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

Introduction / Context:
This problem is similar to an earlier one with three numbers and a special relationship between them. Here the three numbers have a known average, and the first number is specified as one sixth of the sum of the other two. We must form equations from this information and solve for the first number.

Given Data / Assumptions:

• There are three numbers, call them x, y and z.
• Their average is 112.
• The first number x is one sixth of the sum of the other two numbers, so x equals one sixth of (y + z).
• We are asked to find the value of x.

Concept / Approach:
From the average we can compute the total sum of the three numbers. From the relationship x = (y + z) / 6 we can express the sum of the other two numbers in terms of x. Substituting into the total sum equation gives an equation in a single variable x that we can solve using basic algebra.

Step-by-Step Solution:
Step 1: From the average, (x + y + z) / 3 = 112, so x + y + z = 112 * 3 = 336. Step 2: From the relationship, x = (y + z) / 6. Step 3: Let S = y + z. Then x = S / 6. Step 4: Total sum x + y + z becomes x + S = 336. Step 5: Substitute x = S / 6 into x + S = 336 to get S / 6 + S = 336. Step 6: Combine terms: S / 6 + S = S / 6 + 6S / 6 = 7S / 6. Step 7: So 7S / 6 = 336. Multiply both sides by 6 to get 7S = 336 * 6 = 2016. Step 8: Now S = 2016 / 7 = 288. Step 9: Since x = S / 6, we have x = 288 / 6 = 48.

Verification / Alternative check:
If x = 48 and y + z = 288, then the total sum x + y + z is 48 + 288 = 336. Dividing by 3 gives 112, which matches the given average. Also, x is one sixth of the sum of the other two numbers, because 1 / 6 of 288 is 48. Both conditions are satisfied, confirming that the first number is 48.

Why Other Options Are Wrong:
If x were 45, then the sum of the other two numbers would be 336 - 45 = 291, and one sixth of 291 is not 45. Similar contradictions arise with 30 or 15, which would produce incorrect totals or fail the one sixth relationship. Only 48 produces both the correct total sum and the correct proportional relationship to the other two numbers.

Common Pitfalls:
Some learners misinterpret one sixth of the sum and write x = 6 * (y + z) or x = (y + z) / 3 by mistake. Others forget to multiply the average by 3 to get the total sum. Writing each step clearly, introducing S for y + z, and solving the resulting equation systematically will avoid these errors.

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