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GMAT Set 4 Q28

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by Abhijit K » Wed Feb 18, 2015 1:04 am
This year Henry will save a certain amount of his income, and he will spend the rest.
Next year Henry will have no income, but for each dollar that he saves this year, he will
have 1 + r dollars available to spend. In terms of r, what fraction of his income should
Henry save this year so that next year the amount he was available to spend will be equal
to half the amount that he spends this year?
A.1/(r+2)
B.1/(2r+2)
C.1/(3r+2)
D.1/(r+3)
E.1/(2r+3)
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by GMATGuruNY » Wed Feb 18, 2015 3:20 am
Abhijit K wrote:This year Henry will save a certain amount of his income, and he will spend the rest.
Next year Henry will have no income, but for each dollar that he saves this year, he will
have 1 + r dollars available to spend. In terms of r, what fraction of his income should
Henry save this year so that next year the amount he was available to spend will be equal
to half the amount that he spends this year?
A.1/(r+2)
B.1/(2r+2)
C.1/(3r+2)
D.1/(r+3)
E.1/(2r+3)
Let r=2.
Now let's plug in values that follow the words in the problem, step by step.

For each dollar that Henry saves this year, he will have 1+r = 1+2 = 3 dollars available to spend [next year]:
If he saves $1 this year, he can spend $3 next year.

Next year the amount he has available to spend will be equal to half the amount that he spends this year:
Since he has $3 to spend next year, he has $6 to spend this year.

What fraction of his income should Henry save this year?
Since he saves $1 and spends $6, saved/total = 1/(1+6) = 1/7. This is our target.

Now we plug r=2 into the answers to see which yields our target of 1/7.

Only E works:
1/(2r+3) = 1/(2*2 + 3) = 1/7.

The correct answer is E.
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by Brent@GMATPrepNow » Wed Feb 18, 2015 8:32 am
adthedaddy wrote:This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he has available to spend will be equal to half the amount that he spends this year?

(A)1/(r+2)
(B)1/(2r+2)
(C)1/(3r+2)
(D)1/(r+3)
(E)1/(2r+3)
Mitch's input-output approach is the best (fastest) approach, but here's an algebraic solution as well:

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Answer = E

Cheers,
Brent
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by Brent@GMATPrepNow » Wed Feb 18, 2015 8:41 am
By the way, this question is known as a "Variables in the Answer Choices" question.
The good thing about these questions is that there are typically 2 approaches you can take:
1) Algebraic
2) Input-Output

I recently wrote 2 articles on this topic:
- https://www.gmatprepnow.com/articles/var ... approaches
- https://www.gmatprepnow.com/articles/var ... 93-part-ii

Here are some more questions to practice with:
- https://www.beatthegmat.com/what-mistake ... 76293.html
- https://www.beatthegmat.com/car-dealer-s ... 74136.html
- https://www.beatthegmat.com/ps-rate-times-t276107.html
- https://www.beatthegmat.com/to-find-the- ... 73338.html
- https://www.beatthegmat.com/y-dollars-fr ... 73031.html
- https://www.beatthegmat.com/meters-in-x- ... 73854.html
- https://www.beatthegmat.com/a-better-exp ... 79396.html

Cheers,
Brent
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by Scott@TargetTestPrep » Wed Jan 03, 2018 7:15 am
Abhijit K wrote:This year Henry will save a certain amount of his income, and he will spend the rest.
Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?
A.1/(r+2)
B.1/(2r+2)
C.1/(3r+2)
D.1/(r+3)
E.1/(2r+3)
We can let s = the amount of money, in dollars, Henry saves this year and t = his income, in dollars, this year; thus, the amount that he spends this year is (t - s). Since for each dollar that he saves this year he has 1 + r dollars available to spend next year, he has s(1 + r) dollars to spend next year. Furthermore, since this amount is half what he spends this year, we have:

s(1 + r) = (1/2)(t - s)

2s(1 + r) = t - s

2s + 2sr = t - s

2sr + 3s = t

s(2r + 3) = t

s = t/(2r + 3)

s = [1/(2r + 3)] * t

So, the amount he saves this year is 1/(2r + 3) of his income.

Answer: E

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