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 Post subject: GMAT FractionsPosted: Fri Aug 06, 2010 7:55 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
In a group of people, 3/5 have brown hair and 3/4 of the people with brown hair also have brown eyes, while only one quarter of the people who don't have brown hair have brown eyes. What is the ratio of the number of people with both brown hair and brown eyes to the number with neither?
A. 1/2
B. 9/8
C. 3/2
D. 5/2
E. 9/2

(C) 3/4 of 3/5 of the people in the group have both brown hair and brown eyes. This is (3/4) × (3/5) = 9/20 of the people.

Similarly, 2/5 of people don't have brown hair, and 3/4 of these people don't have brown eyes. So (2/5) × (3/4) = 6/20 have neither brown hair nor brown eyes.

The ratio of those that have both to those that have neither is:
= (9/20) / (6/20) = (9/20) × (20/6) = 3/2.

The correct answer is choice (C).

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 Post subject: Re: math: sets, ratioPosted: Fri Aug 06, 2010 8:54 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
When you deal with ratio you should always keep in mind what ratio it is. Is it a ratio of one set to a whole, or is it a ratio of one set to another one?

In order not to get lost you can draw a tree of sets in this case:

Here you clearly see that people who have brown hair and brown eyes make 3/4 of people who have brown hair. In its turn, people who have brown hair make 3/5 of all the people. So you can calculate that a group of people with brown hair and brown eyes make (3/4) × (3/5) = 9/20 of all the people:

So the ratio of those that have both to those that have neither is:
= (9/20) / (6/20) = (9/20) × (20/6) = 3/2.

The key for solving this question is to understand that our question breaks all the people into 4 different groups:
- brown hair & brown eyes
- brown hair & not brown eyes
- not brown hair & brown eyes
- not brown hair & not brown eyes

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