halved: 120 g at 74 days."
  }
 ]
}

[
 {
  "value": 56,
  "misconception": "you read the mass at 12.3 years instead of 24.6 years",
  "feedback": "This answer is wrong — you read the mass at 12.3 years. At 24.6 years the curve sits at 28 mg."
 },
 {
  "value": 14,
  "misconception": "you read the mass at 36.9 years instead of 24.6 years",
  "feedback": "This answer is wrong — you read the mass at 36.9 years. At 24.6 years the curve sits at 28 mg."
 }
]

{
 "keyExplanation": "Go up from 24.6 years on the time axis to the curve. Go straight across from that point to the mass axis. At 24.6 years, 28 mg of hydrogen-3 remains.",
 "anticipatedWrong": [
  {
   "value": 56,
   "misconception": "you read the mass at 12.3 years instead of 24.6 years",
   "feedback": "This answer is wrong — you read the mass at 12.3 years. At 24.6 years the curve sits at 28 mg."
  },
  {
   "value": 14,
   "misconception": "you read the mass at 36.9 years instead of 24.6 years",
   "feedback": "This answer is wrong — you read the mass at 36.9 years. At 24.6 years the curve sits at 28 mg."
  }
 ]
}

[
 {
  "value": 25,
  "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
  "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 25 = 75%."
 },
 {
  "value": 100,
  "misconception": "you added half plus half — the second half-life removes half OF WHAT REMAINS, so a quarter still remains",
  "feedback": "This answer is wrong — you added half plus half. The second half-life removes half OF WHAT REMAINS: 75% decayed, 25% remains."
 },
 {
  "value": 50,
  "misconception": "you froze the decayed share at one halving — the second half-life removes another quarter",
  "feedback": "This answer is wrong — you froze the decayed share at one halving. The second half-life removes another quarter: 75% in total."
 }
]

{
 "keyExplanation": "Remaining first: 100 → 50 → 25, because after each half-life, half of the nuclei that remain decay. Decayed is the rest: 100 − 25 = 75%.",
 "anticipatedWrong": [
  {
   "value": 25,
   "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
   "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 25 = 75%."
  },
  {
   "value": 100,
   "misconception": "you added half plus half — the second half-life removes half OF WHAT REMAINS, so a quarter still remains",
   "feedback": "This answer is wrong — you added half plus half. The second half-life removes half OF WHAT REMAINS: 75% decayed, 25% remains."
  },
  {
   "value": 50,
   "misconception": "you froze the decayed share at one halving — the second half-life removes another quarter",
   "feedback": "This answer is wrong — you froze the decayed share at one halving. The second half-life removes another quarter: 75% in total."
  }
 ]
}

[
 {
  "value": 84.8,
  "misconception": "you used the 16 in 1/16 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 → 1/16 is 4 halvings",
  "feedback": "This answer is wrong — you used the 16 in 1/16 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 → 1/16 is 4 halvings: 21.2 years."
 },
 {
  "value": 15.9,
  "misconception": "you stopped counting at 1/8 — reaching 1/16 takes 4 halvings, not 3",
  "feedback": "This answer is wrong — you stopped counting at 1/8. Reaching 1/16 takes 4 halvings, not 3: 21.2 years."
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8 → 1/16.\nThe sample has halved 4 times.\ntime passed = number of halvings × half-life\ntime passed = 4 × 5.3 years\ntime passed = 21.2 years",
 "anticipatedWrong": [
  {
   "value": 84.8,
   "misconception": "you used the 16 in 1/16 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 → 1/16 is 4 halvings",
   "feedback": "This answer is wrong — you used the 16 in 1/16 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 → 1/16 is 4 halvings: 21.2 years."
  },
  {
   "value": 15.9,
   "misconception": "you stopped counting at 1/8 — reaching 1/16 takes 4 halvings, not 3",
   "feedback": "This answer is wrong — you stopped counting at 1/8. Reaching 1/16 takes 4 halvings, not 3: 21.2 years."
  }
 ]
}

[
 {
  "value": 60,
  "misconception": "you read the time to fall to 50% — the question asks for 25%, which takes two half-lives",
  "feedback": "This answer is wrong — you read the time to fall to 50%. Reaching 25% takes two half-lives: 120 days."
 },
 {
  "value": 180,
  "misconception": "you read the time to fall to 12.5% — that is one half-life too many",
  "feedback": "This answer is wrong — you read the time to fall to 12.5%. The 25% crossing sits at 120 days."
 }
]

{
 "keyExplanation": "Find 25% on the percent axis. Go across to the curve, then down to the time axis. The sample reaches 25% at 120 days.",
 "anticipatedWrong": [
  {
   "value": 60,
   "misconception": "you read the time to fall to 50% — the question asks for 25%, which takes two half-lives",
   "feedback": "This answer is wrong — you read the time to fall to 50%. Reaching 25% takes two half-lives: 120 days."
  },
  {
   "value": 180,
   "misconception": "you read the time to fall to 12.5% — that is one half-life too many",
   "feedback": "This answer is wrong — you read the time to fall to 12.5%. The 25% crossing sits at 120 days."
  }
 ]
}

[
 {
  "value": "1/8",
  "misconception": "you divided the original amount by the number of half-lives — eight half-lives means eight halvings",
  "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Eight half-lives means eight successive halvings, ending at 1/256."
 },
 {
  "value": "1/128",
  "misconception": "you halved one time too few — the eighth halving takes 1/128 to 1/256",
  "feedback": "This answer is wrong — you halved one time too few. The eighth halving takes 1/128 to 1/256."
 },
 {
  "value": "0",
  "misconception": "you emptied the sample — each half-life removes only half of what is left",
  "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/256 still remains."
 }
]

{
 "keyExplanation": "Halve once per half-life: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256 — because after each half-life, half of the nuclei that remain decay. After 8 half-lives, 1/256 of the original barium-140 nuclei remain.",
 "anticipatedWrong": [
  {
   "value": "1/8",
   "misconception": "you divided the original amount by the number of half-lives — eight half-lives means eight halvings",
   "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Eight half-lives means eight successive halvings, ending at 1/256."
  },
  {
   "value": "1/128",
   "misconception": "you halved one time too few — the eighth halving takes 1/128 to 1/256",
   "feedback": "This answer is wrong — you halved one time too few. The eighth halving takes 1/128 to 1/256."
  },
  {
   "value": "0",
   "misconception": "you emptied the sample — each half-life removes only half of what is left",
   "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/256 still remains."
  }
 ]
}

[
 {
  "value": 20,
  "misconception": "you divided 100% by the number of half-lives — five half-lives means five halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Five half-lives means five halvings, ending at 3.125%."
 },
 {
  "value": 6.25,
  "misconception": "you halved one time too few — the fifth halving takes 6.25% down to 3.125%",
  "feedback": "This answer is wrong — you halved one time too few. The fifth halving takes 6.25% down to 3.125%."
 },
 {
  "value": 0.03125,
  "misconception": "you gave the fraction instead of the percentage — 1/32 of the original is 3.125%",
  "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/32 of the original is 3.125%."
 }
]

{
 "keyExplanation": "Halve 100% once per half-life: 100 → 50 → 25 → 12.5 → 6.25 → 3.125. Because after each half-life, half of the nuclei that remain decay. 3.125% of the original nuclei remain.",
 "anticipatedWrong": [
  {
   "value": 20,
   "misconception": "you divided 100% by the number of half-lives — five half-lives means five halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Five half-lives means five halvings, ending at 3.125%."
  },
  {
   "value": 6.25,
   "misconception": "you halved one time too few — the fifth halving takes 6.25% down to 3.125%",
   "feedback": "This answer is wrong — you halved one time too few. The fifth halving takes 6.25% down to 3.125%."
  },
  {
   "value": 0.03125,
   "misconception": "you gave the fraction instead of the percentage — 1/32 of the original is 3.125%",
   "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/32 of the original is 3.125%."
  }
 ]
}

