The average age of three boys is 25 years and their ages are in the proportion 3 : 5 : 7. The age of the youngest boy is

### Concept & Ratio in Ages When ages are given in a ratio, they can be represented as multiples of a common variable. The total sum of these ages can be determined by multiplying the average age by the number of individuals. ### Step-by-Step Solution 1. Let the ages of the three boys be $3x$, $5x$, and $7x$. 2. The average age is given as 25 years. 3. Total sum of their ages = $\text{Average} \times \text{Number of boys} = 25 \times 3 = 75$ years. 4. According to the ratio representation, the sum of their ages is $3x + 5x + 7x = 15x$. 5. Equate the two sums to find $x$: $$ 15x = 75 $$ $$ x = 5 $$ 6. The question asks for the age of the youngest boy, which corresponds to the smallest ratio part ($3x$). 7. Youngest boy's age = $3x = 3(5) = 15$ years. ### Exam Strategy & Shortcut The youngest boy's age corresponds to "3" in the ratio $3:5:7$. Therefore, his age must be a multiple of 3. While all options are multiples of 3, you can find the multiplier quickly mentally: Total sum = $25 \times 3 = 75$. Sum of ratio parts = $3+5+7 = 15$. The multiplier is $75 / 15 = 5$. The youngest is $3 \times 5 = 15$. ### Common Pitfall A common mistake is directly equating the sum of the ratio parts ($15x$) to the average age (25) instead of the total age (75), which leads to an incorrect fractional multiplier. ### Final Answer Therefore, the correct answer is **15 years**.