There are 50 numbers. Each number is subtracted from 43 and the mean of the numbers so obtained is found to be 5. The mean of the given numbers is

Options

A. 38
B. 39
C. 48
D. 49

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have 50 numbers. Each number is subtracted from 43, and the mean of the resulting numbers is 5. We need to find the mean of the original numbers.

### Step 2: Set Up the Equation
Let the original numbers be \( x_1, x_2, \ldots, x_{50} \). When each number is subtracted from 43, we get new numbers:
\[ 43 - x_1, 43 - x_2, \ldots, 43 - x_{50} \]
The mean of these new numbers is given as 5.

### Step 3: Write the Mean Formula
The mean of the new numbers can be expressed as:
\[ \text{Mean} = \frac{(43 - x_1) + (43 - x_2) + \ldots + (43 - x_{50})}{50} \]
This simplifies to:
\[ \text{Mean} = \frac{50 \times 43 - (x_1 + x_2 + \ldots + x_{50})}{50} \]

### Step 4: Substitute the Given Mean
We know that the mean of the new numbers is 5, so we can set up the equation:
\[ 5 = \frac{50 \times 43 - (x_1 + x_2 + \ldots + x_{50})}{50} \]

### Step 5: Multiply Both Sides by 50
To eliminate the denominator, multiply both sides by 50:
\[ 250 = 50 \times 43 - (x_1 + x_2 + \ldots + x_{50}) \]

### Step 6: Calculate \(50 \times 43\)
Calculate \(50 \times 43\):
\[ 50 \times 43 = 2150 \]
So, we can rewrite the equation as:
\[ 250 = 2150 - (x_1 + x_2 + \ldots + x_{50}) \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ x_1 + x_2 + \ldots + x_{50} = 2150 - 250 \]
\[ x_1 + x_2 + \ldots + x_{50} = 1900 \]

### Step 8: Find the Mean of the Original Numbers
Now, we can find the mean of the original numbers:
\[ \text{Mean} = \frac{x_1 + x_2 + \ldots + x_{50}}{50} = \frac{1900}{50} \]
Calculating this gives:
\[ \text{Mean} = 38 \]

### Final Answer
Thus, the mean of the given numbers is **38**.

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