[
 {
  "value": 52,
  "misconception": "you halved only once — 433.2 days holds THREE 144.4-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 433.2 days holds THREE 144.4-day half-lives: 104 → 52 → 26 → 13."
 },
 {
  "value": 34.67,
  "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
  "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 104 → 52 → 26 → 13."
 },
 {
  "value": 26,
  "misconception": "you halved one time too few — the third halving takes 26 g down to 13 g",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 26 g down to 13 g."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 104 g, half-life = 144.4 days, time passed = 433.2 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 433.2 ÷ 144.4 = 3\nHalve the starting amount three times: 104 → 52 → 26 → 13\n13 g of dysprosium-159 remains.",
 "anticipatedWrong": [
  {
   "value": 52,
   "misconception": "you halved only once — 433.2 days holds THREE 144.4-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 433.2 days holds THREE 144.4-day half-lives: 104 → 52 → 26 → 13."
  },
  {
   "value": 34.67,
   "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
   "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 104 → 52 → 26 → 13."
  },
  {
   "value": 26,
   "misconception": "you halved one time too few — the third halving takes 26 g down to 13 g",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 26 g down to 13 g."
  }
 ]
}

[
 {
  "value": 134,
  "misconception": "you halved only once — 7.4 days holds TWO 3.7-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 7.4 days holds TWO 3.7-day half-lives: 268 → 134 → 67."
 },
 {
  "value": 33.5,
  "misconception": "you halved one time too many — two half-lives means exactly two halvings",
  "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 268 → 134 → 67."
 },
 {
  "value": 260.6,
  "misconception": "you subtracted the days from the milligrams — amounts and times never subtract; count the half-lives, then halve",
  "feedback": "This answer is wrong — you subtracted the days from the milligrams. Count the half-lives, then halve: 67 mg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 268 mg, half-life = 3.7 days, time passed = 7.4 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 7.4 ÷ 3.7 = 2\nHalve the starting amount twice: 268 → 134 → 67\n67 mg of rhenium-186 remains.",
 "anticipatedWrong": [
  {
   "value": 134,
   "misconception": "you halved only once — 7.4 days holds TWO 3.7-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 7.4 days holds TWO 3.7-day half-lives: 268 → 134 → 67."
  },
  {
   "value": 33.5,
   "misconception": "you halved one time too many — two half-lives means exactly two halvings",
   "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 268 → 134 → 67."
  },
  {
   "value": 260.6,
   "misconception": "you subtracted the days from the milligrams — amounts and times never subtract; count the half-lives, then halve",
   "feedback": "This answer is wrong — you subtracted the days from the milligrams. Count the half-lives, then halve: 67 mg."
  }
 ]
}

[
 {
  "value": 30.4,
  "misconception": "you used the ratio 60 ÷ 7.5 = 8 as the number of halvings — 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings",
  "feedback": "This answer is wrong — you used the ratio 60 ÷ 7.5 = 8 as the number of halvings. 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings: 11.4 days."
 },
 {
  "value": 7.6,
  "misconception": "you stopped counting one halving early — 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting one halving early. 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings, not 2: 11.4 days."
 }
]

{
 "keyExplanation": "Count the halvings: 60 µg → 30 µg → 15 µg → 7.5 µg.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 3.8 days\ntime passed = 11.4 days",
 "anticipatedWrong": [
  {
   "value": 30.4,
   "misconception": "you used the ratio 60 ÷ 7.5 = 8 as the number of halvings — 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings",
   "feedback": "This answer is wrong — you used the ratio 60 ÷ 7.5 = 8 as the number of halvings. 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings: 11.4 days."
  },
  {
   "value": 7.6,
   "misconception": "you stopped counting one halving early — 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting one halving early. 60 µg → 30 µg → 15 µg → 7.5 µg is 3 halvings, not 2: 11.4 days."
  }
 ]
}

[
 {
  "value": 78.6,
  "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
  "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 39.3 days."
 },
 {
  "value": 157.2,
  "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
  "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 39.3 days."
 }
]

{
 "keyExplanation": "Half of the starting amount: half of 100% is 50%. The curve reaches 50% at 39.3 days. The half-life of ruthenium-103 is 39.3 days.",
 "anticipatedWrong": [
  {
   "value": 78.6,
   "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
   "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 39.3 days."
  },
  {
   "value": 157.2,
   "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
   "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 39.3 days."
  }
 ]
}

[
 {
  "value": 88,
  "misconception": "you read the mass at 15.4 days instead of 30.8 days",
  "feedback": "This answer is wrong — you read the mass at 15.4 days. At 30.8 days the curve sits at 44 mg."
 },
 {
  "value": 22,
  "misconception": "you read the mass at 46.2 days instead of 30.8 days",
  "feedback": "This answer is wrong — you read the mass at 46.2 days. At 30.8 days the curve sits at 44 mg."
 }
]

{
 "keyExplanation": "Go up from 30.8 days on the time axis to the curve, then straight across to the mass axis. At 30.8 days, 44 mg of osmium-191 remains.",
 "anticipatedWrong": [
  {
   "value": 88,
   "misconception": "you read the mass at 15.4 days instead of 30.8 days",
   "feedback": "This answer is wrong — you read the mass at 15.4 days. At 30.8 days the curve sits at 44 mg."
  },
  {
   "value": 22,
   "misconception": "you read the mass at 46.2 days instead of 30.8 days",
   "feedback": "This answer is wrong — you read the mass at 46.2 days. At 30.8 days the curve sits at 44 mg."
  }
 ]
}

[
 {
  "value": 12.5,
  "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage, which is everything else",
  "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 12.5 = 87.5%."
 },
 {
  "value": 37.5,
  "misconception": "you removed one-eighth per half-life — the halvings remove 50, then 25, then 12.5 percent, totaling 87.5%",
  "feedback": "This answer is wrong — you removed one-eighth per half-life. The halvings remove 50, then 25, then 12.5 percent: 87.5% in total."
 },
 {
  "value": 50,
  "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
  "feedback": "This answer is wrong — you froze the decayed share at one halving. Each later half-life keeps removing half of what remains: 87.5%."
 }
]

{
 "keyExplanation": "Remaining first: 100 → 50 → 25 → 12.5, because after each half-life, half of the nuclei that remain decay. Decayed is the rest: 100 − 12.5 = 87.5%.",
 "anticipatedWrong": [
  {
   "value": 12.5,
   "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage, which is everything else",
   "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 12.5 = 87.5%."
  },
  {
   "value": 37.5,
   "misconception": "you removed one-eighth per half-life — the halvings remove 50, then 25, then 12.5 percent, totaling 87.5%",
   "feedback": "This answer is wrong — you removed one-eighth per half-life. The halvings remove 50, then 25, then 12.5 percent: 87.5% in total."
  },
  {
   "value": 50,
   "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
   "feedback": "This answer is wrong — you froze the decayed share at one halving. Each later half-life keeps removing half of what remains: 87.5%."
  }
 ]
}

[
 {
  "value": 33.6,
  "misconception": "you used the 8 in 1/8 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
  "feedback": "This answer is wrong — you used the 8 in 1/8 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 is 3 halvings: 12.6 days."
 },
 {
  "value": 8.4,
  "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting at 1/4. Reaching 1/8 takes 3 halvings, not 2: 12.6 days."
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 4.2 days\ntime passed = 12.6 days",
 "anticipatedWrong": [
  {
   "value": 33.6,
   "misconception": "you used the 8 in 1/8 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
   "feedback": "This answer is wrong — you used the 8 in 1/8 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 is 3 halvings: 12.6 days."
  },
  {
   "value": 8.4,
   "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting at 1/4. Reaching 1/8 takes 3 halvings, not 2: 12.6 days."
  }
 ]
}

