In the expression $\frac{a}{b}+\frac{c}{d}+\frac{e}{f}$ each letter is replaced by a different digit from $1,2,3,4,5$, and 6 . What is the largest possible value of this expression?
(A) $8 \frac{2}{3}$
(B) $9 \frac{5}{6}$
(C) $9 \frac{1}{3}$
(D) $9 \frac{2}{3}$
(E) $10 \frac{1}{3}$
In the expression $\frac{a}{b}+\frac{c}{d}+\frac{e}{f}$ each letter is replaced by a different digit from $1,2,3,4,5$, and 6 . What is the largest possible value of this expression?
(A) $8 \frac{2}{3}$
(B) $9 \frac{5}{6}$
(C) $9 \frac{1}{3}$
(D) $9 \frac{2}{3}$
(E) $10 \frac{1}{3}$

## Solution

To maximize the value of the given expression, we must make the individual fractions as large as possible. We do this by selecting the largest value possible for the numerator of a fraction and then the smallest possible value from the numbers remaining, as its denominator. Using this principle, the first 'fraction' would be $\frac{6}{1}$, the second $\frac{5}{2}$ and the third $\frac{4}{3}$. This gives, $6+\frac{5}{2}+\frac{4}{3}=\frac{36+15+8}{6}=\frac{59}{6}=9 \frac{5}{6}$.

ANSWER: (B)