Income = Save (X) + Spend (Y)

For $1 saved Henry gets $(1+r)
So, next year he will get $(X*(1+r)) by saving $X this year

We are given that Y/2 = X(1+r)
Y = 2X(1+r)

We need to find X/(X+Y)

X / (X + 2X(1+r))
1 / (1+2+2r)
1 / (2r + 3) [option E]

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?

1/(r + 2)
1/(2r + 2)
1/(3r + 2)
1/(r + 3)
1/(2r + 3)
Explanation:

Let the total income of Henry be H and let the amount he saves be s.
⇒ Amount he spends = H − s

Given that for each dollar that he saves this year, he will have 1 + r dollars available to spend.
⇒ Amount Henry will have next year = s(1 + r)

The amount Henry has available to spend next year should be half the amount that he spends this year.
⇒ s(1 + r) = ½ × (H − s)
⇒ 2s(1 + r) = H − s
⇒ H = 2s(1 + r) + s
⇒ H = s(3 + 2r)

We have to find the fraction of his income should Henry save this year, i.e. we have to find s/H. From above equation:
s/H = 1/(3 + 2r)

Answer: E.

income - save = spent

save*(1 + r) = (1/2)*spent

save*(1 + r) = (1/2)*(income - save)

save*(1 + r) + (1/2)*save = (1/2)*income

save (1 + r + 1/2) = 1/2*income

save*(3/2 + r) = 1/2*income

save*(3 + 2r) = income

save / income = 1/(2r + 3)
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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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Bunuel do you have anymore problem links like this to practice?

Posted from my mobile device

Somewhat similar questions:
https://gmatclub.com/forum/mary-decided ... 19290.html
https://gmatclub.com/forum/a-man-saves- ... 18796.html
https://gmatclub.com/forum/last-year-ca ... 67602.html
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Assume he makes 1 dollar this year. That way his income saved will equal the fraction of his income saved.

This year income:1
Saved: S
Spent: 1-S

Next Year:
Spent: (1+R)S

"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?"

1/2(1-S)=(1+R)(S)

.5-.5S=S+SR
.5=1.5S+SR
1=3S+2SR
1=S(3+2R)
S=(1)/(3+2R)

1. Read and understand. We are looking for the fraction \((\frac{x}{I})\)saved such that the amount available to spend \((y)\) this year would be half of the disposed income\((I)\) of yesteryear.

2. Note that the amount saved can be deducted from the amount disposed of and we can then find the amount required to be saved (x), i.e \(I-y\)

Now the best approach can be either algebraic or arithmetic, I will go for a combination of both. I will start with arithmetic, i.e plugging in numbers as this can allow for a quick backsolve if running out of time.

Now to choose numbers just a cursory look at the problem would suggest that multiples of 2 and 3 would make for straightforward (I). Why we are looking for three parts in two periods( I know this may be confusing, but it's not necessary to solve the problem), however, it does help to backsolve. Then I will choose 0 or 1 for \(r\) (easy to manipulate).

Now onto the solution:

Say\(I = 6\) and\(r=0\)

Then \(x= \frac{y}{2}= \frac{(6-x)}{2}\)

\(2x= 6-x; 3x= 6; x=2\)

Our fraction is \(\frac{2}{6}= \frac{1}{3}\)

Only option \(E\) fulfills this when r= 0. Hence, \(E\)

current: income = x saving = y spending = x-y
next: income = 0, saving = 0, spending = y(1+r) (as every dollar saved, (1+r)$ will be spent)

now to determine savings/income = y/x , such that y(1+r) = (x-y)/2
=> 2y(1+r) = x - y
=> y + 2y(1+r) = x
=> y(2r+3) = x => y/x = 1/(2r+3) => (E)

Suppose he saves 50 dollars and spends 50 dollars the first year.

The next year he needs to spend 25 dollars. 50(1+r)=25

r=-.5

He spent 50 dollars and saved 50 dollars his first year so he saved 1/2 of his income. Only choice E gives 1/2 when]

we substitute r=-.5

How did you we get to adding 1 in "2 + 2r + 1 = E/S + 1" ?

Given:
1. This year Henry will save a certain amount of his income, and he will spend the rest.
2. 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.

Asked: 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?

Let the fraction of income saved be x

Income spent this year = I(1-x)
Income saved this year = Ix
Income available to spend next year = Ix(1+r) = I(1-x)/2
\(x(1+r) = \frac{(1-x)}{2}\)
\(\frac{x}{1-x} = \frac{1}{2(1+r)}\)
\(x = \frac{1}{1+2(1+r)} = \frac{1}{3+2r}\)

IMO E
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good question:
Let total income ; I
Saving = S
Expenditure = I-S

given ; 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
find 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?

we can create an eqn;
(I-S)/2 = S(1+r)
so Income ; I = 3s+2sr
fraction of saving ; s/3s+2sr ; \(\frac{1}{2r+3}\)
IMO E

My Approach:

Assume r=2 and next year spending $30.
So, this year spending $30*2 = $60 and saving $30/(1+2)= $10.
Hence, income of this year = Saving + Spending = $10 +$60 = $70.
So, the fraction of saving to income in this year = 10/70=1/7.
Now, substitute the value of r=2 to the answer choices to see which option equals to 1/7.
Easy picking! Yes, it is 'E'.
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Bunuel,
Can you please let me know how do you arrive (1-x) in the equation?

Regards.

x is the fraction of the income Henry saves (for example 3/4). So, 1 - x is the fraction of the income Henry spends (1 - 3/4 = 1/4).
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