M01-31

Official Solution:


If Sarah and Mary had no savings at the beginning of last year and their only income last year was from their salary, what was Sarah's annual salary last year?

(1) The ratio of Sarah's and Mary's annual salaries last year was 4:3. Clearly insufficient.

(2) The ratio of Sarah's and Mary's annual savings last year was 3:2, and their combined spending was $20,000. Clearly insufficient.

(1)+(2) Let S and M denote Sarah's and Mary's salaries, and s and m denote their spending. We have the following equations:

\(\frac{S}{M} = \frac{4}{3}\), which gives \(3S=4M\);

\(\frac{S-s}{M-m} = \frac{3}{2}\), which gives \(2(S - s)=3(M-m)\)

\(s+m = 20000\)

We have three distinct linear equations and four unknowns, with no additional constraints on these unknown, therefore we cannot solve for the individual unknowns and get the value of \(S\). Not sufficient.


Answer: E
_________________

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
_________________

I have an issue related to this question. I solved this question by a different method. it is clear that both statements are insufficient individually. To solve these statements together, I took the salary of Sarah and Mary as 4x and 3x and saving as 3y and 2y respectively. As combined spending is $20,000, it will be

spending of Sarah + spending of Mary= 20,000
(4x-3y) + (3x-2y)= 20,000
7x-5y=20000
When I reached this equation. I got confused as I am not sure whether the unique values of x and y are possible from this equation or not. coz if unique values are possible then the answer will be C and if not then the answer will be E. How to deal with this kind of situation as it is hard sometime to guess numbers that fit in equations?

7x-5y=20000

y = 7x/5 - 4000

Whenever x is large enough multiple of 5, y will turn out to be a positive integer. So, the above equation has more than one set of solutions.

Generally, even though there are some advanced techniques that allow to find whether a linear Diophantine equation of the form ax + by = c has a unique solution, for GMAT purposes it's better to test numbers.
_________________