The mean of 15 numbers is 27. if each number is multiplied by 4, what will be the mean of the new numbers ?

Answer

To solve the problem step by step, we will follow the logical reasoning presented in the video transcript.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that the mean of 15 numbers is 27.

2. **Use the Mean Formula**:
The mean (average) of a set of numbers is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of numbers}} \]
In this case, we can express it as:
\[ 27 = \frac{S}{15} \]
where \( S \) is the sum of the 15 numbers.

3. **Calculate the Sum of the Numbers**:
To find the sum \( S \), we can rearrange the equation:
\[ S = 27 \times 15 \]
Calculating this gives:
\[ S = 405 \]

4. **Multiply Each Number by 4**:
If each of the 15 numbers is multiplied by 4, the new numbers will be:
\[ 4a_1, 4a_2, 4a_3, \ldots, 4a_{15} \]

5. **Calculate the New Sum**:
The new sum of these numbers becomes:
\[ \text{New Sum} = 4a_1 + 4a_2 + 4a_3 + \ldots + 4a_{15} = 4(a_1 + a_2 + a_3 + \ldots + a_{15}) = 4S \]
Substituting the value of \( S \):
\[ \text{New Sum} = 4 \times 405 = 1620 \]

6. **Calculate the New Mean**:
The new mean can be calculated using the new sum and the total number of numbers, which remains 15:
\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Total number of numbers}} = \frac{1620}{15} \]
Performing the division gives:
\[ \text{New Mean} = 108 \]

### Final Answer:
The mean of the new numbers is **108**.

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