Out of three numbers, the first is twice the second and is half of the third. If the average of these three numbers is 56, what is the sum of the highest and the lowest numbers?

Introduction / Context:
This question combines relationships between three unknown numbers with the concept of averages. The information about how the first number is related to the second and third allows us to express all three in terms of a single variable. The given average then helps us determine the exact values and finally compute the required sum of the largest and smallest numbers.

Given Data / Assumptions:

There are three numbers: first, second and third.
The first number is twice the second number.
The first number is half of the third number.
The average of the three numbers is 56.
We must find the sum of the highest and the lowest of these three numbers.

Concept / Approach:
We use algebra to express all three numbers in terms of one variable. If we let the second number be x, then the first number is 2x. Since the first number is half of the third, the third number is 4x. With these expressions we can compute the average in terms of x, set it equal to 56 and solve for x. Once we know the actual values of the three numbers, we identify the smallest and largest and add them together to get the final answer.

Step-by-Step Solution:
Step 1: Represent the numbers using one variable. Let the second number be x. Then the first number = 2x. Since the first number is half of the third, the third number = 4x. Step 2: Use the average condition. Average of the three numbers = (x + 2x + 4x) / 3 = 56. Simplify the numerator: x + 2x + 4x = 7x. So, 7x / 3 = 56. Step 3: Solve for x. Multiply both sides by 3: 7x = 56 * 3 = 168. Therefore, x = 168 / 7 = 24. Step 4: Find the three numbers. Second number = x = 24. First number = 2x = 48. Third number = 4x = 96. Step 5: Identify highest and lowest and find their sum. Lowest number = 24, highest number = 96. Required sum = 24 + 96 = 120.

Verification / Alternative Check:
We can verify by checking the average of 24, 48 and 96. Their sum is 24 + 48 + 96 = 168. Dividing by 3 gives 168 / 3 = 56, which matches the given average. The relationships are also satisfied: first number 48 is indeed twice the second number 24 and is half of the third number 96, confirming that our values are consistent.

Why Other Options Are Wrong:
Option 160 is greater than the sum of any two numbers from the derived set, so it cannot be correct.
Option 80 would be 24 plus 56, but 56 is not one of the three numbers, so it does not match the actual numbers.
Option 60 is 24 plus 36, but 36 is also not one of the actual numbers obtained from the average condition.

Common Pitfalls:
Students sometimes misinterpret the relationships and assign wrong expressions, for example making the third number 2x instead of 4x, which breaks the condition that the first is half of the third. Some also forget to divide by 3 when using the average or incorrectly simplify 7x / 3 = 56, leading to wrong values of x and consequently wrong final answers.

Final Answer:
The sum of the highest and lowest numbers is 120.