The mean of 20 numbers is 32. If 5 is added to each number, then find the new mean

Answer

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

### Step 1: Understand the given information
We know that the mean of 20 numbers is 32. This means that the sum of these 20 numbers can be calculated using the formula for mean:

\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

### Step 2: Calculate the sum of the 20 numbers
Using the mean formula, we can express the sum of the 20 numbers as follows:

\[ \text{Sum} = \text{Mean} \times \text{Number of observations} = 32 \times 20 = 640 \]

### Step 3: Add 5 to each number
If we add 5 to each of the 20 numbers, the new numbers will be:

\[ x_1 + 5, x_2 + 5, \ldots, x_{20} + 5 \]

### Step 4: Calculate the new sum
The new sum of the numbers after adding 5 to each number will be:

\[ \text{New Sum} = (x_1 + 5) + (x_2 + 5) + \ldots + (x_{20} + 5) \]

This can be simplified as:

\[ \text{New Sum} = (x_1 + x_2 + \ldots + x_{20}) + (5 \times 20) = 640 + 100 = 740 \]

### Step 5: Calculate the new mean
Now, we can find the new mean using the new sum and the same number of observations (which is still 20):

\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of observations}} = \frac{740}{20} = 37 \]

### Final Answer
Thus, the new mean after adding 5 to each number is **37**.

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