The average of 75 scores is 45. The average of the first 50 scores is 42 and that of the last 26 scores is 52. What is the 50th score?

Options

A. 60
B. 80
C. 75
D. 77

Answer

To find the 50th score, we can follow these steps:

### Step 1: Calculate the total score for all 75 scores.
The average score of 75 scores is 45. Therefore, the total score can be calculated as:
\[ \text{Total score} = \text{Average} \times \text{Number of scores} = 45 \times 75 = 3375 \]

### Step 2: Calculate the total score for the first 50 scores.
The average score of the first 50 scores is 42. Therefore, the total score for the first 50 scores is:
\[ \text{Total score of first 50} = 42 \times 50 = 2100 \]

### Step 3: Calculate the total score for the last 26 scores.
The average score of the last 26 scores is 52. Therefore, the total score for the last 26 scores is:
\[ \text{Total score of last 26} = 52 \times 26 = 1352 \]

### Step 4: Set up the equation to find the 50th score.
Let \( x_{50} \) be the 50th score. The total score can also be expressed as the sum of the first 50 scores, the 50th score, and the last 25 scores:
\[ \text{Total score} = \text{Total score of first 50} + x_{50} + \text{Total score of last 25} \]
The total score of the last 25 scores can be calculated as:
\[ \text{Total score of last 25} = \text{Total score of last 26} - x_{51} \]
However, we know that \( x_{51} \) is included in the last 26 scores, so we can express the total score of the last 25 scores as:
\[ \text{Total score of last 25} = 1352 - x_{50} \]

### Step 5: Combine the equations.
Now we can set up the equation:
\[ 3375 = 2100 + x_{50} + (1352 - x_{50}) \]
This simplifies to:
\[ 3375 = 2100 + 1352 \]
\[ 3375 = 3452 - x_{50} \]

### Step 6: Solve for \( x_{50} \).
Rearranging gives:
\[ x_{50} = 3452 - 3375 \]
\[ x_{50} = 77 \]

Thus, the 50th score is \( \boxed{77} \).

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