The average of three numbers is 40. First number is `4//3` of the third number. If third number is 20 less than second number, then what is the value of second number ?

Options

A. 60
B. 50
C. 10
D. 20

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Set up the equations based on the given information.
Let the three numbers be:
- First number = \( x_1 \)
- Second number = \( x_2 \)
- Third number = \( x_3 \)

We know the average of these three numbers is 40:
\[ \frac{x_1 + x_2 + x_3}{3} = 40 \]
Multiplying both sides by 3 gives:
\[ x_1 + x_2 + x_3 = 120 \quad \text{(Equation 1)} \]

### Step 2: Express \( x_1 \) in terms of \( x_3 \).
The problem states that the first number is \( \frac{4}{3} \) of the third number:
\[ x_1 = \frac{4}{3} x_3 \quad \text{(Equation 2)} \]

### Step 3: Express \( x_3 \) in terms of \( x_2 \).
It is also given that the third number is 20 less than the second number:
\[ x_3 = x_2 - 20 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equations 2 and 3 into Equation 1.
Substituting \( x_1 \) and \( x_3 \) in Equation 1:
\[ \frac{4}{3}(x_2 - 20) + x_2 + (x_2 - 20) = 120 \]

### Step 5: Simplify the equation.
Expanding the equation:
\[ \frac{4}{3}x_2 - \frac{80}{3} + x_2 + x_2 - 20 = 120 \]
Combining like terms:
\[ \frac{4}{3}x_2 + 2x_2 - \frac{80}{3} - 20 = 120 \]
Convert -20 to a fraction:
\[ -20 = -\frac{60}{3} \]
Thus:
\[ \frac{4}{3}x_2 + 2x_2 - \frac{80}{3} - \frac{60}{3} = 120 \]
Combine the constants:
\[ \frac{4}{3}x_2 + 2x_2 - \frac{140}{3} = 120 \]

### Step 6: Clear the fraction by multiplying through by 3.
\[ 4x_2 + 6x_2 - 140 = 360 \]
Combine like terms:
\[ 10x_2 - 140 = 360 \]

### Step 7: Solve for \( x_2 \).
Adding 140 to both sides:
\[ 10x_2 = 500 \]
Dividing by 10:
\[ x_2 = 50 \]

### Conclusion
The value of the second number \( x_2 \) is **50**.

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