At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 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 babies were in sets of quadruplets?
- A)
- B)
- C)
- D)
- E)
Answer
D
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 described relative to the quadruplets, so let there be sets of quadruplets. Then there are sets of triplets and sets of twins.
Count babies, not sets: each set of quadruplets holds babies, each set of triplets , each set of twins . So
which gives sets of quadruplets.
The question asks for babies in sets of quadruplets: .
The answer is .
Why this works
Chains of "times as many" statements collapse into a single variable when you start from the quantity everything else is compared to. The only subtlety is the two different units in the problem: the ratios are about sets, while the total is about babies, so each set count must be multiplied by its size before adding.
The trap
Answering 25, the number of sets of quadruplets, when the question asks for the number of babies, which is 4 times that.
Common mistakes
- Answering , the number of sets of quadruplets, when the question asks for the number of babies, which is times that.
- Adding the set counts and setting that equal to , mixing sets with babies.
Techniques
Set up the equation/formula and compute; no special trick needed