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Fractions Problem

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Fractions Problem

by taneja.niks » Sat Apr 16, 2011 4:57 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 manpsingh87 » Sat Apr 16, 2011 5:17 am
taneja.niks 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 his income be i and the amount that he spends= s,
amount saved= i-s; next year's income=(i-s)(1+r);
as next year's income is equal to the half of the amount that he spends this year, therefore we have
(i-s)*(1+r)=s/2;
i-s+ir-sr=s/2;
i(1+r)=3s/2+sr;
i(1+r)=((3+2r)/2)s;
s={2(1+r)/(3+2r)}i
required fraction=i-s/i= i(1-{2+2r/3+2r})/i=3+2r-2r-2/3+2r=1/3+2r; hence E
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by Anurag@Gurome » Sat Apr 16, 2011 5:26 am
taneja.niks 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 Henry spends $x and he saves $y. This implies total income = $(x + y)
Then, next year Henry spends $y(1 + r) = x/2 implies x = 2y(1 + r)

We have to find y/(x + y) = y/[2y(1 + r) + y] = y/(3y + 2yr) = 1/(3 + 2r)

The correct answer is E.
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by GMATGuruNY » Sat Apr 16, 2011 5:44 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 has available to spend will be equal to half the amount that he spends this year?
Since there is a variable in the answer choices, and the question is asking only for a fraction, we can plug in our own numbers for all of the unknowns in the problem.

Let r=2.

For each dollar that Henry saves this year, he will have 1+r = 1+2 = 3 dollars available to spend next year:
Saves this year = 1, spends next year = 3.

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

What fraction of his income should Henry save this year?
Total this year = saves this year + spends this year = 1+6 = 7.
Saves/Total = 1/7. This is our target fraction.

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

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

The correct answer is E.
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