If $\frac{2}{3}, \frac{23}{30}, \frac{9}{10}, \frac{11}{15}$, and $\frac{4}{5}$ are written from smallest to largest then the middle fraction will be
(A) $\frac{23}{30}$
(B) $\frac{4}{5}$
(C) $\frac{2}{3}$
(D) $\frac{9}{10}$
(E) $\frac{11}{15}$

## Solution

In order to compare the five fractions, we write them with the same denominator and then compare numerators. Since the lowest common multiple of $3,30,10,15$, and 5 is 30 , we must change each of the fractions into its equivalent form with denominator 30 . The fractions are shown in the following table:

| Given Fraction |
| :---: |
| $\frac{2}{3}$ |
| $\frac{23}{30}$ |
| $\frac{9}{10}$ |
| $\frac{11}{15}$ |
| $\frac{4}{5}$ |

$$
\begin{gathered}
\text { Equivalent Form } \\
\frac{2}{3} \times \frac{10}{10}=\frac{20}{30} \\
\text { Unchanged } \\
\frac{9}{10} \times \frac{3}{3}=\frac{27}{30} \\
\frac{11}{15} \times \frac{2}{2}=\frac{22}{30} \\
\frac{4}{5} \times \frac{6}{6}=\frac{24}{30}
\end{gathered}
$$

In arranging the fractions in the required order they become: $\frac{20}{30}, \frac{22}{30}, \frac{23}{30}, \frac{24}{30}$, and $\frac{27}{30}$.

The middle fraction is $\frac{23}{30}$.
If $\frac{2}{3}, \frac{23}{30}, \frac{9}{10}, \frac{11}{15}$, and $\frac{4}{5}$ are written from smallest to largest then the middle fraction will be
(A) $\frac{23}{30}$
(B) $\frac{4}{5}$
(C) $\frac{2}{3}$
(D) $\frac{9}{10}$
(E) $\frac{11}{15}$