The mean of 20 observations is 19. One more observation is included and the new mean becomes 20. The `21^(st)` observation is:

Options

A. A. 20
B. B. 30
C. C. 40
D. D. 50

Answer

To find the 21st observation given that the mean of 20 observations is 19 and the new mean after including one more observation becomes 20, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Sum of the First 20 Observations:**
The mean of the first 20 observations is given as 19. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Therefore, the sum of the first 20 observations can be calculated as:
\[ \text{Sum of 20 observations} = \text{Mean} \times \text{Number of observations} = 19 \times 20 = 380 \]

2. **Calculate the Sum of the New 21 Observations:**
After including the 21st observation, the new mean becomes 20. The sum of the 21 observations can be calculated as:
\[ \text{Sum of 21 observations} = \text{New Mean} \times \text{New Number of observations} = 20 \times 21 = 420 \]

3. **Determine the 21st Observation:**
Let the 21st observation be \( x \). The sum of the 21 observations is equal to the sum of the first 20 observations plus the 21st observation:
\[ \text{Sum of 21 observations} = \text{Sum of 20 observations} + x \]
Substituting the values we have:
\[ 420 = 380 + x \]
To find \( x \), we rearrange the equation:
\[ x = 420 - 380 = 40 \]

4. **Conclusion:**
Therefore, the 21st observation is \( 40 \).

### Final Answer:
The 21st observation is **40**.

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