Problem Solving

BTGModeratorVI wrote: ↑

Thu Aug 20, 2020 7:08 am

This year, a woman has a lucrative one-year position. 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. Through investments, her husband can turn each dollar she gives him into 1 + r, which will be deposited in a bank account. Their goal is to save & invest enough money so they can live off this money for two years following the end of the wife's position. 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. In terms of r, what should f be?


A. 1/(r+1)

B. 2/(r+2)

C. 2/(2r+1)

D. 2/(r+3)

E. 2/(2r+3)

Answer: D
Source: Magoosh

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

BTGModeratorVI wrote: ↑

Thu Aug 20, 2020 7:08 am

This year, a woman has a lucrative one-year position. 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. Through investments, her husband can turn each dollar she gives him into 1 + r, which will be deposited in a bank account. Their goal is to save & invest enough money so they can live off this money for two years following the end of the wife's position. 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. In terms of r, what should f be?


A. 1/(r+1)

B. 2/(r+2)

C. 2/(2r+1)

D. 2/(r+3)

E. 2/(2r+3)

Answer: D

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