Understanding the Problem on Averages

The question asks us to find the value of a specific result, the 8th result, given information about the average of a total set of results and the averages of two overlapping subsets. We are given:

• The average of 15 results is 52.
• The average of the first 8 results is 56.
• The average of the last 8 results is 55.

Notice that the 8th result is included in both the 'first 8 results' group and the 'last 8 results' group. This overlap is key to solving the problem.

Step-by-Step Solution to Find the 8th Result

To find the value of a specific result when dealing with averages and overlapping sets, we can use the concept that the average multiplied by the number of results gives the total sum of those results. We will calculate the total sum for each group mentioned.

Calculate the Total Sum of Results

1. Total sum of 15 results: The average of 15 results is 52. \[ \text{Sum of 15 results} = \text{Average} \times \text{Number of results} \] \[ \text{Sum of 15 results} = 52 \times 15 \] \[ \text{Sum of 15 results} = 780 \] So, the sum of all 15 results is 780.
2. Total sum of the first 8 results: The average of the first 8 results is 56. \[ \text{Sum of first 8 results} = \text{Average} \times \text{Number of results} \] \[ \text{Sum of first 8 results} = 56 \times 8 \] \[ \text{Sum of first 8 results} = 448 \] The sum of the first 8 results is 448.
3. Total sum of the last 8 results: The average of the last 8 results is 55. \[ \text{Sum of last 8 results} = \text{Average} \times \text{Number of results} \] \[ \text{Sum of last 8 results} = 55 \times 8 \] \[ \text{Sum of last 8 results} = 440 \] The sum of the last 8 results is 440.

Using Overlap to Find the 8th Result

The sum of the first 8 results includes results 1st through 8th. The sum of the last 8 results includes results 8th through 15th. When we add the sum of the first 8 results and the sum of the last 8 results, the 8th result is counted twice.

Let the results be \(R_1, R_2, \dots, R_{15}\). Sum of first 8 results = \(R_1 + R_2 + \dots + R_8\) Sum of last 8 results = \(R_8 + R_9 + \dots + R_{15}\)

Adding these two sums gives:

\( (\text{Sum of first 8}) + (\text{Sum of last 8}) = (R_1 + \dots + R_8) + (R_8 + \dots + R_{15}) \)

This can be rewritten as:

\( (\text{Sum of first 8}) + (\text{Sum of last 8}) = (R_1 + \dots + R_{15}) + R_8 \)

Notice that \(R_1 + \dots + R_{15}\) is the sum of all 15 results. So, we have:

\( (\text{Sum of first 8}) + (\text{Sum of last 8}) = (\text{Sum of 15 results}) + (\text{Value of 8th result}) \)

We can rearrange this equation to find the value of the 8th result:

\[ \text{Value of 8th result} = (\text{Sum of first 8}) + (\text{Sum of last 8}) - (\text{Sum of 15 results}) \]

Calculate the Value of the 8th Result

Now, substitute the sums we calculated:

\[ \text{Value of 8th result} = 448 + 440 - 780 \]

\[ \text{Value of 8th result} = 888 - 780 \]

\[ \text{Value of 8th result} = 108 \]

The value of the 8th result is 108.

Summary of Calculations

Value of 8th result = (Sum of first 8) + (Sum of last 8) - (Sum of 15)

Value of 8th result = \(448 + 440 - 780\)

Value of 8th result = \(888 - 780\)

Value of 8th result = \(108\)

Conclusion

Based on the calculations, the value of the 8th result is 108. This method works because the 8th result is the only one counted in both the group of the first 8 and the group of the last 8. By adding the sums of these two overlapping groups and subtracting the sum of the total group, we isolate the value of the overlapping element.

Revision Table: Averages and Overlapping Sets

Additional Information: Understanding Averages

An average, also known as the arithmetic mean, is a fundamental concept in statistics. It represents a typical value for a set of numbers. The formula for the average of a set of numbers is:

\[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

Averages are useful for summarizing data, comparing different sets of data, and understanding the central tendency of a distribution. In problems involving averages, it's crucial to correctly identify the number of items in each set and the corresponding total sum.

In this specific problem, recognizing the overlap of the 8th result in both subsets is key. The technique used here is applicable to any similar problem where an element is counted in two subsets whose combined size is greater than the total set size.