To increase the mean of 8 numbers by 5, by how much would the sum of the 8 numbers need to increase ?

Options

A. 5
B. 10
C. 20
D. 40

Answer

To solve the problem of how much the sum of 8 numbers needs to increase in order to raise the mean by 5, we can follow these steps:

### Step 1: Understand the Mean
The mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers. For 8 numbers, if we denote the sum of these numbers as \( S \), the mean \( m \) can be expressed as:
\[ m = \frac{S}{8} \]

### Step 2: Set Up the New Mean
We want to increase the mean by 5. Therefore, the new mean will be:
\[ m + 5 = \frac{S + y}{8} \]
where \( y \) is the amount by which the sum \( S \) needs to be increased.

### Step 3: Substitute the Original Mean
From the first step, we know that \( m = \frac{S}{8} \). We can substitute this into our equation for the new mean:
\[ \frac{S}{8} + 5 = \frac{S + y}{8} \]

### Step 4: Clear the Denominator
To eliminate the fraction, we can multiply the entire equation by 8:
\[ S + 40 = S + y \]

### Step 5: Solve for \( y \)
Now, we can simplify the equation:
\[ S + 40 = S + y \]
Subtracting \( S \) from both sides gives us:
\[ 40 = y \]

### Conclusion
Thus, the sum of the 8 numbers needs to increase by **40** in order to raise the mean by 5.