The sum of three numbers is 252. If the first number is thrice the second and third number is two-third of the first, then the second number is

Options

A. 41
B. 21
C. 42
D. 84

Answer

To solve the problem step by step, let's denote the three numbers as follows:

- Let the second number be \( y \).
- The first number, which is thrice the second number, will be \( x = 3y \).
- The third number, which is two-thirds of the first number, will be \( z = \frac{2}{3}x \).

We know from the problem statement that the sum of these three numbers is 252:

\[ x + y + z = 252 \]

Now, substituting the expressions for \( x \) and \( z \) in terms of \( y \):

1. Substitute \( x = 3y \) into the equation:
\[ 3y + y + \frac{2}{3}(3y) = 252 \]

2. Simplify the equation:
\[ 3y + y + 2y = 252 \]
\[ 6y = 252 \]

3. Solve for \( y \):
\[ y = \frac{252}{6} = 42 \]

Thus, the second number is \( 42 \).

### Summary of Steps:
1. Define the variables for the three numbers based on the relationships given.
2. Set up the equation representing the sum of the three numbers.
3. Substitute the expressions for the first and third numbers into the sum equation.
4. Simplify the equation to isolate \( y \).
5. Solve for \( y \) to find the value of the second number.