What will be the average of all the even numbers between 5 and 29?
17
Calculating the Average of Even Numbers Between 5 and 29
The question asks for the average of all even numbers that are greater than 5 and less than 29. To find the average, we first need to identify these even numbers and then use the formula for calculating the average.
Identifying the Even Numbers
Even numbers are integers that are perfectly divisible by 2. We need to list all even numbers that fall strictly between 5 and 29.
- The first even number greater than 5 is 6.
- The last even number less than 29 is 28.
The sequence of even numbers between 5 and 29 is:
6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28
Method 1: Sum and Count
One way to calculate the average is to sum all the numbers in the sequence and then divide by the total count of numbers.
- Sum the numbers: The sum is \(6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28\). Let's group pairs that sum easily: \((6+28) + (8+26) + (10+24) + (12+22) + (14+20) + 16\). Each pair sums to 34. We have 5 pairs summing to 34, plus 16. Sum = \(5 \times 34 + 16 = 170 + 16 = 204\). So, the sum of these even numbers is 204.
- Count the numbers: Let's count how many numbers are in the sequence: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28. There are 12 numbers.
- Calculate the average: The average is the sum divided by the count. Average \( = \frac{\text{Sum}}{\text{Count}} = \frac{204}{12} \). \( \frac{204}{12} = 17 \).
Method 2: Arithmetic Progression Average
The sequence of even numbers (6, 8, 10, ..., 28) is an arithmetic progression because the difference between consecutive terms is constant (which is 2). For an arithmetic progression, the average is simply the average of the first and the last term.
- Identify the first and last terms: First term (a) = 6. Last term (l) = 28.
- Calculate the average: Average \( = \frac{\text{First term} + \text{Last term}}{2} \). Average \( = \frac{6 + 28}{2} = \frac{34}{2} = 17 \).
Both methods yield the same result. The average of all even numbers between 5 and 29 is 17.
Comparing with Options
Let's compare our calculated average with the given options:
| Option | Value |
|---|---|
| 1 | 14 |
| 2 | 22 |
| 3 | 19 |
| 4 | 17 |
Our calculated average, 17, matches Option 4.
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| Even Number | An integer divisible by 2. | 2, 4, 6, 8, ... |
| Average (Mean) | Sum of values divided by the count of values. | Average of 2, 4, 6 is \((2+4+6)/3 = 12/3 = 4\). |
| Arithmetic Progression | A sequence where the difference between consecutive terms is constant. | 6, 8, 10, 12 (common difference is 2). |
| Average of AP | (First term + Last term) / 2 | Average of 6, 8, 10, 12 is \((6+12)/2 = 18/2 = 9\). Also, \((6+8+10+12)/4 = 36/4 = 9\). |
Additional Information on Averages
The average, also known as the mean, is a measure of central tendency. It gives us a single value that represents the center of a set of numbers.
- The average is sensitive to outliers (extreme values).
- For sequences with a constant difference (arithmetic progressions), the average is always the midpoint between the first and last term. It is also the median if the number of terms is odd.
- Understanding the concept of average is fundamental in statistics and data analysis.
In this problem, recognizing the even numbers between 5 and 29 formed an arithmetic progression allowed for a quicker calculation of the average using the first and last terms.
Important Questions from Average
- Find the mean marks of the following distribution.
Marks obtained No. of students 25 5 30 6 35 3 40 1 - The average of first 169 even numbers is
- The average weight (in kg) of a family of five members whose weights are 40 kg, 49 kg, 56 kg, 74 kg and 35 kg is:
- The average of first 152 odd natural numbers, is:
The average runs scored by a batsman in 23 matches is 40. In the next 10 matches, the batsman scored an average of 16 runs. Find his average runs scored (rounded off to two decimals) in all the 33 matches.
