The average of 83 results is 48.5. If the average of the first 59 results is 45.5 and that of the last 23 is 53.5. Find the 60th result.

Options

A. 110.5
B. 111.5
C. 108.75
D. 109.5

Answer

To find the 60th result given the average of 83 results, we can follow these steps:

### Step 1: Calculate the total sum of all 83 results.
The average of 83 results is given as 48.5. The total sum can be calculated using the formula:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Results} \]
\[ \text{Total Sum} = 48.5 \times 83 = 4025.5 \]

### Step 2: Calculate the total sum of the first 59 results.
The average of the first 59 results is given as 45.5. Using the same formula:
\[ \text{Sum of First 59 Results} = 45.5 \times 59 = 2684.5 \]

### Step 3: Calculate the total sum of the last 23 results.
The average of the last 23 results is given as 53.5. Again, using the formula:
\[ \text{Sum of Last 23 Results} = 53.5 \times 23 = 1230.5 \]

### Step 4: Set up the equation to find the 60th result.
Let the 60th result be \( x \). The total sum of all results can also be expressed as:
\[ \text{Total Sum} = \text{Sum of First 59 Results} + x + \text{Sum of Last 23 Results} \]
Substituting the known values:
\[ 4025.5 = 2684.5 + x + 1230.5 \]

### Step 5: Simplify the equation to find \( x \).
Combine the sums of the first 59 and last 23 results:
\[ 4025.5 = 2684.5 + 1230.5 + x \]
\[ 4025.5 = 3915 + x \]
Now, isolate \( x \):
\[ x = 4025.5 - 3915 \]
\[ x = 110.5 \]

### Final Answer:
The 60th result is \( \boxed{110.5} \).

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