The average of X, Y and Z is 24. Also, the ratio X : Y is 2 : 3 and X + Y = 60. What is the value of X − Z?

Introduction / Context:
This problem combines the idea of averages with ratios and simple algebra. You are given the average of three numbers, the ratio of two of them, and the sum of those two numbers. From this information, you must find the difference between one number and the third. Such questions strengthen skills in manipulating equations and ratios in quantitative aptitude exams.

Given Data / Assumptions:

• Average of X, Y and Z is 24.
• Therefore, X + Y + Z = 3 * 24 = 72.
• Ratio X : Y = 2 : 3.
• X + Y = 60.
• We are asked to find X − Z.

Concept / Approach:
First, we use the ratio X : Y = 2 : 3 with the information X + Y = 60 to find the individual values of X and Y. Then, using the total X + Y + Z = 72, we determine Z. Once X and Z are known, computing X − Z is straightforward.

Step-by-Step Solution:
Step 1: From X : Y = 2 : 3 and X + Y = 60, we can write X = 2k and Y = 3k.Step 2: Then X + Y = 2k + 3k = 5k = 60.Step 3: Solve for k: 5k = 60 gives k = 12.Step 4: Therefore, X = 2k = 24 and Y = 3k = 36.Step 5: From X + Y + Z = 72, we get 24 + 36 + Z = 72 so Z = 72 − 60 = 12.Step 6: Now compute X − Z = 24 − 12 = 12.

Verification / Alternative check:
Check averages: (24 + 36 + 12) / 3 = 72 / 3 = 24, which matches the given average. Check ratio: 24 : 36 simplifies to 2 : 3, matching the given ratio. All conditions are satisfied, confirming the internal consistency of the solution.

Why Other Options Are Wrong:
Options 9, 10 or 11 would correspond to different values of Z or X that would not satisfy the average or the ratio conditions simultaneously. Option 8 is even further away and would clearly break the total sum of 72 or the ratio of 2 : 3. Only 12 fits all of the given relationships.

Common Pitfalls:

• Forgetting to multiply the average by 3 to get the total X + Y + Z.
• Misapplying the ratio and not using X + Y = 60 correctly.
• Arithmetic mistakes when solving for k and then for Z.

Final Answer:
The value of X − Z is 12.