Problem Solving

This yr Henry will save a certain amt of his income, and he will spend the rest. Next yr Henry will have no income, but for each dollar that he saves this yr, he will have 1+r dollars available to spend. In terms of r, hat fractions of his income should Henry save this yr so that next yr the amt he was available to spend will be equal to half the amt that he spends this yr?

a) 1/(r+2)
b) 1/(2r+2)
c) 1/(3r+2)
d) 1/(r+3)
e) 1/(2r+3)

tanyajoseph wrote:This yr Henry will save a certain amt of his income, and he will spend the rest. Next yr Henry will have no income, but for each dollar that he saves this yr, he will have 1+r dollars available to spend. In terms of r, hat fractions of his income should Henry save this yr so that next yr the amt he was available to spend will be equal to half the amt that he spends this yr?

a) 1/(r+2)
b) 1/(2r+2)
c) 1/(3r+2)
d) 1/(r+3)
e) 1/(2r+3)

Ok this is really my weak point, along with so many other topics in GMAT, any way my mother finally explained it to me - hope I can solve it in the exam where it counts -

Answer should be E = 1/(2r + 3)

Here is the explaination

The questions asks in terms of r what has he saved to have half of this years spending money for next year.

Now assume his income is 1 and he spends x
so he saves 1 - x

so we have to find the value of 1 - x in terms of r

it is given next year he will have 1+r money to spend for every dollar he saves so next year he will have

(1-x) (1+r)

also this should be half of this years spending money =x/2

so

(1-x) (1+r) = x/2

= 1+r - x -xr =x/2

so x = 2 + 2r - 2x -2xr

so 3x + 2xr = 2 + 2r

so x (3+2r) = (2+2r)

so X = (2+2r) / (3 +2r)

so 1-x = 1 - (2+2r)/ (3+2r)

so 1-x (which is the ans) in terms of r = 1/(2r + 3)

Let "a" be his income this yr

let s denote his saving so a-s is the amt spend this year

now amt availabe for spending next yr is s(1+r)

according to the question this 1/2 of the amt spent this yr

hence s(1+r) = (a-s)/2
i.e 2s + 2sr = a-s
or s(3+2r) =a

s=1/(3+2r) *a

i.e s is 1/(3+2r) of a (annual income)

this is reqd fraction 1/(3+2r)

hence ans should be E

Kindly confirm