The sum of three positive numbers is 252. The first number is three times the second number, and the third number is two third of the first number. What is the value of the second number?

Introduction / Context:
This is a typical linear equation problem involving three related numbers. The question tests the ability to translate proportional relationships such as first is three times the second and third is two third of the first into algebraic equations. Once the equations are written correctly, solving for the unknown numbers is straightforward. Such questions are frequently asked in aptitude tests to examine comfort with basic algebra and proportional reasoning.

Given Data / Assumptions:

• There are three positive numbers: first, second, and third.
• The sum of the three numbers is 252.
• The first number is three times the second number.
• The third number is two third of the first number.
• All numbers are real and positive, and we need the exact value of the second number.

Concept / Approach:
The standard method is to denote the second number by a variable and express the other two numbers in terms of this variable. Then we use the given total sum to form a single linear equation in one variable. Solving this equation gives the value of the second number. This approach ensures there is no confusion between the roles of the three numbers and keeps the algebra neat and systematic.

Step-by-Step Solution:
Let the second number be x.Then the first number, which is three times the second, is 3x.The third number is two third of the first, so it is (2 / 3) * 3x = 2x.Now write the sum condition: first + second + third = 252.So 3x + x + 2x = 252.Combine like terms: 3x + x + 2x = 6x.Thus 6x = 252.Solve for x: x = 252 / 6 = 42.Therefore, the second number is 42.

Verification / Alternative check:
Second number = 42.First number = 3 * 42 = 126.Third number = 2 / 3 of 126 = 84.Check the sum: 126 + 42 + 84 = 252, which matches the given total.All conditions are satisfied, confirming that the second number is 42.

Why Other Options Are Wrong:
21 and 41 do not satisfy the total sum when the corresponding first and third values are computed as per the given ratios.84 is actually equal to the third number in the correct solution, not the second number.Only 42 gives a consistent set of three numbers whose sum is 252 and which obey the given relationships.

Common Pitfalls:
Learners sometimes assign the variable to the wrong number and then mix the relationships, which leads to an incorrect equation.Another common error is misinterpreting two third of the first as first minus two third or some other incorrect expression.Some candidates forget to verify the final numbers by substituting back into the original conditions.

Final Answer:
The value of the second number is 42.