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?
I know that two fractions are equivalent if they have the same ratio of numerator to denominator, or if I can multiply or divide both the numerator and denominator by the same factor to get from one to the other. So, if the fraction is equivalent to $\frac{5}{8},$ then there must be some factor $k$ such that the numerator is $5k$ and the denominator is $8k.$ Then, the sum of the numerator and denominator is $5k + 8k = 13k,$ which equals $91.$ Solving for $k,$ I get $k = \frac{91}{13} = 7.$ Therefore, the numerator is $5 \cdot 7 = 35$ and the denominator is $8 \cdot 7 = 56.$ The difference between them is $56 - 35 = 21.$ 