The sum and difference of the ages of two brothers are in the ratio 5 : 1. If the product of their ages is 96, then what are their ages?

The ratio of the sum and difference of the ages of two brothers is $5:1$.

The product of their ages is $96$.

Let the ages of the two brothers be x and y, with x > y. 

​$$\frac{x+y}{x-y} = \frac{5}{1} \implies x + y = 5(x-y)$$​​

​$$\frac{x}{y} = \frac{3}{2}$$​​

Let x = 3k and y = 2k.
x × y = 96
=> 3k × 2k = 96
=> 6k² = 96 =>
=> k = 4.
Thus, x = 3 × 4 = 12 and y = 2 × 4 = 8.
Thus, the ages of the brothers are 8 and 12 years.