Today one brother is 8 times as old as the other. In 8 years he will be 3 times as old. How old is the older brother now?

Step-by-step solution

1. 1

   Choose the smaller quantity as the variable. Let the younger brother's present age be $x$ years. Then the older brother's present age is $8x$. Naming the smaller age avoids fractions in the setup.

2. 2

   Write both ages eight years later. Everyone ages at the same rate, so in $8$ years the younger is $x+8$ and the older is $8x+8$. Adding $8$ to $8x$, not multiplying, is the key modelling point.

3. 3

   Turn the future condition into an equation. The older will then be three times the younger: $8x+8=3(x+8)$.

4. 4

   Expand and solve. $8x+8=3x+24$, so $8x-3x=24-8$, giving $5x=16$ and $x=\frac{16}{5}$.

5. 5

   Compute the requested age. The older brother is $8x=8\cdot\frac{16}{5}=\frac{128}{5}=25\frac35$ years, i.e. $25.6$ years. The ages are not whole numbers, which is allowed - nothing in the problem requires integer ages.

6. 6

   Check the future condition exactly. In $8$ years the younger is $\frac{16}{5}+8=\frac{56}{5}$ and the older is $\frac{128}{5}+8=\frac{168}{5}$. Since $3\cdot\frac{56}{5}=\frac{168}{5}$, the ratio is exactly $3$.