An older sibling is currently five times as old as a younger sibling. In five years the older will be three times as old as the younger. How old is the older sibling now?

Step-by-step solution

1. 1

   Choose the smaller quantity as the variable. Let $a$ be the younger sibling's age now. Then the older sibling is $5a$ now. Naming the younger age avoids fractions later.

2. 2

   Advance both ages by the same amount. In five years each person is exactly five years older — this is the fact the problem is really testing:

   $\text{younger}\to a+5,\qquad \text{older}\to 5a+5.$

   A common mistake is to write the older sibling's future age as $5(a+5)$, which would keep the ratio at $5$ forever.

3. 3

   Impose the future condition.

   $5a+5=3(a+5).$

4. 4

   Solve the linear equation.

   $5a+5=3a+15\ \Longrightarrow\ 2a=10\ \Longrightarrow\ a=5.$

5. 5

   Answer the question that was asked. The question wants the older sibling's present age:

   $5a=5(5)=25\ \text{years}.$

6. 6

   Check both conditions. Now: $25=5\times 5$ ✓. In five years: the older is $30$ and the younger is $10$, and $30=3\times 10$ ✓. The ratio drops from $5$ to $3$ precisely because the fixed five-year gap matters proportionally more to the younger sibling.