2016 AMC 10B Problems/Problem 13

Problem

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?

$\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160$

Solution

Let us have a system of equations where $a$ is the number of twins, $b$ is the number of triplets, and $c$ is the number of quadruplets. We have: $2a+3b+4c = 1000$ We also have $b=4c$ (Four times as many sets of triplets as quadruplets.) We also have $a=3b$ (Three times as many sets of twins as triplets.) Substituting we have:

We have found our desire $(a,b,c)$ = $(300,100,25)$ There are 25 sets of quadruplets: $A$

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2016 AMC 10B Problems/Problem 13

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