2016 AMC 10B Problem 13

P11-15~2 minPrint

Linear Equations & Word Problems Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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)