Of three numbers whose average is 22, the first number is three eighths of the sum of the other two numbers. What is the value of the first number?

Introduction / Context:
This question tests your understanding of averages and algebraic relationships among three numbers. You are given the average of the three numbers and a specific relationship between the first number and the sum of the other two numbers. By converting the average into a total sum and using the proportional relationship, you can solve for the first number using simple algebra.

Given Data / Assumptions:

• The average of three numbers is 22.
• The first number is three eighths of the sum of the second and third numbers.
• We must find the value of the first number.
• All numbers are real, and most likely integers, given the typical style of aptitude questions.

Concept / Approach:
The average of three numbers is the total sum divided by 3. So, if the average is 22, the sum of the three numbers must be 3 * 22. By assigning variables to the three numbers, we can express the given fractional relationship and the total sum as algebraic equations. Then we solve the equations to find the first number. This is a classic linear equation problem involving fractions and sums.

Step-by-Step Solution:
Step 1: Let the three numbers be a, b, and c. Step 2: The average is 22, so the sum is a + b + c = 3 * 22 = 66. Step 3: The first number a is three eighths of the sum of the other two, so a = (3 / 8) * (b + c). Step 4: From the total sum, we know that b + c = 66 - a. Step 5: Substitute b + c into the relationship for a to get a = (3 / 8) * (66 - a). Step 6: Multiply both sides by 8 to clear the denominator: 8a = 3 * (66 - a). Step 7: Expand the right side: 8a = 198 - 3a. Step 8: Add 3a to both sides to combine like terms: 11a = 198. Step 9: Solve for a: a = 198 / 11 = 18. Step 10: Therefore, the first number is 18.

Verification / Alternative check:
Verify by computing possible values of b and c. From a + b + c = 66 and a = 18, we get b + c = 48. The condition a = (3 / 8) * (b + c) becomes 18 = (3 / 8) * 48. Compute (3 / 8) * 48 = 3 * 6 = 18, which checks out perfectly. This confirms that the first number being 18 satisfies both the average and the fractional relationship.

Why Other Options Are Wrong:
If the first number were 16, then b + c would be 50, and (3 / 8) * 50 would be 18.75, which does not match 16. For 20, b + c would be 46, and (3 / 8) * 46 would be 17.25, again not equal to 20. Similar mismatches occur for 22 and 24. Only 18 satisfies the equation a = (3 / 8) * (66 - a), so only that option is correct.

Common Pitfalls:
Students sometimes confuse the phrase three eighths of the sum of the other two with three eighths of the total sum, which would lead to the wrong equation. Another common error is to mishandle the fraction when clearing denominators or adding terms, resulting in incorrect arithmetic. Writing each step clearly, especially when moving between the sum and the relationship, helps avoid such mistakes. Always verify the final answer by substituting back into the original conditions.

Final Answer:
The first number is 18.