The average of five numbers is 76. The first number is equal to 3/7 of the sum of the other four numbers. What is the value of the first number?

Introduction / Context:
This question combines the concept of average with a proportional relationship between one term and the sum of the remaining terms. You are told that the average of five numbers is 76 and that the first number is three sevenths of the sum of the other four. From this information, you must find the value of the first number. Such problems are very common in aptitude tests and help reinforce algebraic thinking with averages.

Given Data / Assumptions:

- There are five numbers in total. - Their average is 76. - Let the first number be x. - The sum of the other four numbers is 3/7 related to x: specifically, x = (3 / 7) * (sum of the other four). - We need to determine the value of x.

Concept / Approach:
The average of five numbers being 76 means their total sum is 5 * 76. If we call the first number x, then the sum of the remaining four numbers is the total minus x. The condition that the first number is three sevenths of the sum of the other four gives a direct equation involving x and the remaining sum. We set up this equation, substitute the total sum in terms of the average and then solve for x. This approach uses only basic algebra and the definition of average.

Step-by-Step Solution:
Step 1: Let the five numbers be x, a, b, c and d, where x is the first number. Step 2: Their average is 76, so the total sum is 5 * 76 = 380. Step 3: The sum of the other four numbers is a + b + c + d = 380 - x. Step 4: The condition states that the first number x is three sevenths of the sum of the other four numbers: x = (3 / 7) * (380 - x). Step 5: Multiply both sides by 7 to clear the denominator: 7x = 3 * (380 - x). Step 6: Expand the right side: 7x = 1140 - 3x. Step 7: Bring all x terms to one side: 7x + 3x = 1140, so 10x = 1140. Step 8: Solve for x by dividing both sides by 10: x = 114. Step 9: Therefore, the first number is 114.

Verification / Alternative check:
Check the relationship using the calculated value x = 114. The sum of the other four numbers is 380 - 114 = 266. Compute (3 / 7) of 266: 266 / 7 = 38, and 3 * 38 = 114. This matches x, so the proportional condition is satisfied. The average also remains 76 because the total sum is still 380. These checks confirm that 114 is correct.

Why Other Options Are Wrong:
If the first number were 171, the sum of the remaining four would be 380 - 171 = 209, and 3/7 of 209 is not 171. Similar mismatches occur for 76 and 228. None of these values produce a consistent relationship between the first number and the sum of the other four, nor do they maintain the average of 76 in the correct way. Only 114 satisfies both conditions simultaneously.

Common Pitfalls:
One common mistake is to misread the condition and assume the first number is 3/7 of the total sum instead of the sum of the other four. Another error is to forget that the sum of the other four is total minus x, leading to an incorrect equation. Carefully interpreting the phrase and writing each step symbolically helps avoid these misunderstandings.

Final Answer:
The value of the first number is 114, which corresponds to option B.