There are 50 numbers. Each number is subtracted from 53 and the mean of the number so obtained is found to be `=3.5`. The mean of the given number is

Options

A. 46.5
B. 49.5
C. 53.5
D. 56.5

Answer

To find the mean of the original 50 numbers, we can follow these steps:

### Step 1: Understand the given information
We know that there are 50 numbers and each number is subtracted from 53. The mean of the resulting numbers is given as 3.5.

### Step 2: Set up the equation for the mean
The mean of the numbers obtained after subtraction can be expressed as:
\[ \text{Mean} = \frac{(53 - x_1) + (53 - x_2) + (53 - x_3) + \ldots + (53 - x_{50})}{50} \]
This can be simplified to:
\[ \text{Mean} = \frac{53 \times 50 - (x_1 + x_2 + x_3 + \ldots + x_{50})}{50} \]

### Step 3: Substitute the known mean
We know that this mean equals 3.5, so we can set up the equation:
\[ 3.5 = \frac{53 \times 50 - (x_1 + x_2 + x_3 + \ldots + x_{50})}{50} \]

### Step 4: Multiply both sides by 50
To eliminate the denominator, we multiply both sides by 50:
\[ 3.5 \times 50 = 53 \times 50 - (x_1 + x_2 + x_3 + \ldots + x_{50}) \]
Calculating the left side:
\[ 175 = 2650 - (x_1 + x_2 + x_3 + \ldots + x_{50}) \]

### Step 5: Rearrange to find the sum of the original numbers
Now, we can rearrange the equation to find the sum of the original numbers:
\[ x_1 + x_2 + x_3 + \ldots + x_{50} = 2650 - 175 \]
Calculating the right side gives:
\[ x_1 + x_2 + x_3 + \ldots + x_{50} = 2475 \]

### Step 6: Calculate the mean of the original numbers
The mean of the original numbers is given by:
\[ \text{Mean} = \frac{x_1 + x_2 + x_3 + \ldots + x_{50}}{50} \]
Substituting the sum we found:
\[ \text{Mean} = \frac{2475}{50} \]
Calculating this gives:
\[ \text{Mean} = 49.5 \]

### Conclusion
The mean of the given numbers is **49.5**.

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