To decrease the mean of 5 numbers by 3, by how much would the sum of the 5 numbers have to decrease?

Answer

To solve the problem of how much the sum of 5 numbers needs to decrease in order to reduce the mean by 3, we can follow these steps:

### Step 1: Define the Mean
Let the sum of the 5 numbers be denoted as \( x \). The mean \( m \) of these 5 numbers can be expressed as:
\[ m = \frac{x}{5} \]

### Step 2: Express the Sum in Terms of the Mean
From the mean formula, we can rearrange it to express the sum \( x \) in terms of the mean \( m \):
\[ x = 5m \]

### Step 3: Determine the New Mean
We want to decrease the mean by 3. Therefore, the new mean will be:
\[ \text{New Mean} = m - 3 \]

### Step 4: Set Up the Equation for the New Sum
If we denote the amount by which the sum needs to decrease as \( z \), the new sum of the numbers will be:
\[ \text{New Sum} = x - z \]
The new mean can also be expressed in terms of the new sum:
\[ \text{New Mean} = \frac{x - z}{5} \]

### Step 5: Equate the New Mean Expressions
Now we can set the two expressions for the new mean equal to each other:
\[ \frac{x - z}{5} = m - 3 \]

### Step 6: Substitute the Expression for \( x \)
Substituting \( x = 5m \) into the equation gives:
\[ \frac{5m - z}{5} = m - 3 \]

### Step 7: Simplify the Equation
Multiplying both sides by 5 to eliminate the fraction:
\[ 5m - z = 5(m - 3) \]
Expanding the right side:
\[ 5m - z = 5m - 15 \]

### Step 8: Solve for \( z \)
Now, we can isolate \( z \):
\[ -z = -15 \]
Thus, we find:
\[ z = 15 \]

### Conclusion
The sum of the 5 numbers must decrease by 15 in order to reduce the mean by 3.

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