Three numbers have an average of 112. The first number is equal to one sixth of the sum of the other two numbers. What is the value of the first number?

Introduction / Context:
This question combines averages with a simple algebraic condition relating three numbers. You are given the overall average of the three numbers and a relationship between the first number and the sum of the other two numbers. Using algebra, you can determine the actual value of the first number.

Given Data / Assumptions:
• There are three numbers.• Average of the three numbers = 112.• First number is equal to one sixth of the sum of the other two numbers.• All numbers are real numbers, typically expected to be positive in such aptitude questions.

Concept / Approach:
The average of three numbers equals their sum divided by three. Thus, knowing the average gives us their total. Let us denote the first number as a and the other two numbers as b and c. The relation a = (b + c) / 6 provides a second equation. Combining both expressions, we can solve for a directly using substitution.

Step-by-Step Solution:
Step 1: Let the three numbers be a, b, and c.Step 2: Given that their average is 112, so (a + b + c) / 3 = 112.Step 3: Multiply both sides by 3: a + b + c = 336.Step 4: The first number a is one sixth of the sum of the other two: a = (b + c) / 6.Step 5: From this, b + c = 6a.Step 6: Substitute b + c = 6a into the total sum: a + 6a = 336.Step 7: This gives 7a = 336.Step 8: Therefore, a = 336 / 7 = 48.

Verification / Alternative check:
Once we have a = 48, we can compute b + c = 6a = 6 * 48 = 288. Then a + b + c = 48 + 288 = 336, and the average is 336 / 3 = 112, matching the given average. Also, a is indeed one sixth of the sum of the other two, since 48 = 288 / 6. Everything is consistent, confirming the correctness of the solution.

Why Other Options Are Wrong:
45, 30, 15: None of these values satisfy both conditions simultaneously. If you plug any of them into a = (b + c) / 6 and a + b + c = 336, you will not get consistent values for b and c.

Common Pitfalls:
Some students misinterpret the condition and write 6a = b + c incorrectly or forget to multiply the average by the number of values to get the total sum. Others may confuse average with one of the numbers. Correctly setting up the algebraic equations is the key step here.

Final Answer:
The first number is 48.