What fraction of his income should Henry save this year?

Question

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. 1/(r+2)

b. 1/(2r+2)

c. 1/(3r+2)

d. 1/(r+3)

e. 1/(2r+3)

Answer 1

Call Henry's total income this year J.  

Let f be the fraction he saves.  He saves fJ, and he spends (1-f)J.  

Next year, the savings will be worth fJ(1+r).  Assuming he spends all of that, and assuming this is half of what he spend this year, we get the equation

fJ(1+r) = .5(1-f)J

                           divide both sides by J

f(1+r) = .5(1-f)

                           multiply both sides by 2

2f(1+r) = 1 - f

                           distribute on the left side

2f + 2fr = 1 - f

                           add f to both sides

3f + 2fr = 1

                           factor out f

f(3+2r) = 1

                           divide by the expression in parenthesis

f = 1/(3+2r)

That last expression is the fraction Henry needs to save this year.  It equals answer choice (e)