[
 {
  "value": 5.2,
  "misconception": "you read the time for the first halving — 116 g is a quarter of 464 g, which takes two halvings",
  "feedback": "This answer is wrong — you read the time for the first halving. 116 g is a quarter of 464 g: two halvings, 10.4 days."
 },
 {
  "value": 15.6,
  "misconception": "you read one halving too far — the curve reaches 116 g at 10.4 days, not 58 g",
  "feedback": "This answer is wrong — you read one halving too far. The curve reaches 116 g at 10.4 days."
 }
]

{
 "keyExplanation": "Find 116 g on the mass axis. Go across to the curve, then down to the time axis. The sample reaches 116 g at 10.4 days.",
 "anticipatedWrong": [
  {
   "value": 5.2,
   "misconception": "you read the time for the first halving — 116 g is a quarter of 464 g, which takes two halvings",
   "feedback": "This answer is wrong — you read the time for the first halving. 116 g is a quarter of 464 g: two halvings, 10.4 days."
  },
  {
   "value": 15.6,
   "misconception": "you read one halving too far — the curve reaches 116 g at 10.4 days, not 58 g",
   "feedback": "This answer is wrong — you read one halving too far. The curve reaches 116 g at 10.4 days."
  }
 ]
}

[
 {
  "value": "1/9",
  "misconception": "you divided the original amount by the number of half-lives — nine half-lives means nine halvings",
  "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Nine half-lives means nine halvings, ending at 1/512."
 },
 {
  "value": "1/256",
  "misconception": "you halved one time too few — the ninth halving takes 1/256 to 1/512",
  "feedback": "This answer is wrong — you halved one time too few. The ninth halving takes 1/256 to 1/512."
 },
 {
  "value": "0",
  "misconception": "you emptied the sample — each half-life removes only half of what is left",
  "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/512 still remains."
 }
]

{
 "keyExplanation": "Halve once per half-life, nine times over: the ladder runs 1/2, 1/4, ... , 1/256, 1/512 — because after each half-life, half of the nuclei that remain decay. After 9 half-lives, 1/512 of the original rubidium-86 nuclei remain.",
 "anticipatedWrong": [
  {
   "value": "1/9",
   "misconception": "you divided the original amount by the number of half-lives — nine half-lives means nine halvings",
   "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Nine half-lives means nine halvings, ending at 1/512."
  },
  {
   "value": "1/256",
   "misconception": "you halved one time too few — the ninth halving takes 1/256 to 1/512",
   "feedback": "This answer is wrong — you halved one time too few. The ninth halving takes 1/256 to 1/512."
  },
  {
   "value": "0",
   "misconception": "you emptied the sample — each half-life removes only half of what is left",
   "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/512 still remains."
  }
 ]
}

[
 {
  "value": 33.33,
  "misconception": "you divided 100% by the number of half-lives — three half-lives means three halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Three half-lives means three halvings: 100 → 50 → 25 → 12.5."
 },
 {
  "value": 25,
  "misconception": "you halved one time too few — the third halving takes 25% down to 12.5%",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 25% down to 12.5%."
 },
 {
  "value": 0.125,
  "misconception": "you gave the fraction instead of the percentage — 1/8 of the original is 12.5%",
  "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/8 of the original is 12.5%."
 }
]

{
 "keyExplanation": "Halve 100% once per half-life: 100 → 50 → 25 → 12.5. Because after each half-life, half of the nuclei that remain decay. 12.5% of the original nuclei remain.",
 "anticipatedWrong": [
  {
   "value": 33.33,
   "misconception": "you divided 100% by the number of half-lives — three half-lives means three halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Three half-lives means three halvings: 100 → 50 → 25 → 12.5."
  },
  {
   "value": 25,
   "misconception": "you halved one time too few — the third halving takes 25% down to 12.5%",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 25% down to 12.5%."
  },
  {
   "value": 0.125,
   "misconception": "you gave the fraction instead of the percentage — 1/8 of the original is 12.5%",
   "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/8 of the original is 12.5%."
  }
 ]
}

[
 {
  "value": 108,
  "misconception": "you halved only once — 19.8 days holds THREE 6.6-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 19.8 days holds THREE 6.6-day half-lives: 216 → 108 → 54 → 27."
 },
 {
  "value": 72,
  "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
  "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 216 → 108 → 54 → 27."
 },
 {
  "value": 54,
  "misconception": "you halved one time too few — the third halving takes 54 µg down to 27 µg",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 54 µg down to 27 µg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 216 µg, half-life = 6.6 days, time passed = 19.8 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 19.8 ÷ 6.6 = 3\nHalve the starting amount three times: 216 → 108 → 54 → 27\n27 µg of lutetium-177 remains.",
 "anticipatedWrong": [
  {
   "value": 108,
   "misconception": "you halved only once — 19.8 days holds THREE 6.6-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 19.8 days holds THREE 6.6-day half-lives: 216 → 108 → 54 → 27."
  },
  {
   "value": 72,
   "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
   "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 216 → 108 → 54 → 27."
  },
  {
   "value": 54,
   "misconception": "you halved one time too few — the third halving takes 54 µg down to 27 µg",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 54 µg down to 27 µg."
  }
 ]
}

[
 {
  "value": 186,
  "misconception": "you halved only once — 924 days holds TWO 462-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 924 days holds TWO 462-day half-lives: 372 → 186 → 93."
 },
 {
  "value": 46.5,
  "misconception": "you halved one time too many — two half-lives means exactly two halvings",
  "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 372 → 186 → 93."
 },
 {
  "value": 0,
  "misconception": "you emptied the sample after two half-lives — each half-life removes only half of what remains, so a quarter is left",
  "feedback": "This answer is wrong — you emptied the sample after two half-lives. Each half-life removes only half of what remains: 93 ng is left."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 372 ng, half-life = 462 days, time passed = 924 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 924 ÷ 462 = 2\nHalve the starting amount twice: 372 → 186 → 93\n93 ng of cadmium-109 remains.",
 "anticipatedWrong": [
  {
   "value": 186,
   "misconception": "you halved only once — 924 days holds TWO 462-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 924 days holds TWO 462-day half-lives: 372 → 186 → 93."
  },
  {
   "value": 46.5,
   "misconception": "you halved one time too many — two half-lives means exactly two halvings",
   "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 372 → 186 → 93."
  },
  {
   "value": 0,
   "misconception": "you emptied the sample after two half-lives — each half-life removes only half of what remains, so a quarter is left",
   "feedback": "This answer is wrong — you emptied the sample after two half-lives. Each half-life removes only half of what remains: 93 ng is left."
  }
 ]
}

[
 {
  "value": 404,
  "misconception": "you used the ratio 208 ÷ 26 = 8 as the number of halvings — 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings",
  "feedback": "This answer is wrong — you used the ratio 208 ÷ 26 = 8 as the number of halvings. 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings: 151.5 days."
 },
 {
  "value": 101,
  "misconception": "you stopped counting one halving early — 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting one halving early. 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings, not 2: 151.5 days."
 }
]

{
 "keyExplanation": "Count the halvings: 208 µg → 104 µg → 52 µg → 26 µg.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 50.5 days\ntime passed = 151.5 days",
 "anticipatedWrong": [
  {
   "value": 404,
   "misconception": "you used the ratio 208 ÷ 26 = 8 as the number of halvings — 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings",
   "feedback": "This answer is wrong — you used the ratio 208 ÷ 26 = 8 as the number of halvings. 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings: 151.5 days."
  },
  {
   "value": 101,
   "misconception": "you stopped counting one halving early — 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting one halving early. 208 µg → 104 µg → 52 µg → 26 µg is 3 halvings, not 2: 151.5 days."
  }
 ]
}

[
 {
  "value": 120.4,
  "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
  "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 60.2 days."
 },
 {
  "value": 240.8,
  "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
  "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 60.2 days."
 }
]

