The mean of 5 numbers is 15. If one more number is included, the mean of 6 numbers becomes 17. What is the included number?
27
Finding the Included Number Using Mean Calculation
This problem involves understanding the concept of the mean (or average) and how it changes when a new number is added to a set. The mean is calculated by dividing the sum of all numbers by the count of numbers.
Understanding the Concept of Mean
The mean provides a central value for a set of numbers. The formula for the mean is:
\(\text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\)
From this formula, we can also find the sum of numbers if we know the mean and the count:
\(\text{Sum of numbers} = \text{Mean} \times \text{Count of numbers}\)
Step-by-Step Calculation to Find the Included Number
Step 1: Calculate the sum of the initial 5 numbers.
We are given that the mean of 5 numbers is 15.
- Count of initial numbers = 5
- Mean of initial numbers = 15
Using the formula for the sum:
\(\text{Sum of 5 numbers} = \text{Mean of 5 numbers} \times \text{Count of 5 numbers}\)
\(\text{Sum of 5 numbers} = 15 \times 5\)
\(\text{Sum of 5 numbers} = 75\)
So, the sum of the first 5 numbers is 75.
Step 2: Calculate the sum of the 6 numbers after inclusion.
One more number is included, making the total count 6. The new mean of these 6 numbers is 17.
- Count of new numbers = 6
- Mean of new numbers = 17
Using the formula for the sum:
\(\text{Sum of 6 numbers} = \text{Mean of 6 numbers} \times \text{Count of 6 numbers}\)
\(\text{Sum of 6 numbers} = 17 \times 6\)
\(\text{Sum of 6 numbers} = 102\)
So, the sum of the 6 numbers is 102.
Step 3: Find the included number.
The sum of the 6 numbers is the sum of the original 5 numbers plus the one included number. Therefore, the included number is the difference between the sum of 6 numbers and the sum of 5 numbers.
\(\text{Included number} = \text{Sum of 6 numbers} - \text{Sum of 5 numbers}\)
\(\text{Included number} = 102 - 75\)
\(\text{Included number} = 27\)
The included number is 27.
Summary of Calculations
| Description | Calculation | Result |
|---|---|---|
| Sum of 5 numbers | \(15 \times 5\) | 75 |
| Sum of 6 numbers | \(17 \times 6\) | 102 |
| Included number | \(102 - 75\) | 27 |
The value of the included number is 27.
Revision Table: Key Mean Concepts
| Term | Definition | Formula |
|---|---|---|
| Mean (Average) | The sum of a set of numbers divided by the count of numbers in the set. | \(\text{Mean} = \frac{\sum x}{n}\) (where \(\sum x\) is the sum and \(n\) is the count) |
| Sum of Numbers | The total obtained by adding all the numbers in a set. | \(\text{Sum} = \text{Mean} \times \text{Count}\) |
| Count of Numbers | The total number of items or values in a set. | \(n\) |
Additional Information: Properties of Mean
- Adding or removing a number changes the sum and the count, thus affecting the mean.
- If all numbers in a set are increased or decreased by a constant value, the mean also increases or decreases by the same constant value.
- If all numbers in a set are multiplied or divided by a constant value, the mean also gets multiplied or divided by the same constant value.
- The mean is sensitive to extreme values (outliers).
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