AMC 10 · 2016 · #13

Grade 6 number-theory
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking
ratio-proportionsystems-of-equationslinear-equations-one-var convert-to-algebra ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 2 insights
Problem
At a hospital, the babies born in sets of twins, triplets, and quadruplets together total 1000. There were four times as many sets of triplets as sets of quadruplets, and three times as many sets of twins as sets of triplets. Find how many of the 1000 babies were in sets of quadruplets.

Pick an answer.

(A)
25
(B)
40
(C)
64
(D)
100
(E)
160

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Three unknown quantities are chained together by "four times" and "three times," but they all hang off a single peg: the number of quadruplet sets. Tool #4 (Introduce a Variable) names that one quantity q, and then every other count becomes an expression in q, so we never need a second variable. Tool #13 (Convert to Algebra) turns the sentence "these sets account for 1000 babies" into one equation by counting 2 babies per twin set, 3 per triplet set, and 4 per quadruplet set. Solving gives q; tool #3 (Eliminate Possibilities) guards the final move, because the question asks for babies in quadruplet sets — not the value of q itself, which is the planted trap answer (A).

1STEP 1

Name one count, build the rest

Let q be the quadruplet sets; then triplet sets = 4q and twin sets = 3·4q = 12q, so all three counts ride on one unknown.

quad sets=q, triplet sets=4q, twin sets=3(4q)=12q
2STEP 2

Count the babies

Each twin set holds 2 babies, each triplet set 3, each quadruplet set 4, so 2·12q + 3·4q + 4q = 1000 — the given total.

2(12q) + 3(4q) + 4q = 1000
3STEP 3

Solve for the number of quadruplet sets

Combine like terms: 40q = 1000, so q = 25 quadruplet sets.

24q + 12q + 4q = 40q = 1000 → q = 25
4STEP 4

Answer what was actually asked

The question wants babies, not sets: the 25 quadruplet sets hold 4 each, so 4·25 = 100 — choice (D); the bare 25 is the trap (A).

4q = 4 · 25 = 100 = (D)
Answer
100
Check the original sentences against q = 25: 25 quadruplet sets, 4 · 25 = 100 triplet sets, 3 · 100 = 300 twin sets. Babies: 300 · 2 = 600 twins, 100 · 3 = 300 triplets, 25 · 4 = 100 quadruplets, totaling 600 + 300 + 100 = 1000 — exactly the given total. The quadruplet babies, 100, are a clean tenth of all 1000, which matches the answer (D) and confirms the work.
💡Key takeaway

Tie every unknown to one count, turn the words into a single equation — then reread the question so you report the babies it asked for, not just the variable you solved.

  • Name one count, build the rest
  • Count the babies
  • Solve for the number of quadruplet sets
  • Answer what was actually asked

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