The average of 61 numbers is 54 .The average of first 31 numbers is 50.6 and that of last 31 numbers is 60.4 . If the `31^(st)` number is excluded ,then what is the average of the remaining numbers ?

Options

A. 53.55
B. 52.45
C. 51.65
D. 51 . 75

Answer

To solve the problem step by step, we will calculate the necessary sums and averages based on the information provided.

### Step 1: Calculate the sum of all 61 numbers.
The average of 61 numbers is given as 54. The sum can be calculated using the formula:
\[ \text{Sum} = \text{Average} \times \text{Total Numbers} \]
So,
\[ \text{Sum of 61 numbers} = 54 \times 61 = 3294 \]

### Step 2: Calculate the sum of the first 31 numbers.
The average of the first 31 numbers is given as 50.6. Thus, the sum is:
\[ \text{Sum of first 31 numbers} = 50.6 \times 31 = 1568.6 \]

### Step 3: Calculate the sum of the last 31 numbers.
The average of the last 31 numbers is given as 60.4. Thus, the sum is:
\[ \text{Sum of last 31 numbers} = 60.4 \times 31 = 1872.4 \]

### Step 4: Identify the 31st number.
The 31st number is included in both the first and the last 31 numbers. To find the 31st number, we can use the total sum of the first 31 numbers and the total sum of the last 31 numbers:
\[ \text{Sum of 31st number} = \text{Sum of first 31 numbers} + \text{Sum of last 31 numbers} - \text{Sum of 61 numbers} \]
Substituting the values:
\[ \text{31st number} = 1568.6 + 1872.4 - 3294 = 146 \]

### Step 5: Calculate the sum of the remaining numbers after excluding the 31st number.
To find the sum of the remaining numbers, we subtract the 31st number from the total sum of 61 numbers:
\[ \text{Sum of remaining numbers} = \text{Sum of 61 numbers} - \text{31st number} = 3294 - 146 = 3148 \]

### Step 6: Calculate the average of the remaining numbers.
After excluding the 31st number, we have 60 numbers left. The average can be calculated as:
\[ \text{Average of remaining numbers} = \frac{\text{Sum of remaining numbers}}{\text{Total remaining numbers}} = \frac{3148}{60} \approx 52.47 \]

### Conclusion
The average of the remaining numbers after excluding the 31st number is approximately 52.47.