Among the members of the Malmo family, there are three times

This is an OVERLAPPING SETS question. The best way to approach it is with a DOUBLE-SET MATRIX:



If there are three times as many members who do not wear glasses as members who do, we can express the total who wear glasses as x and those who don't wear glasses as 3x:


Since we're given all percents and no concrete values, we can treat the total as 100%. Thus, x = 25 and 3x = 75.

Half of the Malmo family members have blue eyes --> implies that 50% do and 50% do not:


4/5 of the family members who have blue eyes do not wear glasses --> 4/5 of 50 is 40, so the difference must be 10:


What percent of the Malmo family members wear glasses and do not have blue eyes? If 10% wear glasses and have blue eyes, and 25% wear glasses total, then 15% wear glasses but do not have blue eyes:


The answer is B.

It's a pure ratio/percent problem, so we can freely make up a number, and since we're asked for a percent in the end, I'd just use 100.

Then we have:

50 have blue eyes, of whom:
40 have blue eyes and no glasses
10 have blue eyes and glasses

We also know that the ratio of glasses-wearers to non-glasses-wearers in the family as a whole is 1 to 3, so 1/4 of the whole family wears glasses. We know that in the blue eyes group, which is half the family, 20% wear glasses, so to get 25% for the whole family, it must be that 30% of the other half of the family wears glasses (using the fact that the groups are equal in size). So,

50 do not have blue eyes, of whom:
35 do not wear glasses
15 do wear glasses, which is what the question asked for

so the answer is 15%.

Overlapping Sets are a common question type on the GMAT, and the double-set matrix is often the easiest way to solve. For more on how to use the matrix, see:
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swerve wrote:Among the members of the Malmo family, there are three times as many members who do not wear glasses as members who do. Half of the Malmo family members have blue eyes, and 4/5 of the family members who have blue eyes do not wear glasses. What percent of the Malmo family members wear glasses and do not have blue eyes?

A. 10
B. 15
C. 20
D. 25
E. 35

The OA is B

Source: Economist GMAT

We can assume there are 20 members in the Malmo family, so 5 of them wear glasses and 15 don’t. Furthermore, 10 of them have blue eyes and 10 don’t. Lastly, 4/5 x 10 = 8 members who have blue eyes don’t wear glasses. That means 2 members who have blue eyes wear glasses and therefore, 3 members wear glasses and don’t have blue eyes. So the percent of the Malmo family members that wear glasses and do not have blue eyes is 3/20 x 100 = 15.

Answer: B