Every student at Darcy School is in at least… | GMAT Sprint

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PS · Problem Solving

705-805

Fractions/Ratios/Decimals

Every student at Darcy School is in at least one of three clubs: horseback riding, embroidery, and country dancing, which are the only clubs in existence at the school. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3, while the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Which of the following could be the total number of students at Darcy School?

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Explanation

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Official Explanation

Official

Although the setup of this problem seems to call for a Venn Diagram with three circles, this question is actually solely about ratios. We need to figure out the ratio of students who are in only one club to the ratio of students in exactly two clubs to the ratio of students in all three clubs.

Let:

-

x

= number of students in exactly one club

-

y

= number of students in exactly two clubs

-

z

= number of students in all three clubs

We are given:

y : x = 4 : 3

y : (y + z) = 5 : 7

(since "at least two clubs" means exactly two or all three)

From the first ratio:

y = 4a

x = 3a

for some multiplier

a

.…

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What this question tests

What it tests

Your ability to convert between fractions, ratios, and decimals and to reason proportionally.

Common trap

Cross-multiplying incorrectly or mixing up the ratio parts — a part:part ratio is not a fraction of the whole.