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This year Henry will save a certain amount of his income

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New post 12 Sep 2010, 07:21
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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. \(\frac{1}{(r+2)}\)

B. \(\frac{1}{2r+2}\)

C. \(\frac{1}{3r+2}\)

D. \(\frac{1}{r+3}\)

E. \(\frac{1}{2r+3}\)
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This year Henry will save a certain amount of his income  [#permalink]

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New post 12 Sep 2010, 09:01
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udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)


Let \(x\) be the fraction of saving, and \(I\) be the income income.

Set the equation:
\(x*I*(1+r)=\frac{(1-x)*I}{2}\), \(I\) cancels out.

Here LHS is "the amount he has available to spend next year", which according to the stem equals to RHS: "half the amount that he spends this year".

\(x=\frac{1}{3+2r}\)

Answer: E.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 12 Sep 2010, 10:40
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My approach:



Table
Total IncomeSavingSpends
XaX-a


Given: Amount available next year is 'a' times \((r+1)\) => \(a(r+1)\).
Question: 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.

Setting up the equation => \((X-a)/2 = a(r+1)\)
\(X - a = 2ar+ 2a\)
\(X = 2ar +3a\)
Fraction of the income that Henry should save this year is \((a/X)\) which is equal to \(1/(2r+3)\).
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 22 Oct 2010, 11:26
1
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)
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 27 Dec 2010, 19:46
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tonebeeze wrote:
I'm having trouble charting this problem out. Can someone help? Thanks.

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) \(1/2r +3\)


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.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 25 Apr 2012, 11:13
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udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)


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
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 26 Apr 2012, 00:18
Thanks Karishma,

I am always comfortable with number picking however in this, wasn't sure what to pick. You have explained exceptionally well.

Thanks.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 27 Jun 2012, 14:27
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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.

-----
Going over my response now, I realized that I might not follow my above approach again. :)
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 26 Sep 2012, 12:42
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.
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New post 01 Dec 2012, 20:59
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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: \(\frac{S}{S(3+2r)}=\frac{1}{3+2R}\)
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 09 Oct 2013, 00:03
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Bunuel wrote:
udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)


\(x\) fraction of saving,\(I\) income.

Set the equation:
\(x*I*(1+r)=\frac{(1-x)*I}{2}\), \(I\) cancels out.

Here LHS is "the amount he has available to spend next year", which according to the stem equals to RHS: "half the amount that he spends this year".

\(x=\frac{1}{3+2r}\)

Answe: E.


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

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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 09 Oct 2013, 02:40
honchos wrote:
Bunuel wrote:
udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)


\(x\) fraction of saving,\(I\) income.

Set the equation:
\(x*I*(1+r)=\frac{(1-x)*I}{2}\), \(I\) cancels out.

Here LHS is "the amount he has available to spend next year", which according to the stem equals to RHS: "half the amount that he spends this year".

\(x=\frac{1}{3+2r}\)

Answe: E.


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.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 09 Apr 2014, 14:23
You can pick smart numbers.
Let's first set r = .1 (10%)
Let's say he spent $110 this year. We want him to have $55 ($110/2) to spend next year.
The amount he would have to save this year would be given by x(1 + .1) = $55, solving for x gives us x = $50
Therefore his income last year was $110 + $50 = $160
The fraction of his income he saved was 50/160 = 1 / 3.2
Plug in .1 for each of the answer choices and you get E.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 10 Apr 2014, 13:16
honchos wrote:
Bunuel wrote:
udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)


\(x\) fraction of saving,\(I\) income.

Set the equation:
\(x*I*(1+r)=\frac{(1-x)*I}{2}\), \(I\) cancels out.

Here LHS is "the amount he has available to spend next year", which according to the stem equals to RHS: "half the amount that he spends this year".

\(x=\frac{1}{3+2r}\)

Answe: E.


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


Hi - probably look at the question like this that every dollar you save, your bank gives you a return of r on every dollar, so your wealth increases by a factor of (1+r)

if you are saving x part of your income I, the money will increase to xI *(1+r)
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 24 Aug 2014, 10:58
Bunuel wrote:
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.


Hi Bunuel,

The statement "for each dollar that he saves this year, he will have 1 + r dollars available to spend." -- how does this imply that it's (savings)*(1+r) vs. just "1+ savings".

I tried to plug in numbers, or even algebra for that matter and I took 1+r to be a definite. For example:

Income = 100
Savings = 10
Spent = 90

Therefore, to get 45 next year, I did 1+r=45, therefore r=44. That was obviously wrong based on your explanation above but I fail to see why we multiply savings by (1+r)?

Thanks!
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 07 Sep 2014, 00:39
udaymathapati 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. \(\frac{1}{(r+2)}\)
B. \(\frac{1}{2r+2}\)
C. \(\frac{1}{3r+2}\)
D. \(\frac{1}{r+3}\)
E. \(\frac{1}{2r+3}\)



T.I = Sp+Sa..

but for each dollar that he saves this year, he will have 1 + r dollars available to spend --> so Sa(1+r).

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-->

(T.I-Sa)/2 = Sa(1+r) => T.I - Sa = 2Sa(1+r)

T.I = 2Sa(1+r)+Sa ==> Sa(2(1+r) +1)

T.I=Sa(2r+3)

since asked for fraction, Sa/T.I = 1/ (2r+3)..

first i have tried with number plug in with 150 - T.I & saving as 50 but got an wrong answer D later with the help of Bunuel's post got an idea on this.


could someone please share similar problems for practice?
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 06 Oct 2014, 22:53
Let the total income be \(x\) and the saved income be \(y\).
That means the income spent=\(x-y\).
The question asks us to find out \(y/x\) such that \(y(1+r)=(x-y)/2\).
Solving this equation we can get \(y/x=1/(3+2r)\).
Answer E.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 17 Nov 2014, 22:20
The question asks what fraction of income, or in other words, what percent of income henry should save. That means, that income is 100% or income is equal 1.
So the fraction that henry will spend this year will be 1-x, where x is the amount he will save.
The problem says that next year henry will have 1-r for every dollar of x, which can be expressed x(1-r)--> henry's spending dollars for next year. the problem also says, that this amount has to be equal to 1/2 of (1-x).
Thus, the equation will look like this:
(1-x)/2 = x(1+r)
1-x = 2x(1+r)
1 = 2x + 2xr + x
1 = 3x + 2xr
1 = x(3+2r)
1/(3+2r)=x
Answer E
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 04 Jan 2015, 08:43
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.
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Re: This year Henry will save a certain amount of his income  [#permalink]

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New post 04 Jan 2015, 21:53
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pacifist85 wrote:
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.
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Re: This year Henry will save a certain amount of his income &nbs [#permalink] 04 Jan 2015, 21:53

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