M01, #31

michaeldavis,

I was wondering the same thing. I believe the correct answer should be C.

since we get a concrete number for the amount of spending between the two people, we can set up 4x+3x-20000=5x... and we get the unknown multiplier of 10000 and figure out the annual income of both people.

Can anyone confirm that the answer should be C?

I'd say:
Income ratio: 4x/3x ::: total income 7x
Saving ratio: 3x/2x ::: total savings 5x
Total expenditure is 20,000

Total income - total savings = total expenditure
7x - 5x = 20,000 :::: x=10,000
Sara's income: 10,000x4= 40,000

Hi yoonsiklim1, nicasc

The problem is that you are both assuming that the quantity by which each part of the ratio is "scaled" is the same. That is not correct. In doing so, you are using an additional constraint not described in the original problem. Look at the following example.

Sara's income (S) could be 40000, but could also be 20000. If S = 20000, according to the first statement, Mary's income (M) = 3*S/4. Then, if S = 20000, M = 15000.

\(\frac{S}{M} = \frac{20000}{15000} = \frac{4}{3}*\frac{5000}{5000}\)

where that 5000 is the number you called x.

In this case, the spending of Sara (z) is 11000 and the spending of Mary (y) is 9000 (you can get these numbers using all the information given by the statements). Therefore,

\(\frac{(S - z)}{(M - y)} = \frac{(20000 - 11000)}{(15000 - 9000)} = \frac{9000}{6000} = \frac{3}{2} * \frac{3000}{3000}\)

where that 3000 is ALSO the number you called x.

The red numbers are the quantities by which each part of the ratio is multiplied in order to obtain the actual incomes, spendings, and savings (Manhattan GMAT defines this quantity as the "Unknown Multiplier"). You can see that they are different. It happens, however, that in the case of S = 40000 and M = 30000 (the one you described) these quantities (your x´s) are both equal to 10000.