Mean of 100 items is 49. It was discovered that three items which should have been `60, 70,80` were wrongly read as `40, 20, 50` respectively. The correct mean is. (a) `48` (b) `82 1/2` (c) `50` (d) `80`

Options

A. 48
B. 82.5
C. 50
D. 80

Explanation

To find the correct mean after correcting the wrongly read items, we can follow these steps:

### Step 1: Calculate the original sum of the items
Given that the mean of 100 items is 49, we can find the total sum of these items (S) using the formula for mean:
\[ \text{Mean} = \frac{\text{Sum of items}}{\text{Number of items}} \]
Thus,
\[ S = \text{Mean} \times \text{Number of items} = 49 \times 100 = 4900 \]

### Step 2: Identify the wrongly read items and their correct values
The wrongly read items are:
- 40 (should be 60)
- 20 (should be 70)
- 50 (should be 80)

### Step 3: Calculate the sum of the wrongly read items
Now, we calculate the sum of the wrongly read items:
\[ \text{Sum of wrongly read items} = 40 + 20 + 50 = 110 \]

### Step 4: Calculate the sum of the correct items
Next, we calculate the sum of the correct items:
\[ \text{Sum of correct items} = 60 + 70 + 80 = 210 \]

### Step 5: Adjust the total sum to reflect the correct items
To find the new total sum after correcting the items, we subtract the sum of the wrongly read items from the original sum and then add the sum of the correct items:
\[ \text{New Sum} = S - \text{Sum of wrongly read items} + \text{Sum of correct items} \]
Substituting the values:
\[ \text{New Sum} = 4900 - 110 + 210 = 4900 - 110 + 210 = 4900 + 100 = 5000 \]

### Step 6: Calculate the new mean
Now, we can find the new mean using the new total sum:
\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of items}} = \frac{5000}{100} = 50 \]

### Conclusion
The correct mean after adjusting for the wrongly read items is **50**.