If the ratio between the ages of $P$ and $Q$ is $2 : 3$ and that between the ages of $Q$ and $R$ is $4 : 5$, then the ratio between the ages of $P$ and $R$ is

### Concept & Compounding Ratios When given sequential ratios connecting three entities (like $P$ to $Q$, then $Q$ to $R$), the ratio of the first entity to the last can be found by multiplying the fractional forms of the ratios. $$ \frac{P}{R} = \frac{P}{Q} \times \frac{Q}{R} $$ ### Step-by-Step Solution 1. Write the given ratios as fractions: $\frac{P}{Q} = \frac{2}{3}$ $\frac{Q}{R} = \frac{4}{5}$ 2. To find the ratio of $P$ to $R$, multiply the two fractions. The $Q$ variable will cancel out: $\frac{P}{R} = \left(\frac{2}{3}\right) \times \left(\frac{4}{5}\right)$ 3. Multiply the numerators and the denominators: Numerator: $2 \times 4 = 8$ Denominator: $3 \times 5 = 15$ 4. The resulting fraction is $\frac{8}{15}$. 5. Express the fraction as a ratio: $P : R = 8 : 15$ ### Exam Strategy & Shortcut Just like finding the ratio of $a:c$ from $a:b$ and $b:c$, simply multiply the numerical ratios straight across. $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$. No need to find the combined ratio $P:Q:R$. ### Common Pitfall A common error is to try and equalize the $Q$ values ($3$ and $4$) to form a combined ratio $P:Q:R = 8:12:15$ and then making an arithmetic mistake during the conversion. While correct, it introduces unnecessary steps. ### Final Answer Therefore, the correct answer is **$8 : 15$**.