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.

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

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.