The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
26
Understanding the Average Calculation
The problem asks us to find a specific number that was excluded from a set of five numbers, given the average before and after the exclusion. The average of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers.
The formula for average is:
\(\text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\)
We can rearrange this formula to find the sum of numbers:
\(\text{Sum of numbers} = \text{Average} \times \text{Count of numbers}\)
Calculating the Sum of the Original Five Numbers
We are given that the average of five numbers is 30. Using the formula above, we can find the sum of these five numbers.
\(\text{Sum of 5 numbers} = \text{Average of 5 numbers} \times \text{Count of numbers}\)
\(\text{Sum of 5 numbers} = 30 \times 5\)
\(\text{Sum of 5 numbers} = 150\)
So, the sum of the original five numbers is 150.
Calculating the Sum of the Remaining Four Numbers
When one number is excluded, there are now four numbers left. The average of these four numbers becomes 31. We can find the sum of these four numbers using the same formula.
\(\text{Sum of 4 numbers} = \text{Average of 4 numbers} \times \text{Count of numbers}\)
\(\text{Sum of 4 numbers} = 31 \times 4\)
\(\text{Sum of 4 numbers} = 124\)
Thus, the sum of the remaining four numbers is 124.
Finding the Excluded Number
The excluded number is the difference between the sum of the original five numbers and the sum of the remaining four numbers. This is because the sum of the five numbers includes the excluded number, while the sum of the four numbers does not.
\(\text{Excluded number} = \text{Sum of 5 numbers} - \text{Sum of 4 numbers}\)
\(\text{Excluded number} = 150 - 124\)
\(\text{Excluded number} = 26\)
Therefore, the excluded number is 26.
Step-by-Step Solution Summary
- Identify the initial average and count: Average = 30, Count = 5.
- Calculate the sum of the initial numbers: Sum = Average * Count = \(30 \times 5 = 150\).
- Identify the new average and count after exclusion: New Average = 31, New Count = 4.
- Calculate the sum of the numbers after exclusion: New Sum = New Average * New Count = \(31 \times 4 = 124\).
- Find the excluded number by subtracting the new sum from the initial sum: Excluded Number = \(150 - 124 = 26\).
| Description | Calculation | Value |
|---|---|---|
| Initial Count | - | 5 |
| Initial Average | - | 30 |
| Sum of 5 numbers | \(30 \times 5\) | 150 |
| New Count (after exclusion) | - | 4 |
| New Average | - | 31 |
| Sum of 4 numbers | \(31 \times 4\) | 124 |
| Excluded Number | \(150 - 124\) | 26 |
Revision Table: Average Calculations
| Concept | Formula | Notes |
|---|---|---|
| Average | \(\frac{\text{Sum}}{\text{Count}}\) | Sum of values divided by number of values. |
| Sum | \(\text{Average} \times \text{Count}\) | Useful for finding total from average. |
| Finding Excluded/Included Number | Change in Sum | Difference between sums before and after change. |
Additional Information: Properties of Averages
Understanding how averages change when numbers are added or removed is crucial for solving such problems. Here are some key points:
- If a number equal to the current average is added or removed, the average remains unchanged.
- If a number greater than the current average is added, the average increases. If a number greater than the current average is removed, the average decreases.
- If a number less than the current average is added, the average decreases. If a number less than the current average is removed, the average increases.
- In this specific problem, removing a number increased the average from 30 to 31. This indicates that the removed number must have been less than the original average (30). Our calculated excluded number, 26, is indeed less than 30, which makes sense.
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