Among three numbers, the second number is twice the first number and also three times the third number. If the average of the three numbers is 33, what is the value of the largest number?

Introduction / Context:
This aptitude question tests your understanding of number relationships and averages. You are given three numbers linked by ratio like conditions, and you must use the information about their average to identify the actual values. These kinds of problems are common in exams because they combine algebraic thinking with average concepts in a compact way.

Given Data / Assumptions:
- There are three numbers: first, second, and third.
- The second number is twice the first number.
- The second number is also three times the third number.
- The average of the three numbers is 33.
- All numbers are real and positive.
- The task is to find the largest among the three numbers.

Concept / Approach:
The key idea is to represent the three unknown numbers in terms of a single variable using the given relationships. Once they are expressed in terms of one variable, the formula for average allows us to form an equation. Solving that equation gives the value of the variable and therefore the actual numbers. The largest number can then be identified directly.

Step-by-Step Solution:
Step 1: Let the first number be a.Step 2: The second number is twice the first, so it is 2a.Step 3: The second number is also three times the third, so 2a = 3c, where c is the third number. This gives c = (2a / 3).Step 4: The average of the three numbers is 33. The sum of the three numbers is a + 2a + (2a / 3).Step 5: Combine like terms: a + 2a + (2a / 3) = 3a + (2a / 3) = (9a / 3) + (2a / 3) = (11a / 3).Step 6: Average = (sum of numbers) / 3, so (11a / 3) / 3 = 33. This simplifies to 11a / 9 = 33.Step 7: Multiply both sides by 9: 11a = 33 * 9 = 297.Step 8: Divide both sides by 11: a = 297 / 11 = 27.Step 9: The first number is 27, the second number is 2a = 54, and the third number is (2a / 3) = (54 / 3) = 18.Step 10: Among 27, 54, and 18, the largest number is 54.

Verification / Alternative check:
If the three numbers are 27, 54, and 18, their sum is 27 + 54 + 18 = 99. The average is 99 / 3 = 33, which matches the given average. The second number is twice the first because 54 = 2 * 27, and it is three times the third because 54 = 3 * 18. Since all given conditions are satisfied, the solution is consistent and correct.

Why Other Options Are Wrong:
Values like 36, 45, 62, and 72 do not fit the relationships among the three numbers when you try to reconstruct a compatible set. If 36 or 72 were taken as the largest number, it would not be possible to choose two other numbers so that the second is both twice the first and three times the third while keeping the average equal to 33. Only 54 fits perfectly into this structure.

Common Pitfalls:
A common error is to misinterpret the relationships and assume, for example, that the first is twice the second or that the third is three times the second. Another mistake is to confuse the formula for average and divide by the wrong number of terms. Always read relationship statements slowly and translate them carefully into algebraic equations before starting any calculation.

Final Answer:
The largest number is 54.