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?

y=1+r-2a where y=amount saved this year and a = amount available to be spent next year.

Explanation:for this year ,let actual money spent by Henry be x and savings by Henry be y

So x+y

=1+r

Remember 1+r I the money available for Henry to spend this year.so x=1+r-y is the expression for actual money spent by Henry.

So for money available to be spend next year a ,to be half of actual spend this year

Means a=x/2=1+r-y/2 ,so

2a=1+r-y

y=1+r-2a

1/2r+3

Explanation: let i = income this year

and x1 = amount spend this year

y = amount saved this year

x2 = amount available to spend next year

Equations derived from the statement are as follows

(1) i = x1 + y

(2) x2 = y(1+r) = y + yr

(3) x2 = x1 / 2

From equation(3),

x2 = x1 / 2

x1 = 2(x2)

Substituting 2(×2) for x1 in equation (1), we get,

i = x1 + y

i = 2(x2) + y

Now we can use equation (2) and substitute into (1) for the x2 variable:

i = 2(x2) + y

i = 2(y+yr) + y

i = 2y + 2yr + y

i = 3y + 2yr

i = y(3+2r)

So in answering the question: y/i = 1/(3+2r)

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