When do you multiply both numerator and denominator vs numerator only?

Hello, adkor95. I think you may be getting your theory mixed up and looking to take shortcuts instead. Neither answer you calculated above is correct. You need a common denominator to add or subtract fractions, but either approach--finding a common denominator for all fractions in the expression or finding one for each of the top and bottom fractions--will lead you to the same answer if you make your conversions correctly. Consider:

a) Choose a common denominator for all fractions:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{30}{30}-\frac{1}{5}*\frac{12}{12}}{\frac{1}{3}*\frac{20}{20}-\frac{1}{4}*\frac{15}{15}}\)

\(\frac{\frac{30}{60}-\frac{12}{60}}{\frac{20}{60}-\frac{15}{60}}\)

\(\frac{\frac{18}{60}}{\frac{5}{60}}\)

\(\frac{18}{60}*\frac{60}{5}\)

\(\frac{18}{5}\)

b) Choose a common denominator for the numerator and the denominator separately:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{5}{5}-\frac{1}{5}*\frac{2}{2}}{\frac{1}{3}*\frac{4}{4}-\frac{1}{4}*\frac{3}{3}}\)

\(\frac{\frac{5}{10}-\frac{2}{10}}{\frac{4}{12}-\frac{3}{12}}\)

\(\frac{\frac{3}{10}}{\frac{1}{12}}\)

\(\frac{3}{10}*\frac{12}{1}\)

\(\frac{36}{10}\)

\(\frac{18}{5}\)

I would refer you to any post by Quant Expert Bunuel for further information about theory. (Just look in the signature for a link to the Ultimate GMAT Quant Megathread.) Good luck with your studies.

- Andrew