Hard problem solving question from OG 12th edition

Problem Solving

OG 12th edition Q163
My students believe this is one of the hardest questions in the new edition of the Offical Guide.

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)

Students find it difficult to fathom where to start. Always focus on what you are being asked to solve. Here it is the ratio of savings to income this year. It is helpful to set up a table, inserting the given information and variables, letting i = income for this year, and s = amount saved this year. So, the amount spent this year must be i-s.

This Year Next Year
Spend i-s (1+r)s
Save s 0
Income i 0

The critical cell is Spend Next Year. We are given that for each s, Henry will have 1+r dollars available to spend (You can interpret r as the interest rate). So insert (1+r)s in the critical cell.

The last sentence of the question should be set up algebraically as:
(1+r)s = (i-s)/2
Recall that we need to solve for the ratio of saving to income, that is s/i.
Use your middle school algebra skills to solve for s/i.
2s(1+r) = i-s
2s + 2rs = i-s
3s +2rs = i
s(3+2r) = i
s/i = 1/(3+2r)

Many students will find completing this within two minutes challenging. But if you can do so, you will likely be on your way to a 98 + %ile score.