Two numbers are in the ratio 5 : 8. When 9 is added to each of them, the ratio becomes 2 : 3. Find the smaller number.

Let the numbers be $5x$ and $8x$. Then $\frac{5x+9}{8x+9}=\frac{2}{3}$ → $3(5x+9)=2(8x+9)$ → $15x+27=16x+18$ → $x=9$, so the smaller number is $5x=45$. Check: 54 : 81 = 2 : 3. The trap 9 is the value of $x$ (not a number in the ratio) and 72 is the larger number.