Of four numbers whose average is 60, the first is one-fourth of the sum of the last three. The first number is

### Concept & Logic When a question describes relationships among fractions of a sum, express the entire group in terms of a single variable. The total sum is easily calculated from the average and the number of items. $$ \text{Sum of Terms} = \text{Average} \times \text{Number of Terms} $$ ### Step-by-Step Solution * **Given:** * Total numbers = $4$ * Average of the four numbers = $60$ * Let the four numbers be $a, b, c$, and $d$. * Sum of all four numbers: $a + b + c + d = 60 \times 4 = 240$ * **Deduction:** * Condition: The first number is one-fourth of the sum of the last three. * $a = \frac{1}{4}(b + c + d)$ * Rearranging this gives: $b + c + d = 4a$ * **Calculation:** * We know the total sum is $a + (b + c + d) = 240$ * Substitute $(b + c + d)$ with $4a$: * $a + 4a = 240$ * $5a = 240$ * $a = \frac{240}{5} = 48$ ### Exam Strategy & Shortcut Use the "Ratio Method" for extreme speed. The ratio of the first number to the sum of the other three is $1:4$. This means the total sum is split into $1 + 4 = 5$ equal "parts". Total sum $= 60 \times 4 = 240$. Value of $1$ part (the first number) $= \frac{240}{5} = 48$. ### Common Pitfall Many students mistakenly calculate $\frac{1}{4}$ of the total average or $\frac{1}{4}$ of the total sum directly, yielding $15$ or $60$, rather than recognizing that the $1:4$ relationship applies to the parts making up the total ($5$ parts). ### Final Answer **Therefore, the correct answer is 48.**