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# Dollars available to spend

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Manager
Joined: 06 May 2009
Posts: 70

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09 Jul 2009, 10:51
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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+1)
C 1/(3r+2)
D 1/(r+3)
E 1/(2r+3)
Current Student
Joined: 03 Aug 2006
Posts: 112
Re: Dollars available to spend [#permalink]

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09 Jul 2009, 12:06

Let
$$I$$ = Total Income
$$Sa$$ = Income Saved this year
$$I-Sa$$ = Income Spent this year

Given:
For each dollar saved this year, there will be $$1 + r$$dollars available to spend next year i.e. $$Sa(1+r)$$
Also the amount available to spend next year will be equal to half the amount spent this year

$$\frac{Sa}{I} = ?$$

$$\Rightarrow \frac{I-Sa}{2} = Sa(1+r)$$

$$\Rightarrow I-Sa = 2 \times Sa(1+r)$$

$$\Rightarrow I-Sa = Sa(2+2r)$$

$$\Rightarrow I = Sa(2+2r)+Sa$$

$$\Rightarrow I = Sa[(2+2r)+1]$$

$$\Rightarrow I = Sa(2+2r+1)$$

$$\Rightarrow I = Sa(2r+3)$$

$$\Rightarrow \frac{Sa}{I} = \frac{1}{2r+3}$$

Re: Dollars available to spend   [#permalink] 09 Jul 2009, 12:06
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