My mother's age today is 3 times my own. Ten years ago she was 5 times my age. How old am I now?

Step-by-step solution

1. 1

   Choose the smaller age as the variable. Letting the smaller quantity be $x$ keeps every expression a whole number instead of a fraction. Let

   $x=\text{my age now}\quad\Longrightarrow\quad 3x=\text{my mother's age now}$

2. 2

   Age both people by the same amount. This is the step people get wrong: ten years ago each person was ten years younger, so subtract $10$ from both ages, not from the multiple:

   $x-10=\text{my age then},\qquad 3x-10=\text{my mother's age then}$

   Writing $3(x-10)$ for her past age would silently assume the ratio stayed $3$, which contradicts the problem.

3. 3

   Turn the second condition into an equation. Ten years ago her age was $5$ times mine:

   $3x-10=5(x-10)$

4. 4

   Expand and solve.

   $3x-10=5x-50$

   $50-10=5x-3x$

   $40=2x$

   $x=20$

5. 5

   Check both conditions, not just one. Today: I am $20$ and my mother is $3\cdot 20=60$, and $60=3\times 20$ $\checkmark$. Ten years ago: I was $10$ and she was $50$, and

   $50=5\times 10\quad\checkmark$

   Both statements hold, so I am $20$ and my mother is $60$.