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2016 AMC 10B Problem 13

2016 AMC 10B Problem 13AMC 10 Step by Step

Name: Date: October 28, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P11-15

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?

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

Answer

Key insight

With q sets of quadruplets there are 4q sets of triplets and 12q sets of twins, so the babies number 4q + 12q + 24q = 40q = 1000.

Solution

Everything is compared to the quadruplets, so start with them.

1. Let $q$ be the number of sets of quadruplets.
2. There are $4$ times as many sets of triplets: $4q$ sets.
3. There are $3$ times as many sets of twins as sets of triplets: $3 \cdot 4q = 12q$ sets.
4. The $1000$ counts babies, not sets. A set of quadruplets is $4$ babies, triplets $3$ , and twins $2$ . So
$4 \cdot q + 3 \cdot 4q + 2 \cdot 12q = 4q + 12q + 24q = 40q = 1000.$
5. So $q = 25$ sets of quadruplets.
6. The question asks for babies in those sets: $4 \cdot 25 = 100$ .

The answer is $\boxed{\textbf{(D)}\ 100}$ .

Why this works

A chain of "times as many" facts becomes one equation if you name the thing everything else is compared to. The only catch is units: the ratios are about sets, but the total is about babies. Multiply each number of sets by the size of the set before adding.

The trap

Answering 25, the number of sets of quadruplets. The question asks for babies, which is 4 times as many.

Concise solution

Everything is described relative to the quadruplets, so let there be $q$ sets of quadruplets. Then there are $4q$ sets of triplets and $3 \cdot 4q = 12q$ sets of twins.

Count babies, not sets: each set of quadruplets holds $4$ babies, each set of triplets $3$ , each set of twins $2$ . So
$4q + 3(4q) + 2(12q) = 4q + 12q + 24q = 40q = 1000,$
which gives $q = 25$ sets of quadruplets.

The question asks for babies in sets of quadruplets: $4q = 100$ .

The answer is $\boxed{\textbf{(D)}\ 100}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

Techniques

Set up the equation/formula and compute; no special trick needed