Understanding the Problem: Sum of Three Numbers

The question asks us to find the value of the second number among three numbers. We are given the total sum of these three numbers and the relationships between them. Let's break down the given information:

• The sum of the three numbers is 252.
• The first number is thrice the second number.
• The third number is two-third of the first number.
• We need to find the value of the second number.

Setting up Equations to Solve for the Second Number

To solve this problem, we can use algebraic equations. Let's represent the three numbers using variables:

• Let the first number be \(x\).
• Let the second number be \(y\).
• Let the third number be \(z\).

Based on the problem statement, we can write the following equations:

1. Sum of the three numbers: \(x + y + z = 252\)
2. First number is thrice the second: \(x = 3y\)
3. Third number is two-third of the first: \(z = \frac{2}{3}x\)

Our goal is to find the value of \(y\).

Step-by-Step Calculation to Find the Second Number

We can use the relationships given (equations 2 and 3) to express \(x\) and \(z\) in terms of \(y\). Then, we can substitute these expressions into the sum equation (equation 1) to solve for \(y\).

Step 1: Express \(z\) in terms of \(y\)

We know that \(z = \frac{2}{3}x\) and \(x = 3y\). Substitute the expression for \(x\) into the equation for \(z\):

\(z = \frac{2}{3} \times (3y)\)

\(z = \frac{2 \times 3y}{3}\)

\(z = \frac{6y}{3}\)

\(z = 2y\)

So, the third number (\(z\)) is twice the second number (\(y\)).

Step 2: Substitute \(x\) and \(z\) into the sum equation

Now we have expressions for \(x\) and \(z\) in terms of \(y\):

• \(x = 3y\)
• \(z = 2y\)

Substitute these into the equation \(x + y + z = 252\):

\((3y) + y + (2y) = 252\)

Step 3: Solve the equation for \(y\)

Combine the terms involving \(y\):

\(3y + y + 2y = 252\)

\(6y = 252\)

Now, isolate \(y\) by dividing both sides by 6:

\(y = \frac{252}{6}\)

To perform the division:

\(252 \div 6\)

\(240 \div 6 = 40\)

\(12 \div 6 = 2\)

\(40 + 2 = 42\)

So, \(y = 42\).

The second number is 42.

Verifying the Numbers

Let's find the other two numbers using \(y = 42\) and check if their sum is 252.

• Second number (\(y\)) = 42
• First number (\(x\)) = \(3y = 3 \times 42 = 126\)
• Third number (\(z\)) = \(2y = 2 \times 42 = 84\)

Now, let's check the sum:

\(x + y + z = 126 + 42 + 84\)

\(126 + 42 = 168\)

\(168 + 84 = 252\)

The sum is indeed 252. The relationships also hold:

• First number (126) is thrice the second (42): \(126 = 3 \times 42\) (True)
• Third number (84) is two-third of the first (126): \(84 = \frac{2}{3} \times 126 = 2 \times 42\) (True)

Thus, the second number is 42.

Conclusion on Finding the Second Number

By setting up equations based on the problem description and using substitution, we found that the second number is 42. This matches one of the given options.

Revision Table: Understanding Number Problems

Additional Information: Solving Word Problems

Word problems like this require translating real-world scenarios into mathematical models (equations). Here are some general tips for solving such problems:

• Read Carefully: Understand what information is given and what is being asked.
• Assign Variables: Choose letters to represent the unknown quantities. Clearly define what each variable stands for.
• Write Equations: Translate the relationships given in the problem into mathematical equations using the variables.
• Solve the Equations: Use algebraic methods (like substitution or elimination) to find the values of the variables.
• Answer the Question: Make sure your final answer addresses what the question specifically asked for (e.g., the second number, not necessarily all three).
• Check Your Work: Substitute your answers back into the original word problem or equations to ensure everything fits.

Practicing different types of word problems involving numbers, ages, distances, etc., helps improve your skill in setting up and solving equations correctly.