Of three numbers whose average is 40, the first number is one third 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 relationship between three numbers. You are given the average of the numbers and a condition that connects the first number with the sum of the other two. The task is to translate the words into equations and then solve for the unknown first number.

Given Data / Assumptions:

• There are three numbers, call them x, y and z.
• Their average is 40.
• The first number x is one third of the sum of the other two numbers, so x is equal to one third of (y + z).
• We need to find the value of the first number x.

Concept / Approach:
The average of three numbers is given by (x + y + z) / 3. We turn the average information into an equation for the sum x + y + z. The second condition links x directly to y + z. By substituting this relationship into the total sum, we reduce the problem to a single variable equation in x and solve it using basic algebra.

Step-by-Step Solution:
Step 1: From the average, (x + y + z) / 3 = 40, so x + y + z = 40 * 3 = 120. Step 2: From the condition, x is one third of the sum of the other two numbers, so x = (y + z) / 3. Step 3: Let S = y + z. Then x = S / 3 and x + y + z = x + S = 120. Step 4: Substitute x = S / 3 into x + S = 120 to get S / 3 + S = 120. Step 5: Combine terms: S / 3 + S = S / 3 + 3S / 3 = 4S / 3, so 4S / 3 = 120. Step 6: Multiply both sides by 3 to get 4S = 360, so S = 360 / 4 = 90. Step 7: Now x = S / 3 = 90 / 3 = 30.

Verification / Alternative check:
Take x = 30. Then y + z must be 90 to satisfy x = one third of (y + z). The total x + y + z is 30 + 90 = 120, and the average is 120 / 3 = 40, which matches the given information. Thus the value 30 is fully consistent with both conditions.

Why Other Options Are Wrong:
If x were 20, then y + z would be 60, giving a total of 80 and an average well below 40. If x were 50, then y + z would be 150, giving a total of 200 and an average far above 40. If x were 25, the equations would also fail to produce the given average of 40. Only x = 30 satisfies both the average condition and the one third relationship.

Common Pitfalls:
Some learners confuse one third of the sum with the sum being three times the first number and write incorrect equations. Others forget to multiply the average by 3 to get the total. Writing the equations carefully and introducing a helper symbol like S for y + z makes the algebra easier to manage.

Final Answer:
The first number is 30.