The mean of three numbers is `50`. If two of them are `30` and `50`, what is the third number?

Options

A. `65`
B. `70`
C. `72`
D. `79`

Answer

To find the third number when the mean of three numbers is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean Formula**: The mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers. For three numbers, the formula is:
\[ \text{Mean} = \frac{x_1 + x_2 + x_3}{3} \]
where \(x_1\), \(x_2\), and \(x_3\) are the three numbers.

2. **Set Up the Equation**: We know that the mean of the three numbers is 50. Therefore, we can set up the equation:
\[ \frac{x_1 + x_2 + x_3}{3} = 50 \]

3. **Substitute Known Values**: We are given that two of the numbers are 30 and 50. Let's substitute these values into the equation:
\[ \frac{30 + 50 + x_3}{3} = 50 \]

4. **Simplify the Equation**: First, calculate the sum of the known numbers:
\[ 30 + 50 = 80 \]
Now, substitute this back into the equation:
\[ \frac{80 + x_3}{3} = 50 \]

5. **Eliminate the Denominator**: To eliminate the fraction, multiply both sides of the equation by 3:
\[ 80 + x_3 = 150 \]
(since \(3 \times 50 = 150\))

6. **Solve for the Third Number**: Now, isolate \(x_3\) by subtracting 80 from both sides:
\[ x_3 = 150 - 80 \]
\[ x_3 = 70 \]

7. **Conclusion**: Therefore, the third number is:
\[ \text{The third number is } 70. \]