Let T = the woman's total salary for the year

During this year, she will give a fraction f of her salary to her husband, a private investor, to invest and they will live this year on the remainder.
So, the amount SAVED (and then given to huspand)= fT

This means the amount SPENT = T - fT (total salary minus the amount saved)
In other words, T - fT = living expenses for ONE year

Through investments, her husband can turn each dollar she gives him into 1 + r, which will be deposited in a bank account.
Many will struggle converting this info to an algebraic expression.
When this happens, start by examining a few easier scenarios and then try to generalize. Here's what I mean:
If the husband is given 1 dollar, then he will turn that into 1 + r dollars
If the husband is given 2 dollars, then he will turn that into 2 + 2r dollars
If the husband is given 3 dollars, then he will turn that into 3 + 3r dollars
.
.
.

So, If the husband is given fT dollars, then he will turn that into fT + fTr dollars

They want to choose f such that the amount in the account at the end of the year is twice what they lived off this year. In terms of r, what should f be?
We want: (invested money) = 2(living expenses for 1 year)
Rewrite as: (fT + fTr) = 2(T - fT)
Expand right side: fT + fTr = 2T - 2fT
Divide both sides by T to get: f + fr = 2 - 2f
Add 2f to both sides: 3f + fr = 2
Factor right side: f(3 + r) = 2
Divide both sides by (3 + r) to get: f = 2/(3 + r)

Answer: D
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One of the ways to go about this question is assuming numbers

Let the value of f be \(\frac{1}{4}\) and Woman's dollar amount income is $10000.

Now, the woman gives $2500 to her husband and he makes that initial investment ($2500)(1 + r)
The couple will have $7500 for living this year. Since they need to choose f such that the amount
will be twice what they lived off this year, they would need to save $15000

($2500)(1 + r) = $15000 -> r = $12500/$2500 = 5.

Evaluating answer options to find out which of the answer options gives \(\frac{1}{4}\) for r=5

A. 1/(r+1) = 1/6
B. 2/(r+2) = 2/7
C. 2/(2r+1) = 2/11
D. 2/(r+3) = 2/8 = 1/4 (We have a match)

Therefore, the value of f in terms of r is (2/r+3 - Option D)

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Let S be salary, then f*S was given to husband. They lived off of (1-f)*S

After investments, the account had f*S*(1+r)

Hence, as per question,

f*S*(1+r) = 2*(1-f)*S

=> f + f*r = 2 - 2*f
=> f(1+r+2) = 2

f = 2/(r+3)

Option D

Dear GMATPrepNow

I feel a bit confused.
Does the highlighted part in prompt have any use in formulating the equations?
Thanks

That highlighted part tells us that the money they SAVE must be enough to last them for two years.
In other words, the amount they SAVE must be equal to two times the amount of money they SPEND in one year

Does that help?

Cheers,
Brent
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Dear Brent,
Thanks for your quick response. Let me open the discussion a bit.

Is there any connection of they can live off this money for two years and Toward this end, they want to choose f such that the amount in the account at the end of the year is twice what they lived off this year?? i.e is it mandatory to be twice because they an it for 2 years?

Could be the case that Toward this end, they want to choose f such that the amount in the account at the end of the year is three times what they lived off this year but Their goal is to save & invest enough money so they can live off this money for two years? In this case, on which statment should I depend to interpret it mathematically?

I hope my questions are clear to you

Thanks in advance

Great questions/points!!

There are some necessary assumptions here.
For example, we must assume that the couple wants to save exactly 2 years worth of money, otherwise the couple could save (as you suggest) 3 years worth of money, or 10 years worth, etc, which would mean there could be several correct answers.
Also, the question does not state that the couple's expenses for the next 2 years would be the SAME as the 1 year in which the wife works. Otherwise, there could be more than 1 correct answers.

Since the test-makers take every measure to avoid such ambiguity, you will not encounter this kind of question on test day.

Cheers,
Brent
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Thanks Brent for you help. However, I have another thought or assumption.

The most important condition in the prompt is:Toward this end, they want to choose f such that the amount in the account at the end of the year is twice what they lived off this year. If you do not have this statement: they can live off this money for two years. Would this affect you calculation or mathematical interpretation?? I do not think at all. The couple wants to save over the year twice what they spend. Does it matter if they spend this amount over 2, 6, 10 years. This only will affect the annual spending that we can assume it is equal for sake of simplicity.

What I want to say is that this is redudant statement: The statement is useless: they can live off this money for two years.

Please correct me if I go wrong.

Thanks in advance

Arrgg! I don't know how I didn't see that ("Toward this end, they want to choose f such that the amount in the account at the end of the year is twice what they lived off this year") each time I re-read it.

On a somewhat related note, I saw this post on Reddit's Showerthoughts yesterday "You will always this read wrong" and it took me at least 7 re-reads to get it.


Now that I've read the question AGAIN, I think the original wording is fine. They want to save exactly twice as much as they spent this year.

Cheers,
Brent
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So do you agree with me about that the other statement "so they can live off this money for two years following the end of the wife's position" is redundant? or is it a must with other statement ("Toward this end, they want to choose f such that the amount in the account at the end of the year is twice what they lived off this year") to solve the question?

You actually missed my real question

Ha - you're right!!!

Yes, that first piece of info is somewhat redundant.

Cheers,
Brent
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I dont understand if we take Salary = 1,000, f = 500(1/2), then we will get r = 1. In that case both option A and D is right if we proceed in your way. Really confused..

Solution:

Let the wife’s salary be x. Then, she gives xf to her husband and they live on x(1 - f) for the year. The husband will turn xf dollars into xf(1 + r) dollars. We want xf(1 + r) to be twice x(1 - f); thus:

xf(1 + r) = 2x(1 - f)

f(1 + r) = 2(1 - f)

f + rf = 2 - 2f

3f + rf = 2

f(3 + r) = 2

f = 2/(3 + r)

So, in terms of r, f should be chosen as 2/(r + 3).

Answer: D
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