Of three numbers whose average is 64, the first number is equal to one-third of the sum of the other two numbers. What is the value of this first number?

Introduction / Context:
This problem tests the concept of averages and how to convert a verbal condition about three numbers into algebraic equations. Questions of this type are common in aptitude exams because they check whether the learner can translate sentences into equations and then solve them logically. Understanding how the average relates to the total sum is particularly important for many quantitative reasoning topics.

Given Data / Assumptions:

• There are three numbers, let them be a, b, and c.
• The average of the three numbers is 64.
• The first number a is equal to one-third of the sum of the other two numbers, so a = (b + c) / 3.
• All numbers are assumed to be real numbers, and we are asked to find the value of the first number a.

Concept / Approach:
The key ideas are: average = (sum of numbers) / (number of numbers), and the ability to express the verbal condition as an equation. Once we know the average, we can compute the total sum of the three numbers. Then we use the relationship between a, b, and c to form a system of equations and solve for a. Because there are two equations involving a, b, and c, we can eliminate b and c to get a single equation in a, which is easy to solve.

Step-by-Step Solution:
Step 1: Let the three numbers be a, b, and c.Step 2: The average is 64, so (a + b + c) / 3 = 64.Step 3: Multiply both sides by 3 to get the total sum: a + b + c = 64 * 3 = 192.Step 4: The condition says the first number is one-third of the sum of the other two, so a = (b + c) / 3.Step 5: Rearrange this to get b + c = 3a.Step 6: Substitute b + c = 3a into the sum equation a + b + c = 192, giving a + 3a = 192.Step 7: Simplify to get 4a = 192.Step 8: Divide both sides by 4 to obtain a = 192 / 4 = 48.

Verification / Alternative check:
We can check the result by constructing one possible set of numbers. Suppose a = 48. Then b + c = 3a = 144. One simple example is b = 70 and c = 74, whose sum is 144. Then a + b + c = 48 + 70 + 74 = 192. The average is 192 / 3 = 64, which matches the given condition. This confirms that a = 48 is consistent with the problem statement.

Why Other Options Are Wrong:
Option 32: If a = 32, then b + c = 3a = 96 and a + b + c = 128, so the average would be 128 / 3, which is not 64.
Option 72: If a = 72, then b + c = 216 and a + b + c = 288, giving an average of 96, not 64.
Option 96: If a = 96, then b + c = 288 and a + b + c = 384, giving an average of 128, again not 64.
Option 56: If a = 56, then b + c = 168 and a + b + c = 224, giving an average of 224 / 3, which is not 64.

Common Pitfalls:
A common mistake is to treat the phrase one-third of the sum of the other two numbers as a = (b + c) / 2 or a = b / 3 + c, which is incorrect. Another error is to confuse average with sum and forget to multiply the average by the number of terms. Learners might also incorrectly assume symmetry between the three numbers and think they are all equal, which violates the given condition about the first number. Careful translation of the language into equations is essential.

Final Answer:
The value of the first number is 48.