{
 "keyExplanation": "Half of the starting amount: half of 100% is 50%. The curve reaches 50% at 60.2 days. The half-life of antimony-124 is 60.2 days.",
 "anticipatedWrong": [
  {
   "value": 120.4,
   "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
   "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 60.2 days."
  },
  {
   "value": 240.8,
   "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
   "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 60.2 days."
  }
 ]
}

[
 {
  "value": 124,
  "misconception": "you read the mass at 53.2 days instead of 106.4 days",
  "feedback": "This answer is wrong — you read the mass at 53.2 days. At 106.4 days the curve sits at 62 µg."
 },
 {
  "value": 31,
  "misconception": "you read the mass at 159.6 days instead of 106.4 days",
  "feedback": "This answer is wrong — you read the mass at 159.6 days. At 106.4 days the curve sits at 62 µg."
 }
]

{
 "keyExplanation": "Go up from 106.4 days on the time axis to the curve, then straight across to the mass axis. At 106.4 days, 62 µg of beryllium-7 remains.",
 "anticipatedWrong": [
  {
   "value": 124,
   "misconception": "you read the mass at 53.2 days instead of 106.4 days",
   "feedback": "This answer is wrong — you read the mass at 53.2 days. At 106.4 days the curve sits at 62 µg."
  },
  {
   "value": 31,
   "misconception": "you read the mass at 159.6 days instead of 106.4 days",
   "feedback": "This answer is wrong — you read the mass at 159.6 days. At 106.4 days the curve sits at 62 µg."
  }
 ]
}

[
 {
  "value": 6.25,
  "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
  "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 6.25 = 93.75%."
 },
 {
  "value": 25,
  "misconception": "you divided 100% by the number of half-lives — four half-lives means four halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Four halvings leave 6.25% remaining, so 93.75% has decayed."
 },
 {
  "value": 50,
  "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
  "feedback": "This answer is wrong — you froze the decayed share at one halving. After four halvings, 93.75% has decayed."
 }
]

{
 "keyExplanation": "Remaining first: 100 → 50 → 25 → 12.5 → 6.25, because after each half-life, half of the nuclei that remain decay. Decayed is the rest: 100 − 6.25 = 93.75%.",
 "anticipatedWrong": [
  {
   "value": 6.25,
   "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
   "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 6.25 = 93.75%."
  },
  {
   "value": 25,
   "misconception": "you divided 100% by the number of half-lives — four half-lives means four halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Four halvings leave 6.25% remaining, so 93.75% has decayed."
  },
  {
   "value": 50,
   "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
   "feedback": "This answer is wrong — you froze the decayed share at one halving. After four halvings, 93.75% has decayed."
  }
 ]
}

[
 {
  "value": 10.68,
  "misconception": "you used the 4 in 1/4 as the number of halvings — 1 → 1/2 → 1/4 is 2 halvings",
  "feedback": "This answer is wrong — you used the 4 in 1/4 as the number of halvings. 1 → 1/2 → 1/4 is 2 halvings: 5.34 days."
 },
 {
  "value": 2.67,
  "misconception": "you counted only one halving — reaching 1/4 takes 2 halvings",
  "feedback": "This answer is wrong — you counted only one halving. Reaching 1/4 takes 2 halvings: 5.34 days."
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4.\nThe sample has halved 2 times.\ntime passed = number of halvings × half-life\ntime passed = 2 × 2.67 days\ntime passed = 5.34 days",
 "anticipatedWrong": [
  {
   "value": 10.68,
   "misconception": "you used the 4 in 1/4 as the number of halvings — 1 → 1/2 → 1/4 is 2 halvings",
   "feedback": "This answer is wrong — you used the 4 in 1/4 as the number of halvings. 1 → 1/2 → 1/4 is 2 halvings: 5.34 days."
  },
  {
   "value": 2.67,
   "misconception": "you counted only one halving — reaching 1/4 takes 2 halvings",
   "feedback": "This answer is wrong — you counted only one halving. Reaching 1/4 takes 2 halvings: 5.34 days."
  }
 ]
}

[
 {
  "value": 26.8,
  "misconception": "you read the time for the first halving — 94 µg is a quarter of 376 µg, which takes two halvings",
  "feedback": "This answer is wrong — you read the time for the first halving. 94 µg is a quarter of 376 µg: two halvings, 53.6 hours."
 },
 {
  "value": 80.4,
  "misconception": "you read one halving too far — the curve reaches 94 µg at 53.6 hours, not 47 µg",
  "feedback": "This answer is wrong — you read one halving too far. The curve reaches 94 µg at 53.6 hours."
 }
]

{
 "keyExplanation": "Find 94 µg on the mass axis. Go across to the curve, then down to the time axis. The sample reaches 94 µg at 53.6 hours.",
 "anticipatedWrong": [
  {
   "value": 26.8,
   "misconception": "you read the time for the first halving — 94 µg is a quarter of 376 µg, which takes two halvings",
   "feedback": "This answer is wrong — you read the time for the first halving. 94 µg is a quarter of 376 µg: two halvings, 53.6 hours."
  },
  {
   "value": 80.4,
   "misconception": "you read one halving too far — the curve reaches 94 µg at 53.6 hours, not 47 µg",
   "feedback": "This answer is wrong — you read one halving too far. The curve reaches 94 µg at 53.6 hours."
  }
 ]
}

[
 {
  "value": "1/10",
  "misconception": "you divided the original amount by the number of half-lives — ten half-lives means ten halvings",
  "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Ten half-lives means ten halvings, ending at 1/1024."
 },
 {
  "value": "1/512",
  "misconception": "you halved one time too few — the tenth halving takes 1/512 to 1/1024",
  "feedback": "This answer is wrong — you halved one time too few. The tenth halving takes 1/512 to 1/1024."
 },
 {
  "value": "0",
  "misconception": "you emptied the sample — each half-life removes only half of what is left",
  "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/1024 still remains."
 }
]

{
 "keyExplanation": "Halve once per half-life, ten times over: 1/2, 1/4, ... , 1/512, 1/1024 — because after each half-life, half of the nuclei that remain decay. After 10 half-lives, 1/1024 of the original niobium-95 nuclei remain.",
 "anticipatedWrong": [
  {
   "value": "1/10",
   "misconception": "you divided the original amount by the number of half-lives — ten half-lives means ten halvings",
   "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Ten half-lives means ten halvings, ending at 1/1024."
  },
  {
   "value": "1/512",
   "misconception": "you halved one time too few — the tenth halving takes 1/512 to 1/1024",
   "feedback": "This answer is wrong — you halved one time too few. The tenth halving takes 1/512 to 1/1024."
  },
  {
   "value": "0",
   "misconception": "you emptied the sample — each half-life removes only half of what is left",
   "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/1024 still remains."
  }
 ]
}

[
 {
  "value": 16.67,
  "misconception": "you divided 100% by the number of half-lives — six half-lives means six halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Six half-lives means six halvings, ending at 1.5625%."
 },
 {
  "value": 3.125,
  "misconception": "you halved one time too few — the sixth halving takes 3.125% down to 1.5625%",
  "feedback": "This answer is wrong — you halved one time too few. The sixth halving takes 3.125% down to 1.5625%."
 },
 {
  "value": 0.015625,
  "misconception": "you gave the fraction instead of the percentage — 1/64 of the original is 1.5625%",
  "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/64 of the original is 1.5625%."
 }
]

{
 "keyExplanation": "Halve 100% once per half-life, six times: 100 → 50 → 25 → 12.5 → 6.25 → 3.125 → 1.5625. Because after each half-life, half of the nuclei that remain decay. 1.5625% of the original nuclei remain.",
 "anticipatedWrong": [
  {
   "value": 16.67,
   "misconception": "you divided 100% by the number of half-lives — six half-lives means six halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Six half-lives means six halvings, ending at 1.5625%."
  },
  {
   "value": 3.125,
   "misconception": "you halved one time too few — the sixth halving takes 3.125% down to 1.5625%",
   "feedback": "This answer is wrong — you halved one time too few. The sixth halving takes 3.125% down to 1.5625%."
  },
  {
   "value": 0.015625,
   "misconception": "you gave the fraction instead of the percentage — 1/64 of the original is 1.5625%",
   "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/64 of the original is 1.5625%."
  }
 ]
}

