All, this Q has been discussed many times. I'm interested in plugging numbers approach since solving this Q can result in arithmetic errors. Any takers?
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?
Well, in my opinion, it's not that hard to just solve in a direct way
Let T=total amount he earns in first year
A = proportion he saves (and this is what we wanna find)
Thus, amount he saves in first year = TA
amount he spends = T(1-A)
in second year, amount he saves will become TA(1+r)
we get equation TA(1+r) = T(1-A)/2
2TA(1+r) = T(1-A)
2A(1+r) = 1-A
2A+2Ar = 1-A
3A+2Ar = 1
A(3+2r) = 1
A = 1/(3+2r)