The mean of 20 numbers is 18. if 3 is added to each of the first ten numbers, find the mean of the new set of 20 numbers.

Explanation

To solve the problem step by step, we will follow the mathematical principles of mean calculation.

### Step 1: Understand the given information
We know that the mean of 20 numbers is 18. This means that the sum of these 20 numbers can be calculated using the formula for mean:
\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total numbers}} \]
From this, we can derive the sum of the 20 numbers.

### Step 2: Calculate the sum of the 20 numbers
Using the mean formula:
\[ 18 = \frac{\text{Sum of 20 numbers}}{20} \]
To find the sum of the 20 numbers, we can rearrange this equation:
\[ \text{Sum of 20 numbers} = 18 \times 20 = 360 \]

### Step 3: Adjust the first ten numbers
According to the problem, we add 3 to each of the first ten numbers. Therefore, the total increase in the sum due to this adjustment is:
\[ \text{Increase} = 3 \times 10 = 30 \]

### Step 4: Calculate the new sum of the 20 numbers
Now, we add this increase to the original sum of the 20 numbers:
\[ \text{New sum} = \text{Original sum} + \text{Increase} = 360 + 30 = 390 \]

### Step 5: Calculate the new mean
Now that we have the new sum of the 20 numbers, we can find the new mean:
\[ \text{New Mean} = \frac{\text{New sum}}{\text{Total numbers}} = \frac{390}{20} = 19.5 \]

### Final Answer
The mean of the new set of 20 numbers is **19.5**.

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