1. **State the problem:** We are given the ratio of the present ages of two people, $a$ and $b$, as $3:4$. Five years ago, the ratio of their ages was $2:3$. We need to find their present ages. 2. **Set up variables:** Let the present ages be $a = 3x$ and $b = 4x$ for some positive number $x$. 3. **Use the information about ages five years ago:** Five years ago, their ages were $a - 5$ and $b - 5$. The ratio then was $2:3$, so $$\frac{a - 5}{b - 5} = \frac{2}{3}$$ Substitute $a = 3x$ and $b = 4x$: $$\frac{3x - 5}{4x - 5} = \frac{2}{3}$$ 4. **Solve for $x$:** Cross-multiply: $$3(3x - 5) = 2(4x - 5)$$ $$9x - 15 = 8x - 10$$ Subtract $8x$ from both sides: $$9x - 8x - 15 = -10$$ $$x - 15 = -10$$ Add 15 to both sides: $$x = 5$$ 5. **Find present ages:** $$a = 3x = 3 \times 5 = 15$$ $$b = 4x = 4 \times 5 = 20$$ 6. **Answer:** The present ages of $a$ and $b$ are 15 and 20 years respectively.