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## Official Guide Explanation:Problem Solving #163

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Question: 163
Page: 175
Difficulty: 7 (Very Difficult)
Category 1: Algebra > Equations >
Category 2: Word Problems > Other >

Explanation: This is a complicated question. Let's dive right in.

Call Henry's income i, the amount he saves v, and the amount he spends p . Thus:

i = v + p

Next year, for every dollar he saves (v), he'll have 1 + r available to spend, or:

v(1 + r)

The question wants the amount he has available to spend (v(1 + r)) to be half the amount he spends this year (p):

v(1 + r) = (1/2)p

The question is asking what fraction of his income he should save in order for that to be true, or:

((v)/(i))?

And since we just invented i, p, and v, it needs to be in terms of r!

Multiply both sides of the second equation by 2 to isolate p:

2v(1 + r) = p

Now plug that in to the first equation so we're focusing on income:

i = v + 2v(1 + r)

i = v(1 + 2(1 + r))

i = v(3 + 2r)

((i)/(v)) = 3 + 2r

((v)/(i)) = (1/(3 + 2r))

That it! Choice (E) is correct.

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