[
 {
  "value": 148,
  "misconception": "you halved only once — 9.9 days holds THREE 3.3-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 9.9 days holds THREE 3.3-day half-lives: 296 → 148 → 74 → 37."
 },
 {
  "value": 98.67,
  "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
  "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 296 → 148 → 74 → 37."
 },
 {
  "value": 74,
  "misconception": "you halved one time too few — the third halving takes 74 µg down to 37 µg",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 74 µg down to 37 µg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 296 µg, half-life = 3.3 days, time passed = 9.9 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 9.9 ÷ 3.3 = 3\nHalve the starting amount three times: 296 → 148 → 74 → 37\n37 µg of gallium-67 remains.",
 "anticipatedWrong": [
  {
   "value": 148,
   "misconception": "you halved only once — 9.9 days holds THREE 3.3-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 9.9 days holds THREE 3.3-day half-lives: 296 → 148 → 74 → 37."
  },
  {
   "value": 98.67,
   "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
   "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 296 → 148 → 74 → 37."
  },
  {
   "value": 74,
   "misconception": "you halved one time too few — the third halving takes 74 µg down to 37 µg",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 74 µg down to 37 µg."
  }
 ]
}

[
 {
  "value": 178,
  "misconception": "you halved only once — 167.6 days holds TWO 83.8-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 167.6 days holds TWO 83.8-day half-lives: 356 → 178 → 89."
 },
 {
  "value": 44.5,
  "misconception": "you halved one time too many — two half-lives means exactly two halvings",
  "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 356 → 178 → 89."
 },
 {
  "value": 188.4,
  "misconception": "you subtracted the days from the micrograms — amounts and times never subtract; count the half-lives, then halve",
  "feedback": "This answer is wrong — you subtracted the days from the micrograms. Count the half-lives, then halve: 89 µg."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 356 µg, half-life = 83.8 days, time passed = 167.6 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 167.6 ÷ 83.8 = 2\nHalve the starting amount twice: 356 → 178 → 89\n89 µg of scandium-46 remains.",
 "anticipatedWrong": [
  {
   "value": 178,
   "misconception": "you halved only once — 167.6 days holds TWO 83.8-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 167.6 days holds TWO 83.8-day half-lives: 356 → 178 → 89."
  },
  {
   "value": 44.5,
   "misconception": "you halved one time too many — two half-lives means exactly two halvings",
   "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 356 → 178 → 89."
  },
  {
   "value": 188.4,
   "misconception": "you subtracted the days from the micrograms — amounts and times never subtract; count the half-lives, then halve",
   "feedback": "This answer is wrong — you subtracted the days from the micrograms. Count the half-lives, then halve: 89 µg."
  }
 ]
}

[
 {
  "value": 36,
  "misconception": "you used the ratio 272 ÷ 34 = 8 as the number of halvings — 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings",
  "feedback": "This answer is wrong — you used the ratio 272 ÷ 34 = 8 as the number of halvings. 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings: 13.5 days."
 },
 {
  "value": 9,
  "misconception": "you stopped counting one halving early — 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting one halving early. 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings, not 2: 13.5 days."
 }
]

{
 "keyExplanation": "Count the halvings: 272 µg → 136 µg → 68 µg → 34 µg.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 4.5 days\ntime passed = 13.5 days",
 "anticipatedWrong": [
  {
   "value": 36,
   "misconception": "you used the ratio 272 ÷ 34 = 8 as the number of halvings — 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings",
   "feedback": "This answer is wrong — you used the ratio 272 ÷ 34 = 8 as the number of halvings. 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings: 13.5 days."
  },
  {
   "value": 9,
   "misconception": "you stopped counting one halving early — 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting one halving early. 272 µg → 136 µg → 68 µg → 34 µg is 3 halvings, not 2: 13.5 days."
  }
 ]
}

[
 {
  "value": 257.2,
  "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
  "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 128.6 days."
 },
 {
  "value": 514.4,
  "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
  "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 128.6 days."
 }
]

{
 "keyExplanation": "Half of the starting amount: half of 100% is 50%. The curve reaches 50% at 128.6 days. The half-life of thulium-170 is 128.6 days.",
 "anticipatedWrong": [
  {
   "value": 257.2,
   "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
   "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 128.6 days."
  },
  {
   "value": 514.4,
   "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
   "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 128.6 days."
  }
 ]
}

[
 {
  "value": 66,
  "misconception": "you read the mass at 72.3 days instead of 144.6 days",
  "feedback": "This answer is wrong — you read the mass at 72.3 days. At 144.6 days the curve sits at 33 mg."
 },
 {
  "value": 16.5,
  "misconception": "you read the mass at 216.9 days instead of 144.6 days",
  "feedback": "This answer is wrong — you read the mass at 216.9 days. At 144.6 days the curve sits at 33 mg."
 }
]

{
 "keyExplanation": "Go up from 144.6 days on the time axis to the curve, then straight across to the mass axis. At 144.6 days, 33 mg of terbium-160 remains.",
 "anticipatedWrong": [
  {
   "value": 66,
   "misconception": "you read the mass at 72.3 days instead of 144.6 days",
   "feedback": "This answer is wrong — you read the mass at 72.3 days. At 144.6 days the curve sits at 33 mg."
  },
  {
   "value": 16.5,
   "misconception": "you read the mass at 216.9 days instead of 144.6 days",
   "feedback": "This answer is wrong — you read the mass at 216.9 days. At 144.6 days the curve sits at 33 mg."
  }
 ]
}

[
 {
  "value": 3.125,
  "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
  "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 3.125 = 96.875%."
 },
 {
  "value": 20,
  "misconception": "you divided 100% by the number of half-lives — five half-lives means five halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Five halvings leave 3.125% remaining, so 96.875% has decayed."
 },
 {
  "value": 50,
  "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
  "feedback": "This answer is wrong — you froze the decayed share at one halving. After five halvings, 96.875% has decayed."
 }
]

{
 "keyExplanation": "Remaining first: 100 → 50 → 25 → 12.5 → 6.25 → 3.125, because after each half-life, half of the nuclei that remain decay. Decayed is the rest: 100 − 3.125 = 96.875%.",
 "anticipatedWrong": [
  {
   "value": 3.125,
   "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
   "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 3.125 = 96.875%."
  },
  {
   "value": 20,
   "misconception": "you divided 100% by the number of half-lives — five half-lives means five halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Five halvings leave 3.125% remaining, so 96.875% has decayed."
  },
  {
   "value": 50,
   "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
   "feedback": "This answer is wrong — you froze the decayed share at one halving. After five halvings, 96.875% has decayed."
  }
 ]
}

[
 {
  "value": 23.2,
  "misconception": "you used the 8 in 1/8 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
  "feedback": "This answer is wrong — you used the 8 in 1/8 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 is 3 halvings: 8.7 days."
 },
 {
  "value": 5.8,
  "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting at 1/4. Reaching 1/8 takes 3 halvings, not 2: 8.7 days."
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 2.9 days\ntime passed = 8.7 days",
 "anticipatedWrong": [
  {
   "value": 23.2,
   "misconception": "you used the 8 in 1/8 as the number of halvings — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
   "feedback": "This answer is wrong — you used the 8 in 1/8 as the number of halvings. 1 → 1/2 → 1/4 → 1/8 is 3 halvings: 8.7 days."
  },
  {
   "value": 5.8,
   "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting at 1/4. Reaching 1/8 takes 3 halvings, not 2: 8.7 days."
  }
 ]
}

