The average of four numbers is 60. The first number is equal to one-fourth of the sum of the other three numbers. What is the value of the first number?

Introduction / Context:
This question involves both the concept of average and a special relationship among four numbers. You are told that their average is 60 and that the first number is one-fourth of the sum of the remaining three. Using these two conditions, you must find the value of the first number by setting up simple algebraic equations.

Given Data / Assumptions:

• There are four numbers: let them be a, b, c and d.
• The average of the four numbers is 60.
• The first number a is one-fourth of the sum of the last three numbers, i.e. a = (b + c + d) / 4.
• We need to find a.

Concept / Approach:
From the average, we find the total sum of all four numbers. Then we translate the condition relating a to b, c and d into an equation. Combining the two equations allows us to solve for a directly. The key step is to express the sum of the last three numbers in terms of a and the total.

Step-by-Step Solution:
Step 1: Average of four numbers = 60. So total sum S = 4 * 60 = 240. Step 2: Given that a = (b + c + d) / 4. This implies b + c + d = 4a. Step 3: But we know that a + b + c + d = 240. Substitute b + c + d = 4a into this: a + (4a) = 240 → 5a = 240. Step 4: Solve for a: a = 240 / 5 = 48.
Verification / Alternative check:
If a = 48, then b + c + d = 4a = 4 * 48 = 192. Total sum = a + b + c + d = 48 + 192 = 240. Average = 240 / 4 = 60, which matches the given average. Also, a is indeed one-fourth of the sum of the last three numbers, since (b + c + d) / 4 = 192 / 4 = 48 = a.
Why Other Options Are Wrong:
Option A (17), B (29), C (36), and E (60) do not satisfy both conditions simultaneously. For example, if a were 60, then b + c + d = 4 * 60 = 240, making total 300 and the average 75, not 60.
Common Pitfalls:
Some students mistakenly assume that a is directly equal to the average, which is not given. Others misinterpret the condition and write a = 4(b + c + d) instead of (b + c + d) / 4.
Final Answer:
The first number is 48.