This year Henry will save a certain amount of his income

Responding to a pm:

There is no 'best' way to solve a problem, in my opinion. The best way for you depends on what you are comfortable with. You can follow Bunuel's algebraic approach here (by taking x as the fraction of saving) or you can plug in values for r and check (or do something else... I would like to plug in values for r as shown in my second method)

Say r = 0
Whatever he saves this year, he has only that next year so he must save 1/3 this year (so that he spends 2/3 this year) Only options D and E give 1/3 when r = 0.

Say r = 1
Whatever he saves this year, it becomes double. This double should be half of what he spends this year. So what he spends this year should be 4 times what he saves i.e. he should save 1/5 of his income this year. Out of D and E, only E gives you 1/5

Answer E

OR, preferably, look for a value of r which gives a different answer for each option right in the beginning. I would choose r = 2.
Whatever he saves, it becomes 3 times. This 3 times amount must be half of what he spends this year. So what he spends this year must be 6 times of what he saves. Therefore, he saves 1/7 of his income. Only option E gives 1/7

Saving this year = S
Spending this year = E

Next year spend = S (1+r)

Given, S(1+r) = E/2

2 + 2r = E/S

2 + 2r + 1 = E/S + 1
3+2r = E+S/S

S/E+S = 1/(3+2r)

Let's say that his income is I. So he saves S and spends E.

I = E + S

For each dollar he saves today he'll have 1+r dollars to spend next year, so if he saves S, he'll have \(S(1+r)\) dollars to spend next year.

But its given that \(S(1+r) = \frac{E}{2}\) which means that \(2S (1+r) = E\).

Thus, \(S = \frac{E}{2(1+r)}\) and \(S+E = (E)(\frac{1}{2+2r} + 1) = (E)(\frac{3+2r}{2+2r})\)

Savings as a fraction of his income is \(\frac{S}{I} = \frac{S}{S+E} = \frac{\frac{E}{2+2r}}{\frac{E(3+2r)}{2+2r} = \frac{1}{3 + 2r}\)

Hope this is clear.

Even I went for number picking. I chose slightly different numbers and approach. I am not sure if I chose the correct path but ended up getting the same answer. It took me about 3 minutes to get there finally so I do not think my approach was the optimum one. I am going over Karishma's approach again to see if I can apply it.

Let total income = 100.

This year Henry saved : 50
This year Henry spent : 50

Next Year amount available to spend = 50(1+r) ---> (I)

Per the last statement in the question stem, this amount is half of his amount spent this year. Hence, overall amount that we should get for a value of r is (1/2)

Also we get that, therefore 50(1+r)=25. Solving for r gives r= (-1/2)

Plug in r=(-1/2) in the answer choices to see which option gives answer as (1/2). Only E gives this answer.

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Going over my response now, I realized that I might not follow my above approach again.

There is one more way to these kind of question :By using nbr.

Lets say u have saved 10$.
The next year it will be lets saey y=10(1+1)=20 [r=1]
As question suggest the spending is twice lets say Z=2*20=40
So the total income will be 40+10=50$
and 10/50=1/5=1/(2*1+3)=1/(2*r+3)..

It may looks lenghthy while explaining but the very simple..

Please give ur comments or suggestion.

Let \(I\) be the income and \(S\) be the savings this year.
Let \(0\) be the income and \(S(1+r)\) the money available for spending next year.

The money available next year is half of the money spent this year.

\(S(1+r)=\frac{I-S}{2}\)

Calculate I: \(3S + 2Sr=I\)

We are looking for S/I: \(\frac{S}{S(3+2r)}=\frac{1}{3+2R}\)

How did you make the LHS equation, whats the logic behind it?

We are told that "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". So, if he saves $10, then the amount available to spend next year is 10(1+r).

The amount saved this year = x*I, where x is the fraction of savings (say 1/2) and I is the income.
The amount available to spend next year = x*I*(1+r).

Hope it's clear.

This is what I didn't inderstand: Next year spend = S (1+r)

If someone would like to explain. Why do we multiply?

Thank you.

"but for each dollar that he saves this year, he will have 1 + r dollars available to spend."

Say, he saves 1 dollar this year; next year he has 1 + r dollars to spend.
Say, he saves 2 dollars this year; next year he has (1+r)+(1+r) = 2*(1+r) dollars to spend.
Say, he saves 3 dollars this year; next year he has (1+r)+(1+r)+(1+r) = 3*(1+r) dollars to spend.

So if he saves S dollars, next year he will have S*(1+r) dollars to spend.

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.

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

Bunuel do you have anymore problem links like this to practice?

Posted from my mobile device

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https://gmatclub.com/forum/a-man-saves- ... 18796.html
https://gmatclub.com/forum/last-year-ca ... 67602.html

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).

avigutman can you please explain this ques?

how we can reason from the stem of the question?