[
 {
  "value": 58.5,
  "misconception": "you read the time for the first halving — 114 g is a quarter of 456 g, which takes two halvings",
  "feedback": "This answer is wrong — you read the time for the first halving. 114 g is a quarter of 456 g: two halvings, 117 days."
 },
 {
  "value": 175.5,
  "misconception": "you read one halving too far — the curve reaches 114 g at 117 days, not 57 g",
  "feedback": "This answer is wrong — you read one halving too far. The curve reaches 114 g at 117 days."
 }
]

{
 "keyExplanation": "Find 114 g on the mass axis. Go across to the curve, then down to the time axis. The sample reaches 114 g at 117 days.",
 "anticipatedWrong": [
  {
   "value": 58.5,
   "misconception": "you read the time for the first halving — 114 g is a quarter of 456 g, which takes two halvings",
   "feedback": "This answer is wrong — you read the time for the first halving. 114 g is a quarter of 456 g: two halvings, 117 days."
  },
  {
   "value": 175.5,
   "misconception": "you read one halving too far — the curve reaches 114 g at 117 days, not 57 g",
   "feedback": "This answer is wrong — you read one halving too far. The curve reaches 114 g at 117 days."
  }
 ]
}

[
 {
  "value": "1/11",
  "misconception": "you divided the original amount by the number of half-lives — eleven half-lives means eleven halvings",
  "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Eleven half-lives means eleven halvings, ending at 1/2048."
 },
 {
  "value": "1/1024",
  "misconception": "you halved one time too few — the eleventh halving takes 1/1024 to 1/2048",
  "feedback": "This answer is wrong — you halved one time too few. The eleventh halving takes 1/1024 to 1/2048."
 },
 {
  "value": "0",
  "misconception": "you emptied the sample — each half-life removes only half of what is left",
  "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/2048 still remains."
 }
]

{
 "keyExplanation": "Halve once per half-life, eleven times over: 1/2, 1/4, ... , 1/1024, 1/2048 — because after each half-life, half of the nuclei that remain decay. After 11 half-lives, 1/2048 of the original europium-152 nuclei remain.",
 "anticipatedWrong": [
  {
   "value": "1/11",
   "misconception": "you divided the original amount by the number of half-lives — eleven half-lives means eleven halvings",
   "feedback": "This answer is wrong — you divided the original amount by the number of half-lives. Eleven half-lives means eleven halvings, ending at 1/2048."
  },
  {
   "value": "1/1024",
   "misconception": "you halved one time too few — the eleventh halving takes 1/1024 to 1/2048",
   "feedback": "This answer is wrong — you halved one time too few. The eleventh halving takes 1/1024 to 1/2048."
  },
  {
   "value": "0",
   "misconception": "you emptied the sample — each half-life removes only half of what is left",
   "feedback": "This answer is wrong — you emptied the sample. Each half-life removes only half of what is left, so 1/2048 still remains."
  }
 ]
}

[
 {
  "value": 50,
  "misconception": "you divided 100% by the number of half-lives (or halved once) — two half-lives means two halvings of what remains",
  "feedback": "This answer is wrong — you halved only once (or divided 100% by the number of half-lives). Two half-lives means two successive halvings of what remains: 100 → 50 → 25."
 },
 {
  "value": 0,
  "misconception": "you emptied the sample by adding half plus half — the second half-life removes half OF WHAT REMAINS, leaving a quarter",
  "feedback": "This answer is wrong — you emptied the sample by adding half plus half. The second half-life removes half OF WHAT REMAINS, leaving 25%."
 },
 {
  "value": 0.25,
  "misconception": "you gave the fraction instead of the percentage — 1/4 of the original is 25%",
  "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/4 of the original is 25%."
 }
]

{
 "keyExplanation": "Halve 100% once per half-life: 100 → 50 → 25. Because after each half-life, half of the nuclei that remain decay. 25% of the original nuclei remain.",
 "anticipatedWrong": [
  {
   "value": 50,
   "misconception": "you divided 100% by the number of half-lives (or halved once) — two half-lives means two halvings of what remains",
   "feedback": "This answer is wrong — you halved only once (or divided 100% by the number of half-lives). Two half-lives means two successive halvings of what remains: 100 → 50 → 25."
  },
  {
   "value": 0,
   "misconception": "you emptied the sample by adding half plus half — the second half-life removes half OF WHAT REMAINS, leaving a quarter",
   "feedback": "This answer is wrong — you emptied the sample by adding half plus half. The second half-life removes half OF WHAT REMAINS, leaving 25%."
  },
  {
   "value": 0.25,
   "misconception": "you gave the fraction instead of the percentage — 1/4 of the original is 25%",
   "feedback": "This answer is wrong — you gave the fraction instead of the percentage. 1/4 of the original is 25%."
  }
 ]
}

[
 {
  "value": 236,
  "misconception": "you halved only once — 8.4 days holds THREE 2.8-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 8.4 days holds THREE 2.8-day half-lives: 472 → 236 → 118 → 59."
 },
 {
  "value": 157.33,
  "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
  "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 472 → 236 → 118 → 59."
 },
 {
  "value": 118,
  "misconception": "you halved one time too few — the third halving takes 118 ng down to 59 ng",
  "feedback": "This answer is wrong — you halved one time too few. The third halving takes 118 ng down to 59 ng."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 472 ng, half-life = 2.8 days, time passed = 8.4 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 8.4 ÷ 2.8 = 3\nHalve the starting amount three times: 472 → 236 → 118 → 59\n59 ng of indium-111 remains.",
 "anticipatedWrong": [
  {
   "value": 236,
   "misconception": "you halved only once — 8.4 days holds THREE 2.8-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 8.4 days holds THREE 2.8-day half-lives: 472 → 236 → 118 → 59."
  },
  {
   "value": 157.33,
   "misconception": "you divided the starting amount by the number of half-lives — three half-lives means three halvings, not division by 3",
   "feedback": "This answer is wrong — you divided the starting amount by the number of half-lives. Three half-lives means three halvings: 472 → 236 → 118 → 59."
  },
  {
   "value": 118,
   "misconception": "you halved one time too few — the third halving takes 118 ng down to 59 ng",
   "feedback": "This answer is wrong — you halved one time too few. The third halving takes 118 ng down to 59 ng."
  }
 ]
}

[
 {
  "value": 158,
  "misconception": "you halved only once — 27.2 days holds TWO 13.6-day half-lives",
  "feedback": "This answer is wrong — you halved only once. 27.2 days holds TWO 13.6-day half-lives: 316 → 158 → 79."
 },
 {
  "value": 39.5,
  "misconception": "you halved one time too many — two half-lives means exactly two halvings",
  "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 316 → 158 → 79."
 },
 {
  "value": 288.8,
  "misconception": "you subtracted the days from the nanograms — amounts and times never subtract; count the half-lives, then halve",
  "feedback": "This answer is wrong — you subtracted the days from the nanograms. Count the half-lives, then halve: 79 ng."
 }
]

{
 "keyExplanation": "Write down the values in the question:\nstarting amount = 316 ng, half-life = 13.6 days, time passed = 27.2 days\nWrite down the equation:\nnumber of half-lives = time passed ÷ half-life\nSubstitute in the values, and calculate:\nnumber of half-lives = 27.2 ÷ 13.6 = 2\nHalve the starting amount twice: 316 → 158 → 79\n79 ng of praseodymium-143 remains.",
 "anticipatedWrong": [
  {
   "value": 158,
   "misconception": "you halved only once — 27.2 days holds TWO 13.6-day half-lives",
   "feedback": "This answer is wrong — you halved only once. 27.2 days holds TWO 13.6-day half-lives: 316 → 158 → 79."
  },
  {
   "value": 39.5,
   "misconception": "you halved one time too many — two half-lives means exactly two halvings",
   "feedback": "This answer is wrong — you halved one time too many. Two half-lives means exactly two halvings: 316 → 158 → 79."
  },
  {
   "value": 288.8,
   "misconception": "you subtracted the days from the nanograms — amounts and times never subtract; count the half-lives, then halve",
   "feedback": "This answer is wrong — you subtracted the days from the nanograms. Count the half-lives, then halve: 79 ng."
  }
 ]
}

