In a group of people, 70% are men and 30% are women. The average age of the men is 50 years and that of the women is 40 years. Find the average age of all the members in the group (in years).
47
Calculating the Average Age of a Mixed Group
This problem involves finding the average age of an entire group when we know the proportions and average ages of its subgroups (men and women). This requires using the concept of a weighted average.
Understanding Weighted Average
A weighted average is used when different data points contribute differently to the final average. In this case, the contribution of the men's average age and the women's average age to the overall group average is weighted by their respective proportions in the group.
The formula for a weighted average is:
\(\text{Weighted Average} = \frac{\sum (\text{Weight} \times \text{Value})}{\sum \text{Weight}}\)
In this problem:
- The 'values' are the average ages of the subgroups (men and women).
- The 'weights' are the proportions or percentages of these subgroups in the total group.
Given Information
We are given the following information:
- Percentage of men in the group = 70%
- Percentage of women in the group = 30%
- Average age of men = 50 years
- Average age of women = 40 years
We need to find the average age of all the members in the group.
Step-by-Step Calculation of Average Age
Let \(P_m\) be the percentage of men and \(A_m\) be the average age of men.
Let \(P_w\) be the percentage of women and \(A_w\) be the average age of women.
The total percentage of the group is \(P_m + P_w = 70\% + 30\% = 100\%\).
Using the weighted average formula, the average age of the group (\(A_{group}\)) is:
\(A_{group} = \frac{(P_m \times A_m) + (P_w \times A_w)}{P_m + P_w}\)
Since the percentages are out of 100, we can use the percentage values directly in the numerator and divide by the total percentage (100), or use decimal equivalents (0.70 and 0.30) and divide by 1 (0.70 + 0.30).
Using percentages:
\(A_{group} = \frac{(70 \times 50) + (30 \times 40)}{70 + 30}\)
\(A_{group} = \frac{(70 \times 50) + (30 \times 40)}{100}\)
Now, perform the multiplications:
\(70 \times 50 = 3500\)
\(30 \times 40 = 1200\)
Substitute these values back into the formula:
\(A_{group} = \frac{3500 + 1200}{100}\)
\(A_{group} = \frac{4700}{100}\)
\(A_{group} = 47\)
So, the average age of all the members in the group is 47 years.
Alternatively, using decimal proportions:
\(P_m = 0.70\)
\(P_w = 0.30\)
\(A_{group} = \frac{(0.70 \times 50) + (0.30 \times 40)}{0.70 + 0.30}\)
\(A_{group} = \frac{35 + 12}{1.00}\)
\(A_{group} = \frac{47}{1}\)
\(A_{group} = 47\)
The result is the same: 47 years.
Conclusion
The average age of all the members in the group is 47 years.
| Subgroup | Percentage | Average Age | Weighted Contribution |
|---|---|---|---|
| Men | 70% (or 0.70) | 50 years | \(70 \times 50 = 3500\) (or \(0.70 \times 50 = 35\)) |
| Women | 30% (or 0.30) | 40 years | \(30 \times 40 = 1200\) (or \(0.30 \times 40 = 12\)) |
| Total Group | 100% (or 1.00) | Average Age = ? | Sum = \(3500 + 1200 = 4700\) (or \(35 + 12 = 47\)) |
Average Age = \(\frac{\text{Sum of Weighted Contributions}}{\text{Total Percentage or Proportion}}\) = \(\frac{4700}{100}\) or \(\frac{47}{1}\) = 47 years.
Revision Table: Average Age Calculation
| Concept | Explanation | Formula Used |
|---|---|---|
| Average Age | Sum of ages divided by the number of people. | \( \text{Average} = \frac{\text{Sum of Ages}}{\text{Number of People}} \) |
| Weighted Average | Average considering the relative importance (weights) of different values. | \( \text{Weighted Avg} = \frac{\sum (\text{Weight} \times \text{Value})}{\sum \text{Weight}} \) |
| This Problem | Finding the overall average age of a group from subgroup averages and proportions. | \( A_{group} = \frac{(P_m \times A_m) + (P_w \times A_w)}{P_m + P_w} \) |
Additional Information: Applications of Weighted Average
Weighted averages are commonly used in various fields, not just for calculating average age. Here are a few examples:
- Academic Grades: Calculating a final grade where different assignments (tests, homework, quizzes) have different weights.
- Finance: Calculating the average cost of stocks purchased at different prices, or portfolio returns based on the proportion invested in each asset.
- Statistics: Calculating population parameters from sampled data where different strata have different sizes.
Understanding weighted averages helps in accurately representing the central tendency of data points when their contributions are not equal.
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?
