Three numbers are in the ratio $4 : 5 : 6$ and their average is $25$. The largest number is

### Concept & Formula When numbers are given in a ratio, we can represent them as multiples of a common variable (e.g., $4x, 5x, 6x$). The average is the sum of the items divided by the number of items. $$ \text{Average} = \frac{\text{Sum of Terms}}{\text{Number of Terms}} $$ ### Step-by-Step Solution * **Given:** The ratio of the three numbers is $4:5:6$. Let the numbers be $4x$, $5x$, and $6x$. The average of these three numbers is $25$. * **Calculation:** Set up the average formula: $$ \frac{4x + 5x + 6x}{3} = 25 $$ Sum the terms in the numerator: $$ \frac{15x}{3} = 25 $$ Simplify the fraction: $$ 5x = 25 $$ Solve for $x$: $$ x = \frac{25}{5} = 5 $$ The problem asks for the largest number. Based on the ratio ($4x, 5x, 6x$), the largest number is $6x$. Largest number $= 6 \times 5 = 30$. ### Exam Strategy & Shortcut Notice that the numbers are in an arithmetic progression (ratio $4:5:6$ implies the terms are evenly spaced). For an odd number of evenly spaced terms, the **average is always the middle term**. Therefore, the middle term is exactly $25$. Since the ratio is $4:5:6$, and the "5" part represents $25$, the multiplier is $5$. The largest part is "6". So, $6 \times 5 = 30$. You can solve this entirely in your head in seconds. ### Common Pitfall A frequent trap is calculating $x = 5$ and then mistakenly selecting an option if $5$ were present, forgetting to multiply by $6$ to find the *largest* number. Always reread what the question specifically demands. ### Final Answer **Therefore, the correct answer is 30.**