The average of 11 results is 50. If the average of first six results is 51 and that of the last six is 59, then find the sixth result.
110
Understanding Average and Calculating the Sixth Result
This question involves the concept of average. The average of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers. We are given the average of a total set of results and the averages of two overlapping subsets (the first six and the last six). Our goal is to find the value of the element that is common to both subsets, which is the sixth result.
Calculating Total Sum from Average
We know the average of 11 results is 50. Using the definition of average, the total sum of these 11 results can be found by multiplying the average by the number of results.
Total Sum of 11 results = Average $\times$ Number of results
Total Sum of 11 results = $50 \times 11 = 550$
Calculating Sums of Subsets
Next, we find the sum of the first six results and the sum of the last six results using their respective averages.
- Sum of first 6 results = Average of first 6 $\times$ Number of first 6 results
- Sum of first 6 results = $51 \times 6 = 306$
- Sum of last 6 results = Average of last 6 $\times$ Number of last 6 results
- Sum of last 6 results = $59 \times 6 = 354$
Analyzing the Overlap to Find the Sixth Result
We have 11 results in total. The first six results are results 1, 2, 3, 4, 5, and 6. The last six results are results 6, 7, 8, 9, 10, and 11. Notice that the sixth result is included in both sets.
When we add the sum of the first six results and the sum of the last six results, the sixth result is counted twice. The sum of the first six results and the sum of the last six results combined covers all 11 results, but with the sixth result being counted extra.
Sum of (First 6 results) + Sum of (Last 6 results) = (Result 1 + ... + Result 6) + (Result 6 + ... + Result 11)
This sum is equivalent to (Sum of all 11 results) + (Sixth result).
Therefore, to find the value of the sixth result, we can add the sum of the first six results and the sum of the last six results, and then subtract the total sum of all 11 results.
Sixth result = (Sum of first 6 results) + (Sum of last 6 results) - (Total Sum of 11 results)
Sixth result = $306 + 354 - 550$
Sixth result = $660 - 550$
Sixth result = $110$
Summary of Calculations
| Item | Calculation | Value |
|---|---|---|
| Total Sum of 11 results | $11 \times 50$ | 550 |
| Sum of First 6 results | $6 \times 51$ | 306 |
| Sum of Last 6 results | $6 \times 59$ | 354 |
| Sum of First 6 + Last 6 | $306 + 354$ | 660 |
| Sixth result | $660 - 550$ | 110 |
The sixth result is 110.
Revision Table: Key Average Concepts
| Concept | Formula | Explanation |
|---|---|---|
| Average | $\text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}}$ | Represents the central value of a set of numbers. |
| Sum of Observations | $\text{Sum} = \text{Average} \times \text{Number of observations}$ | Useful for finding the total quantity when average is known. |
| Finding Overlapping Element | Sum(Subset1) + Sum(Subset2) - Sum(Total Set) | Applicable when two subsets overlap, and the overlapping element is the only one counted twice in the combined subset sum. |
Additional Information: Types of Averages
The question deals with the arithmetic mean, which is the most common type of average. However, there are other types of averages used in mathematics and statistics:
- Arithmetic Mean: The sum of all values divided by the number of values. (This is what was used in the problem).
- Geometric Mean: Used for averages of ratios or growth rates. It is the nth root of the product of n values.
- Harmonic Mean: Used for averages of rates, particularly speed or ratios. It is the reciprocal of the arithmetic mean of the reciprocals of the values.
- Median: The middle value in a dataset that is ordered from least to greatest.
- Mode: The value that appears most frequently in a dataset.
Understanding how averages are calculated and how subsets relate to the total set is crucial for solving problems like this one, especially those involving overlapping data points.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
