The average of 65 numbers is 137.8. The average of first 32 numbers is 132.6 and average of last 32 numbber is 140.5. Find the 33th numbers ?

Options

A. 215.4
B. 217.8
C. 213.5
D. 219.6

Answer

To find the 33rd number, we can follow these steps:

### Step 1: Calculate the total sum of all 65 numbers.
The average of 65 numbers is given as 137.8. The total sum can be calculated using the formula:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Items} \]
So,
\[ \text{Total Sum} = 137.8 \times 65 \]
Calculating this gives:
\[ \text{Total Sum} = 8967 \]

### Step 2: Calculate the sum of the first 32 numbers.
The average of the first 32 numbers is given as 132.6. Thus, the sum of the first 32 numbers is:
\[ \text{Sum of First 32} = 132.6 \times 32 \]
Calculating this gives:
\[ \text{Sum of First 32} = 4243.2 \]

### Step 3: Calculate the sum of the last 32 numbers.
The average of the last 32 numbers is given as 140.5. Therefore, the sum of the last 32 numbers is:
\[ \text{Sum of Last 32} = 140.5 \times 32 \]
Calculating this gives:
\[ \text{Sum of Last 32} = 4480 \]

### Step 4: Set up the equation to find the 33rd number.
Let the 33rd number be \( x \). The total sum of all 65 numbers can also be expressed as the sum of the first 32 numbers, the last 32 numbers, and the 33rd number:
\[ \text{Total Sum} = \text{Sum of First 32} + \text{Sum of Last 32} + x \]
Substituting the values we calculated:
\[ 8967 = 4243.2 + 4480 + x \]

### Step 5: Solve for \( x \).
Now, we can solve for \( x \):
\[ 8967 = 8723.2 + x \]
Subtracting 8723.2 from both sides gives:
\[ x = 8967 - 8723.2 \]
Calculating this gives:
\[ x = 243.8 \]

### Conclusion:
The 33rd number is \( 243.8 \).

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