Three numbers have an average of 40. The first number is equal to one third of the sum of the other two numbers. What is the value of the first number?

Introduction / Context:
This question involves three numbers linked by an average and a special relationship. We are told that the average of the three numbers is 40 and that the first number is equal to one third of the sum of the remaining two. The task is to find the first number. This requires setting up a pair of equations using the average and the relationship between the numbers and then solving for the unknowns.

Given Data / Assumptions:

• Let the three numbers be x, y and z.
• The average of x, y and z is 40.
• The first number x is one third of the sum of the other two: x = (y + z) / 3.
• We must find the value of x.

Concept / Approach:
From the average, we know that (x + y + z) / 3 = 40, so x + y + z = 120. From the second condition, y + z = 3x. Substituting y + z in the first equation gives x + 3x = 120, which directly yields x. This is a straightforward system of linear equations. The problem emphasizes recognizing how to combine the average equation with the relationship equation effectively.

Step-by-Step Solution:
Step 1: Convert the average into a sum equation. (x + y + z) / 3 = 40. So x + y + z = 3 * 40 = 120. Step 2: Use the relationship between the first number and the other two. x = (y + z) / 3. Multiply both sides by 3 to remove the fraction. 3x = y + z. Step 3: Substitute y + z into the sum equation. From x + y + z = 120 and y + z = 3x, we get x + 3x = 120. 4x = 120. Step 4: Solve for x. x = 120 / 4 = 30.
Verification / Alternative check:
If x = 30, then y + z = 3x = 90. Total sum x + y + z = 30 + 90 = 120, which gives an average of 120 / 3 = 40, matching the given average. Also, x is indeed one third of y + z because (y + z) / 3 = 90 / 3 = 30.
Why Other Options Are Wrong:
If x were 20, then y + z would be 60, giving a total of 80 and an average of 80 / 3, which is not 40. If x were 25, 35 or 50, similar checks show that the total and average would not satisfy both given conditions.
Common Pitfalls:
Some students mistakenly set x equal to one third of the total of all three numbers instead of one third of (y + z). Another frequent error is to forget to multiply the average by 3 when converting it into a sum equation, leading to a wrong total.
Final Answer:
The first number is 30.