The average of five numbers is 72. The first number is equal to one-eighth of the sum of the other four numbers. What is the value of this first number?

Introduction / Context:
This question mixes the concept of averages with a special relationship between one number and the sum of the remaining numbers. We are told the overall average, and that the first number is one-eighth of the sum of the other four numbers. Using algebra, we can derive the value of the first number.

Given Data / Assumptions:

• There are 5 numbers in total.
• The average of these 5 numbers is 72.
• The first number is one-eighth of the sum of the other four numbers.
• Let the first number be x.

Concept / Approach:
From the average, we can obtain the total sum of all five numbers. The special condition tells us how x relates to the total of the other four numbers. By expressing the sum of the other four numbers as total minus x, we can create an equation and solve for x.

Step-by-Step Solution:
Step 1: Compute total sum of the five numbers. Average = 72, number of numbers = 5. Total sum = 72 * 5 = 360. Step 2: Express the relationship using x. Let the first number be x. Sum of the other four numbers = 360 - x. Given that x is one-eighth of this sum: x = (1 / 8) * (360 - x). Step 3: Solve this equation for x. x = (360 - x) / 8. Multiply both sides by 8: 8 * x = 360 - x. 8x + x = 360. 9x = 360. x = 360 / 9 = 40.

Verification / Alternative check:
If the first number is 40, then the sum of the other four numbers is 360 - 40 = 320. One-eighth of 320 is 320 / 8 = 40, which matches x. So the relationship is satisfied. The average is (40 + sum of other four) / 5 = 360 / 5 = 72, which matches the given average. Hence x = 40 is correct.

Why Other Options Are Wrong:
If x = 60, then the sum of the other four numbers would be 360 - 60 = 300, and one-eighth of 300 is 37.5, not 60. For x = 26, one-eighth of the remaining sum does not match 26. For x = 80, the remaining sum would be 280 and one-eighth of 280 is 35, not 80. Only x = 40 satisfies both the total sum and the one-eighth relation simultaneously.

Common Pitfalls:
A common mistake is to misread the condition as the first number being one-eighth of the average or one-eighth of the total, rather than one-eighth of the sum of the other four. Another pitfall is not forming the equation correctly or forgetting to multiply out the fraction. Always translate verbal relationships into clear algebraic equations and solve them step by step.

Final Answer:
The first number is 40.