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)
74.7
Solving the Average Problem with an Excluded Number
This problem involves calculating the average of a set of numbers and then finding the average of the remaining numbers after one specific number is removed. We are given the average of the total set, the average of the first few numbers, and the average of the last few numbers, with one number being common to both partial sets.
Understanding the Given Information
We are provided with the following data:
- Total number of values = 28
- Average of all 28 numbers = 77
- Average of the first 14 numbers = 74
- Average of the last 15 numbers = 84
Notice that the sum of the counts of the partial sets (14 + 15 = 29) is one more than the total number of values (28). This indicates that one number is included in both the first 14 and the last 15 sets. Based on standard problem phrasing, this common number is the 14th number when the numbers are ordered from 1 to 28.
Step-by-Step Calculation
Step 1: Calculate the total sum of all 28 numbers
The sum of a set of numbers is equal to their average multiplied by the count of numbers.
\(\text{Sum of 28 numbers} = \text{Average of 28 numbers} \times \text{Total count}\)
\(\text{Sum of 28 numbers} = 77 \times 28\)
\(\text{Sum of 28 numbers} = 2156\)
Step 2: Calculate the sum of the first 14 numbers
\(\text{Sum of first 14 numbers} = \text{Average of first 14 numbers} \times \text{Count}\)
\(\text{Sum of first 14 numbers} = 74 \times 14\)
\(\text{Sum of first 14 numbers} = 1036\)
Step 3: Calculate the sum of the last 15 numbers
\(\text{Sum of last 15 numbers} = \text{Average of last 15 numbers} \times \text{Count}\)
\(\text{Sum of last 15 numbers} = 84 \times 15\)
\(\text{Sum of last 15 numbers} = 1260\)
Step 4: Find the value of the 14th number
The sum of the first 14 numbers and the sum of the last 15 numbers together include the 14th number twice (once in the first 14 and once in the last 15), while all other numbers (1 to 13 and 15 to 28) are included only once. Therefore, the sum of the first 14 and the last 15 equals the sum of all 28 numbers plus the value of the 14th number.
\(\text{Sum of first 14} + \text{Sum of last 15} = \text{Sum of all 28} + \text{Value of 14th number}\)
We can rearrange this to find the value of the 14th number:
\(\text{Value of 14th number} = (\text{Sum of first 14} + \text{Sum of last 15}) - \text{Sum of all 28}\)
\(\text{Value of 14th number} = (1036 + 1260) - 2156\)
\(\text{Value of 14th number} = 2296 - 2156\)
\(\text{Value of 14th number} = 140\)
Step 5: Calculate the sum of the remaining 27 numbers
When the 14th number is excluded from the set of 28 numbers, the remaining count is 27. The sum of these 27 numbers is the total sum of the 28 numbers minus the value of the excluded 14th number.
\(\text{Sum of remaining 27 numbers} = \text{Sum of all 28 numbers} - \text{Value of 14th number}\)
\(\text{Sum of remaining 27 numbers} = 2156 - 140\)
\(\text{Sum of remaining 27 numbers} = 2016\)
Step 6: Calculate the average of the remaining 27 numbers
The average of the remaining numbers is the sum of these numbers divided by their count.
\(\text{Average of remaining 27 numbers} = \frac{\text{Sum of remaining 27 numbers}}{\text{Count of remaining numbers}}\)
\(\text{Average of remaining 27 numbers} = \frac{2016}{27}\)
Performing the division:
\(\frac{2016}{27} \approx 74.666...\)
The question asks for the answer correct to one decimal place. Rounding 74.666... to one decimal place gives 74.7.
Final Answer
The average of the remaining 27 numbers after excluding the 14th number is approximately 74.7.
Revision Table: Key Concepts for Average Problems
| Concept | Formula / Definition | Notes |
|---|---|---|
| Average (Mean) | \( \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \) | Represents the central value of a dataset. |
| Sum of Values | \( \text{Sum} = \text{Average} \times \text{Number of values} \) | Useful for calculating the total when average and count are known. |
| Handling Overlapping Sets | \( \text{Sum}(\text{A} \cup \text{B}) = \text{Sum}(\text{A}) + \text{Sum}(\text{B}) - \text{Sum}(\text{A} \cap \text{B}) \) | If elements are double-counted (like the 14th number here), their value is added twice when summing overlapping sets. Subtracting the total sum isolates the value of the overlapping element(s). |
Additional Information on Average Calculations
The average is a fundamental concept in statistics used to summarize a dataset with a single value. It is sensitive to outliers (extremely large or small values). In problems like this, understanding how partial sums relate to the total sum, especially when there is overlap, is crucial.
When a value is removed from a dataset, the sum and the count of values change. A new average must be calculated based on these new values.
- If a value greater than the original average is removed, the new average of the remaining numbers will be lower than the original average.
- If a value less than the original average is removed, the new average of the remaining numbers will be higher than the original average.
- If a value equal to the original average is removed, the new average of the remaining numbers will remain the same as the original average (assuming the number of values removed is small compared to the total).
In our problem, the original average was 77, and the excluded 14th number was 140. Since 140 is much greater than 77, the average of the remaining numbers (74.7) is lower than the original average, which aligns with the concept.
Important Questions from Average
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?
- The average weight of P and his three friends is 55 kg. If P is 4 kg more than the average weight of his three friends, what is P's weight (in kg)?
