Understanding the Average and Sum

The average of a set of numbers is calculated by dividing the sum of all numbers by the count of numbers. Mathematically, this can be represented as:

\(\text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\)

From this, we can also find the sum of numbers if we know the average and the count:

\(\text{Sum of numbers} = \text{Average} \times \text{Count of numbers}\)

Calculating the Total Sum of 25 Numbers

We are given that the average of 25 numbers is 36. Using the formula for the sum of numbers:

\(\text{Sum of 25 numbers} = \text{Average of 25 numbers} \times \text{Count of numbers}\)

\(\text{Sum of 25 numbers} = 36 \times 25\)

Let's calculate this value:

\(36 \times 25 = 900\)

So, the total sum of the 25 numbers is 900.

Calculating the Sum of the First 13 Numbers

The problem states that the average of the first 13 numbers is 32. Using the sum formula again:

\(\text{Sum of first 13 numbers} = \text{Average of first 13 numbers} \times \text{Count of numbers}\)

\(\text{Sum of first 13 numbers} = 32 \times 13\)

Calculating this sum:

\(32 \times 13 = 416\)

The sum of the first 13 numbers is 416.

Calculating the Sum of the Last 13 Numbers

We are also given that the average of the last 13 numbers is 39. Using the sum formula:

\(\text{Sum of last 13 numbers} = \text{Average of last 13 numbers} \times \text{Count of numbers}\)

\(\text{Sum of last 13 numbers} = 39 \times 13\)

Calculating this sum:

\(39 \times 13 = 507\)

The sum of the last 13 numbers is 507.

Finding the 13th Number

Consider the set of 25 numbers as \(n_1, n_2, \dots, n_{13}, \dots, n_{25}\).

The sum of the first 13 numbers is \(n_1 + n_2 + \dots + n_{13}\).

The sum of the last 13 numbers is \(n_{13} + n_{14} + \dots + n_{25}\).

If we add the sum of the first 13 numbers and the sum of the last 13 numbers, we get:

\((n_1 + \dots + n_{13}) + (n_{13} + \dots + n_{25})\)

Notice that the 13th number (\(n_{13}\)) is included in both sums. So, adding the sum of the first 13 and the sum of the last 13 gives us the sum of all 25 numbers PLUS the 13th number itself (because it was counted twice).

Let Sum(1-13) be the sum of the first 13 numbers, Sum(13-25) be the sum of the last 13 numbers, and Sum(1-25) be the sum of all 25 numbers.

The relationship is:

\(\text{Sum}(1-13) + \text{Sum}(13-25) = \text{Sum}(1-25) + n_{13}\)

To find the 13th number (\(n_{13}\)), we can rearrange the formula:

\(n_{13} = \text{Sum}(1-13) + \text{Sum}(13-25) - \text{Sum}(1-25)\)

Now, substitute the calculated sums:

• Sum of first 13 numbers = 416
• Sum of last 13 numbers = 507
• Sum of 25 numbers = 900

\(n_{13} = 416 + 507 - 900\)

\(n_{13} = 923 - 900\)

\(n_{13} = 23\)

Therefore, the 13th number is 23.

Step-by-Step Solution

1. Calculate the total sum of 25 numbers: \(36 \times 25 = 900\).
2. Calculate the sum of the first 13 numbers: \(32 \times 13 = 416\).
3. Calculate the sum of the last 13 numbers: \(39 \times 13 = 507\).
4. Add the sum of the first 13 and the sum of the last 13: \(416 + 507 = 923\).
5. Subtract the total sum of 25 numbers from the result of step 4 to find the 13th number: \(923 - 900 = 23\).

Sum of First 13 + Sum of Last 13 = \(416 + 507 = 923\)

This sum (923) includes the 13th number twice. The total sum of all 25 numbers (900) includes the 13th number only once. The difference between these two sums is the 13th number.

13th number = (Sum of First 13 + Sum of Last 13) - (Sum of all 25)

13th number = \(923 - 900 = 23\)

Revision Table: Key Concepts

Additional Information: Average Problems

Problems involving averages often test your understanding of the relationship between the average, the sum of the items, and the number of items. When dealing with overlapping sets of numbers, like the first 13 and last 13 from a list of 25, identifying the element(s) counted multiple times is crucial. In this problem, the 13th number is part of both the "first 13" set and the "last 13" set.

• When the sum of two overlapping sub-groups is calculated, the overlapping elements are included in both sums.
• The total sum of these sub-group sums will be greater than the sum of the entire group by the sum of the overlapping elements.
• If there is only one overlapping element (like the 13th number here, when dealing with first N and last N numbers from a 2N-1 list), its value is the difference between the sum of the sub-groups and the sum of the total group.

This method is particularly useful when dealing with problems where a central element is included in two overlapping ranges that cover the entire set plus the overlap.