The mean of 12 numbers is 10. If one more number is added, the mean remains unaltered. Find the number that is added.

Answer

To solve the problem step by step, let's follow the reasoning provided in the video transcript.

### Step 1: Understand the Mean
The mean (average) of a set of numbers is calculated by dividing the sum of those numbers by the count of the numbers.

### Step 2: Calculate the Total Sum of the Original Numbers
Given that the mean of 12 numbers is 10, we can find the total sum of these numbers using the formula for mean:

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

Let \( S \) be the sum of the 12 numbers. We can set up the equation:

\[ 10 = \frac{S}{12} \]

To find \( S \), we can multiply both sides by 12:

\[ S = 10 \times 12 = 120 \]

### Step 3: Introduce the New Number
Let the new number that is added be \( a \). The new total sum of the numbers after adding this new number will be:

\[ \text{New Sum} = S + a = 120 + a \]

### Step 4: Calculate the New Mean
After adding the new number, the total count of numbers becomes 13. The mean remains unchanged at 10, so we can set up the equation:

\[ 10 = \frac{120 + a}{13} \]

### Step 5: Solve for \( a \)
To find \( a \), we can multiply both sides of the equation by 13:

\[ 10 \times 13 = 120 + a \]

Calculating the left side:

\[ 130 = 120 + a \]

Now, isolate \( a \) by subtracting 120 from both sides:

\[ a = 130 - 120 = 10 \]

### Conclusion
The number that is added is \( 10 \).

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