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2016 AMC 10B Problem 13

Problem 13 of 25IntermediateAlgebra

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 10001000 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 10001000 babies were in sets of quadruplets?

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Solution

Let w,r,qw,r,q be the numbers of sets of twins, triplets, and quadruplets. Then 2w+3r+4q=1000.2w+3r+4q=1000. The statement gives r=4qr=4q and w=3r=12qw=3r=12q. Substituting gives 2(12q)+3(4q)+4q2(12q)+3(4q)+4q =40q=1000=40q=1000, so q=25q=25. The number of babies in sets of quadruplets is 4q=1004q=100. Thus, the correct answer is D.
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