Which of these values is the largest?
(A) $\frac{4}{2-\frac{1}{4}}$
(B) $\frac{4}{2+\frac{1}{4}}$
(C) $\frac{4}{2-\frac{1}{3}}$
(D) $\frac{4}{2+\frac{1}{3}}$
(E) $\frac{4}{2-\frac{1}{2}}$
Solution 1

Since the numerators of the five fractions are the same, the largest fraction will be the one with the smallest denominator.

To get the smallest denominator, we must subtract (not add) the largest quantity from 2.

Therefore, the largest fraction is $\frac{4}{2-\frac{1}{2}}$.

## Solution 2

We evaluate each of the choices:

$$
\begin{aligned}
\frac{4}{2-\frac{1}{4}} & =\frac{4}{\frac{7}{4}}=\frac{16}{7} \approx 2.29 \\
\frac{4}{2+\frac{1}{4}} & =\frac{4}{\frac{9}{4}}=\frac{16}{9} \approx 1.78 \\
\frac{4}{2-\frac{1}{3}} & =\frac{4}{\frac{5}{3}}=\frac{12}{5}=2.4 \\
\frac{4}{2+\frac{1}{3}} & =\frac{4}{\frac{7}{3}}=\frac{12}{7} \approx 1.71 \\
\frac{4}{2-\frac{1}{2}} & =\frac{4}{\frac{3}{2}}=\frac{8}{3} \approx 2.67
\end{aligned}
$$

The largest of the five fractions is $\frac{4}{2-\frac{1}{2}}$.

ANSWER: (E)