Finding the Ratio of Boys to Girls by Average Age

This problem involves calculating the ratio of the number of boys to the number of girls in a school, given their individual average ages and the combined average age of all students.

Understanding the Problem: Average Age Calculation

The average age of a group is calculated by dividing the sum of the ages of all members in the group by the total number of members. When dealing with two different groups combined, the overall average age is a weighted average based on the number of individuals in each group.

Let's denote:

• $B$ = Number of boys
• $G$ = Number of girls
• $A_B$ = Average age of boys = 15.4 years
• $A_G$ = Average age of girls = 18.2 years
• $A_T$ = Combined average age of boys and girls = 16.8 years

The total age of all boys is $B \times A_B$.

The total age of all girls is $G \times A_G$.

The total age of all students (boys and girls) is $(B \times A_B) + (G \times A_G)$.

The total number of students is $B + G$.

The combined average age $A_T$ is given by the formula:

$$A_T = \frac{(B \times A_B) + (G \times A_G)}{B + G}$$

Step-by-Step Solution using Weighted Average Formula

Substitute the given values into the formula:

$$16.8 = \frac{(B \times 15.4) + (G \times 18.2)}{B + G}$$

Multiply both sides by $(B + G)$ to remove the denominator:

$$16.8 \times (B + G) = (B \times 15.4) + (G \times 18.2)$$

$$16.8B + 16.8G = 15.4B + 18.2G$$

Now, rearrange the equation to group terms involving $B$ and terms involving $G$:

Subtract $15.4B$ from both sides:

$$16.8B - 15.4B + 16.8G = 18.2G$$

$$1.4B + 16.8G = 18.2G$$

Subtract $16.8G$ from both sides:

$$1.4B = 18.2G - 16.8G$$

$$1.4B = 1.4G$$

To find the ratio $B : G$, divide both sides by $G$ and by 1.4:

$$\frac{B}{G} = \frac{1.4}{1.4}$$

$$\frac{B}{G} = 1$$

This can be written as:

$$\frac{B}{G} = \frac{1}{1}$$

So, the ratio of the number of boys to the number of girls is $1 : 1$.

Alternative Method: Using Alligation

The alligation method can also be used for mixture problems like this. We place the individual averages and the combined average as follows:

According to the alligation rule, the ratio of the quantity of the first item (boys) to the quantity of the second item (girls) is the ratio of the diagonal differences.

Ratio of Boys : Girls = (Difference related to Girls) : (Difference related to Boys)

Ratio of Boys : Girls = $(18.2 - 16.8) : (16.8 - 15.4)$

Ratio of Boys : Girls = $1.4 : 1.4$

Divide both parts of the ratio by 1.4:

Ratio of Boys : Girls = $\frac{1.4}{1.4} : \frac{1.4}{1.4}$

Ratio of Boys : Girls = $1 : 1$

Both methods yield the same result. The ratio of the number of boys to girls in the school is $1 : 1$.

Ratio of Boys to Girls Summary

By using either the weighted average formula or the alligation method, we found that the number of boys is equal to the number of girls, resulting in a ratio of $1 : 1$.

Revision Table: Average Age and Ratio Concepts

Additional Information: Weighted Averages in Real Life

Weighted averages are used in many real-life situations beyond school ages:

• Calculating GPA: Each course grade is weighted by the number of credit hours.
• Financial Portfolios: The average return of a portfolio is the weighted average of the returns of individual assets, weighted by the amount invested in each.
• Mixing Solutions: Calculating the final concentration when mixing solutions of different concentrations and volumes.
• Population Statistics: Finding the average income or average age for a region composed of different demographic groups.

Understanding weighted averages is crucial for analyzing data where different components contribute unequally to the overall value.