The mean of 10 numbers is 24. If one more number is included, the new mean is 25. Find the included number.

Answer

To solve the problem step by step, we will follow the mathematical principles of mean and sum of observations.

### Step 1: Calculate the sum of the first 10 numbers.
Given that the mean of 10 numbers is 24, we can find the sum of these 10 numbers using the formula for mean:

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

Substituting the known values:

\[ 24 = \frac{\text{Sum of 10 numbers}}{10} \]

To find the sum of the 10 numbers, we multiply both sides by 10:

\[ \text{Sum of 10 numbers} = 24 \times 10 = 240 \]

### Step 2: Calculate the sum of the 11 numbers.
When one more number is included, the new mean becomes 25. Now we have 11 numbers. We can again use the mean formula:

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

Substituting the new mean and the new number of observations:

\[ 25 = \frac{\text{Sum of 11 numbers}}{11} \]

To find the sum of the 11 numbers, we multiply both sides by 11:

\[ \text{Sum of 11 numbers} = 25 \times 11 = 275 \]

### Step 3: Find the included number.
The included number can be found by subtracting the sum of the first 10 numbers from the sum of the 11 numbers:

\[ \text{Included number} = \text{Sum of 11 numbers} - \text{Sum of 10 numbers} \]

Substituting the values we calculated:

\[ \text{Included number} = 275 - 240 = 35 \]

### Final Answer:
The included number is **35**.

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