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To solve this problem, we need to find the 13th number in a sequence of 25 numbers given certain conditions about their means.

1. The mean of 25 numbers is 36.
2. The mean of the first 13 numbers is 32.
3. The mean of the last 13 numbers is 39.

First, let's calculate the total sum of the 25 numbers:
\[ \text{Sum of 25 numbers} = 25 \times 36 = 900 \]

Next, we calculate the sum of the first 13 numbers:
\[ \text{Sum of first 13 numbers} = 13 \times 32 = 416 \]

Similarly, we calculate the sum of the last 13 numbers:
\[ \text{Sum of last 13 numbers} = 13 \times 39 = 507 \]

Since the 13th number is counted in both the first 13 numbers and the last 13 numbers, it is included twice in these sums. Let’s denote the 13th number as \( x \).

The sum of the first 12 numbers plus the 13th number is 416. Similarly, the sum of the last 12 numbers plus the 13th number is 507.

So, the sum of all 25 numbers can also be expressed as:
\[ \text{Sum of first 12 numbers} + x + \text{Sum of next 12 numbers} + x = 900 \]

Simplifying the equation:
\[ 416 + 507 - x = 900 \]

We subtract \( x \) because it is counted twice. Solving for \( x \):
\[ 923 - x = 900 \]
\[ x = 923 - 900 \]
\[ x = 23 \]

Thus, the 13th number is 23.

Would you like details or have any questions? Here are five related questions to further explore this topic:

1. How would you find the mean if one of the numbers in the sequence was removed?
2. What is the variance of these 25 numbers?
3. If the sequence is extended to 30 numbers, how would that affect the calculations?
4. Can you determine the median of these 25 numbers with the given data?
5. How would you approach this problem if the means of overlapping subsets were given?

**Tip:** Always double-check your calculations when working with means and sums to ensure accuracy, especially when dealing with overlapping subsets.

The mean of 25 numbers is 36. If the mean of the first 13 numbers is 32 and that of the last 13 numbers is 39, find the 13th number?

Learn how to solve a problem involving the mean of 25 numbers, where the mean of the first 13 numbers is 32, and the mean of the last 13 numbers is 39. This problem demonstrates key statistical concepts and algebraic reasoning. Suitable for students in Grades 8-10.

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