Official Guide Explanation:
Problem Solving #163

Background

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Solution and Metadata

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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