A fraction is equivalent to $\frac{5}{8}$. Its denominator and numerator add up to 91 . What is the difference between the denominator and numerator of this fraction?
(A) 21
(B) 3
(C) 33
(D) 13
(E) 19
Solution 1

We obtain fractions equivalent to $\frac{5}{8}$ by multiplying the numerator and denominator by the same number.

The sum of the numerator and denominator of $\frac{5}{8}$ is 13 , so when we multiply the numerator and denominator by the same number, the sum of the numerator and denominator is also multiplied by this same number.

Since $91=13 \times 7$, then we should multiply the numerator and denominator both by 7 to get a fraction $\frac{5 \times 7}{8 \times 7}=\frac{35}{56}$ equivalent to $\frac{5}{8}$ whose numerator and denominator add up to 91 .

The difference between the denominator and numerator in this fraction is $56-35=21$.

## Solution 2

We make a list of the fractions equivalent to $\frac{5}{8}$ by multiplying the numerator and denominator by the same number, namely $2,3,4$, and so on:

$$
\frac{5}{8}, \frac{10}{16}, \frac{15}{24}, \frac{20}{32}, \frac{25}{40}, \frac{30}{48}, \frac{35}{56}, \ldots
$$

Since the numerator and denominator of $\frac{35}{56}$ add to 91 (since $35+56=91$ ), then this is the fraction for which we are looking.

The difference between the denominator and numerator is $56-35=21$.

## Solution 3

We obtain fractions equivalent to $\frac{5}{8}$ by multiplying the numerator and denominator by the same number.

If this number is $n$, then a fraction equivalent to $\frac{5}{8}$ is $\frac{5 n}{8 n}$.

For the numerator and denominator to add up to 91 , we must have $5 n+8 n=91$ or $13 n=91$ or $n=7$.

Therefore, the fraction for which we are looking is $\frac{5 \times 7}{8 \times 7}=\frac{35}{56}$.

The difference between the denominator and numerator is $56-35=21$.

ANSWER: (A)

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