2016 AMC 10B Problem 13

Question

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 were three times as many sets of twins as sets of triplets. How many of these

1000

babies were in sets of quadruplets?

\ 25

\ 40

\ 64

\ 100

\ 160

Answer: D

Concepts:system of equations

Difficulty rating: 1140

Small hint:

Let

w,r,q

be the numbers of twin, triplet, and quadruplet sets

Big hint:

Use

r=4q

and

w=3r

Solution:

Let

w,r,q

be the numbers of sets of twins, triplets, and quadruplets. Then

2w+3r+4q=1000.

The statement gives

r=4q

and

w=3r=12q

.

Substituting gives

2(12q)+3(4q)+4q

=40q=1000

, so

q=25

. The number of babies in sets of quadruplets is

4q=100

.

Thus, the correct answer is D.