[
 {
  "value": 339.2,
  "misconception": "you used the ratio 328 ÷ 41 = 8 as the number of halvings — 328 g → 164 g → 82 g → 41 g is 3 halvings",
  "feedback": "This answer is wrong — you used the ratio 328 ÷ 41 = 8 as the number of halvings. 328 g → 164 g → 82 g → 41 g is 3 halvings: 127.2 days."
 },
 {
  "value": 84.8,
  "misconception": "you stopped counting one halving early — 328 g → 164 g → 82 g → 41 g is 3 halvings, not 2",
  "feedback": "This answer is wrong — you stopped counting one halving early. 328 g → 164 g → 82 g → 41 g is 3 halvings, not 2: 127.2 days."
 }
]

{
 "keyExplanation": "Count the halvings: 328 g → 164 g → 82 g → 41 g.\nThe sample has halved 3 times.\ntime passed = number of halvings × half-life\ntime passed = 3 × 42.4 days\ntime passed = 127.2 days",
 "anticipatedWrong": [
  {
   "value": 339.2,
   "misconception": "you used the ratio 328 ÷ 41 = 8 as the number of halvings — 328 g → 164 g → 82 g → 41 g is 3 halvings",
   "feedback": "This answer is wrong — you used the ratio 328 ÷ 41 = 8 as the number of halvings. 328 g → 164 g → 82 g → 41 g is 3 halvings: 127.2 days."
  },
  {
   "value": 84.8,
   "misconception": "you stopped counting one halving early — 328 g → 164 g → 82 g → 41 g is 3 halvings, not 2",
   "feedback": "This answer is wrong — you stopped counting one halving early. 328 g → 164 g → 82 g → 41 g is 3 halvings, not 2: 127.2 days."
  }
 ]
}

[
 {
  "value": 228.8,
  "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
  "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 114.4 days."
 },
 {
  "value": 457.6,
  "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
  "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 114.4 days."
 }
]

{
 "keyExplanation": "Half of the starting amount: half of 100% is 50%. The curve reaches 50% at 114.4 days. The half-life of tantalum-182 is 114.4 days.",
 "anticipatedWrong": [
  {
   "value": 228.8,
   "misconception": "you read the time for the sample to fall to a quarter — that is two half-lives, not one",
   "feedback": "This answer is wrong — you read the time to fall to a quarter. That is two half-lives; one half-life is 114.4 days."
  },
  {
   "value": 457.6,
   "misconception": "you read the last time labeled on the axis instead of the 50% crossing",
   "feedback": "This answer is wrong — you read the last time on the axis. The 50% crossing sits at 114.4 days."
  }
 ]
}

[
 {
  "value": 86,
  "misconception": "you read the mass at 70 days instead of 105 days",
  "feedback": "This answer is wrong — you read the mass at 70 days. At 105 days the curve sits at 43 mg."
 },
 {
  "value": 21.5,
  "misconception": "you read the mass at 140 days instead of 105 days",
  "feedback": "This answer is wrong — you read the mass at 140 days. At 105 days the curve sits at 43 mg."
 }
]

{
 "keyExplanation": "Go up from 105 days on the time axis to the curve, then straight across to the mass axis. At 105 days, 43 mg of argon-37 remains.",
 "anticipatedWrong": [
  {
   "value": 86,
   "misconception": "you read the mass at 70 days instead of 105 days",
   "feedback": "This answer is wrong — you read the mass at 70 days. At 105 days the curve sits at 43 mg."
  },
  {
   "value": 21.5,
   "misconception": "you read the mass at 140 days instead of 105 days",
   "feedback": "This answer is wrong — you read the mass at 140 days. At 105 days the curve sits at 43 mg."
  }
 ]
}

[
 {
  "value": 1.5625,
  "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
  "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 1.5625 = 98.4375%."
 },
 {
  "value": 16.67,
  "misconception": "you divided 100% by the number of half-lives — six half-lives means six halvings of what remains",
  "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Six halvings leave 1.5625% remaining, so 98.4375% has decayed."
 },
 {
  "value": 50,
  "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
  "feedback": "This answer is wrong — you froze the decayed share at one halving. After six halvings, 98.4375% has decayed."
 }
]

{
 "keyExplanation": "Remaining first: 100 → 50 → 25 → 12.5 → 6.25 → 3.125 → 1.5625, because after each half-life, half of the nuclei that remain decay. Decayed is the rest: 100 − 1.5625 = 98.4375%.",
 "anticipatedWrong": [
  {
   "value": 1.5625,
   "misconception": "you gave the percentage that REMAINS — the question asks for the decayed percentage",
   "feedback": "This answer is wrong — you gave the percentage that REMAINS. Decayed is the rest: 100 − 1.5625 = 98.4375%."
  },
  {
   "value": 16.67,
   "misconception": "you divided 100% by the number of half-lives — six half-lives means six halvings of what remains",
   "feedback": "This answer is wrong — you divided 100% by the number of half-lives. Six halvings leave 1.5625% remaining, so 98.4375% has decayed."
  },
  {
   "value": 50,
   "misconception": "you froze the decayed share at one halving — each later half-life keeps removing half of what remains",
   "feedback": "This answer is wrong — you froze the decayed share at one halving. After six halvings, 98.4375% has decayed."
  }
 ]
}

[
 {
  "value": 37,
  "misconception": "you used the 4 in 1/4 as the number of halvings — 1 → 1/2 → 1/4 is 2 halvings",
  "feedback": "This answer is wrong — you used the 4 in 1/4 as the number of halvings. 1 → 1/2 → 1/4 is 2 halvings: 18.5 days."
 },
 {
  "value": 9.25,
  "misconception": "you counted only one halving — reaching 1/4 takes 2 halvings",
  "feedback": "This answer is wrong — you counted only one halving. Reaching 1/4 takes 2 halvings: 18.5 days."
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4.\nThe sample has halved 2 times.\ntime passed = number of halvings × half-life\ntime passed = 2 × 9.25 days\ntime passed = 18.5 days",
 "anticipatedWrong": [
  {
   "value": 37,
   "misconception": "you used the 4 in 1/4 as the number of halvings — 1 → 1/2 → 1/4 is 2 halvings",
   "feedback": "This answer is wrong — you used the 4 in 1/4 as the number of halvings. 1 → 1/2 → 1/4 is 2 halvings: 18.5 days."
  },
  {
   "value": 9.25,
   "misconception": "you counted only one halving — reaching 1/4 takes 2 halvings",
   "feedback": "This answer is wrong — you counted only one halving. Reaching 1/4 takes 2 halvings: 18.5 days."
  }
 ]
}

[
 {
  "value": 32.8,
  "misconception": "you read the time for the first halving — 98 g is a quarter of 392 g, which takes two halvings",
  "feedback": "This answer is wrong — you read the time for the first halving. 98 g is a quarter of 392 g: two halvings, 65.6 days."
 },
 {
  "value": 98.4,
  "misconception": "you read one halving too far — the curve reaches 98 g at 65.6 days, not 49 g",
  "feedback": "This answer is wrong — you read one halving too far. The curve reaches 98 g at 65.6 days."
 }
]

{
 "keyExplanation": "Find 98 g on the mass axis. Go across to the curve, then down to the time axis. The sample reaches 98 g at 65.6 days.",
 "anticipatedWrong": [
  {
   "value": 32.8,
   "misconception": "you read the time for the first halving — 98 g is a quarter of 392 g, which takes two halvings",
   "feedback": "This answer is wrong — you read the time for the first halving. 98 g is a quarter of 392 g: two halvings, 65.6 days."
  },
  {
   "value": 98.4,
   "misconception": "you read one halving too far — the curve reaches 98 g at 65.6 days, not 49 g",
   "feedback": "This answer is wrong — you read one halving too far. The curve reaches 98 g at 65.6 days."
  }
 ]
}

[
 {
  "value": 7.2,
  "misconception": "you used the 2 in 1/2 as the number of half-lives — reaching 1/2 takes exactly one halving",
  "feedback": null
 },
 {
  "value": 1.8,
  "misconception": "you halved the half-life itself — 1/2 remaining means one full half-life has passed",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2. The isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 3.6 billion years\nage = 3.6 billion years"
}

[
 {
  "value": 183360,
  "misconception": "you used the 32 in 1/32 as the number of half-lives — 1 → 1/2 → 1/4 → 1/8 → 1/16 → 1/32 is 5 halvings",
  "feedback": null
 },
 {
  "value": 22920,
  "misconception": "you stopped counting at 1/16 — reaching 1/32 takes 5 halvings, not 4",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8 → 1/16 → 1/32. The carbon-14 has halved 5 times.\nage = number of half-lives × half-life\nage = 5 × 5730 years\nage = 28650 years"
}

[
 {
  "value": 5650,
  "misconception": "you used 12.5 as the number of half-lives — turn the percent into a fraction first: 12.5% is 1/8, which is 3 halvings",
  "feedback": null
 },
 {
  "value": 904,
  "misconception": "you stopped counting at 1/4 (25%) — reaching 12.5% takes 3 halvings, not 2",
  "feedback": null
 }
]

{
 "keyExplanation": "Turn the percent into a fraction: 12.5% is 1/8. Count the halvings: 1 → 1/2 → 1/4 → 1/8 — the isotope has halved 3 times.\nage = number of half-lives × half-life\nage = 3 × 452 million years\nage = 1356 million years"
}

[
 {
  "value": 3.8,
  "misconception": "you used the 2 in 1/2 as a count of two half-lives — one halving reaches 1/2",
  "feedback": null
 },
 {
  "value": 0.95,
  "misconception": "you divided the half-life by two — a full half-life passes to reach 1/2",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2. The isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 1.9 billion years\nage = 1.9 billion years"
}

[
 {
  "value": 3456,
  "misconception": "you used the 64 in 1/64 as the number of half-lives — 1 → 1/2 → 1/4 → 1/8 → 1/16 → 1/32 → 1/64 is 6 halvings",
  "feedback": null
 },
 {
  "value": 270,
  "misconception": "you stopped counting at 1/32 — reaching 1/64 takes 6 halvings, not 5",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8 → 1/16 → 1/32 → 1/64. The isotope has halved 6 times.\nage = number of half-lives × half-life\nage = 6 × 54 million years\nage = 324 million years"
}

[
 {
  "value": 16025,
  "misconception": "you used 25 as the number of half-lives — turn the percent into a fraction first: 25% is 1/4, which is 2 halvings",
  "feedback": null
 },
 {
  "value": 641,
  "misconception": "you counted only one halving — reaching 25% takes 2 halvings",
  "feedback": null
 }
]

{
 "keyExplanation": "Turn the percent into a fraction: 25% is 1/4. Count the halvings: 1 → 1/2 → 1/4 — the isotope has halved 2 times.\nage = number of half-lives × half-life\nage = 2 × 641 million years\nage = 1282 million years"
}

[
 {
  "value": 5.4,
  "misconception": "you used the 2 in 1/2 as the number of half-lives — reaching 1/2 is a single halving",
  "feedback": null
 },
 {
  "value": 1.35,
  "misconception": "you halved the half-life itself — 1/2 remaining means one full half-life has passed",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2. The isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 2.7 billion years\nage = 2.7 billion years"
}

[
 {
  "value": 10.0,
  "misconception": "you used the 8 in 1/8 as the number of half-lives — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
  "feedback": null
 },
 {
  "value": 2.5,
  "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8. The potassium-40 has halved 3 times.\nage = number of half-lives × half-life\nage = 3 × 1.25 billion years\nage = 3.75 billion years"
}

[
 {
  "value": 950,
  "misconception": "you used 6.25 as the number of half-lives — turn the percent into a fraction first: 6.25% is 1/16, which is 4 halvings",
  "feedback": null
 },
 {
  "value": 456,
  "misconception": "you stopped counting one halving early — reaching 6.25% takes 4 halvings, not 3",
  "feedback": null
 }
]

{
 "keyExplanation": "Turn the percent into a fraction: 6.25% is 1/16. Count the halvings: 1 → 1/2 → 1/4 → 1/8 → 1/16 — the isotope has halved 4 times.\nage = number of half-lives × half-life\nage = 4 × 152 million years\nage = 608 million years"
}

[
 {
  "value": 4.3,
  "misconception": "you read the 2 in 1/2 as two half-lives — a single halving lands on 1/2",
  "feedback": null
 },
 {
  "value": 1.075,
  "misconception": "you split the half-life in half — reaching 1/2 costs one entire half-life",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2. The isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 2.15 billion years\nage = 2.15 billion years"
}

[
 {
  "value": 11.28,
  "misconception": "you used the 8 in 1/8 as the number of half-lives — 1 → 1/2 → 1/4 → 1/8 is 3 halvings",
  "feedback": null
 },
 {
  "value": 2.82,
  "misconception": "you stopped counting at 1/4 — reaching 1/8 takes 3 halvings, not 2",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4 → 1/8. The isotope has halved 3 times.\nage = number of half-lives × half-life\nage = 3 × 1.41 billion years\nage = 4.23 billion years"
}

[
 {
  "value": 290.625,
  "misconception": "you used 3.125 as the number of half-lives — turn the percent into a fraction first: 3.125% is 1/32, which is 5 halvings",
  "feedback": null
 },
 {
  "value": 372,
  "misconception": "you stopped counting one halving early — reaching 3.125% takes 5 halvings, not 4",
  "feedback": null
 }
]

{
 "keyExplanation": "Turn the percent into a fraction: 3.125% is 1/32. Count the halvings: 1 → 1/2 → 1/4 → 1/8 → 1/16 → 1/32 — the isotope has halved 5 times.\nage = number of half-lives × half-life\nage = 5 × 93 million years\nage = 465 million years"
}

[
 {
  "value": 6.3,
  "misconception": "you used the 2 in 1/2 as the number of half-lives — reaching 1/2 is a single halving",
  "feedback": null
 },
 {
  "value": 1.575,
  "misconception": "you halved the half-life itself — 1/2 remaining means one full half-life has passed",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2. The isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 3.15 billion years\nage = 3.15 billion years"
}

[
 {
  "value": 4.2,
  "misconception": "you used the 4 in 1/4 as the number of half-lives — 1 → 1/2 → 1/4 is 2 halvings",
  "feedback": null
 },
 {
  "value": 1.05,
  "misconception": "you counted only one halving — reaching 1/4 takes 2 halvings",
  "feedback": null
 }
]

{
 "keyExplanation": "Count the halvings: 1 → 1/2 → 1/4. The isotope has halved 2 times.\nage = number of half-lives × half-life\nage = 2 × 1.05 billion years\nage = 2.10 billion years"
}

[
 {
  "value": 1790,
  "misconception": "you used the 2 in 1/2 as the number of half-lives — 50% is 1/2, which is a single halving",
  "feedback": null
 },
 {
  "value": 447.5,
  "misconception": "you halved the half-life itself — 50% remaining means one full half-life has passed",
  "feedback": null
 }
]

{
 "keyExplanation": "Turn the percent into a fraction: 50% is 1/2. Count the halvings: 1 → 1/2 — the isotope has halved once.\nage = number of half-lives × half-life\nage = 1 × 895 million years\nage = 895 million years"
}