Understanding the Average Age Problem

This question asks us to find the percentage of boys within a group, given the average ages of the boys, the girls, and the entire group combined. This is a typical weighted average problem.

We are provided with the following information:

• The average age of the entire group (boys and girls) is 18 years.
• The average age of the boys is 20 years.
• The average age of the girls is 15 years.

Calculating the Ratio of Boys to Girls

To solve this, we can set up an equation based on the definition of average age. Let's define:

• $N_B$ = the number of boys in the group
• $N_G$ = the number of girls in the group

The total age contributed by the boys is $N_B \times 20$ years.

The total age contributed by the girls is $N_G \times 15$ years.

The total age of the entire group is the sum of the ages of boys and girls: $(N_B \times 20) + (N_G \times 15)$ years.

The total number of individuals in the group is $N_B + N_G$.

The average age of the group is calculated as:

$ \text{Average Age of Group} = \frac{\text{Total Age of Group}}{\text{Total Number of People}} $

Substituting the given values:

$ 18 = \frac{(20 \times N_B) + (15 \times N_G)}{N_B + N_G} $

Now, we rearrange the equation to find the relationship (ratio) between the number of boys and girls:

$ 18 \times (N_B + N_G) = 20 N_B + 15 N_G $

Distribute the 18 on the left side:

$ 18 N_B + 18 N_G = 20 N_B + 15 N_G $

To find the ratio, we group the terms with $N_B$ on one side and $N_G$ on the other:

$ 18 N_G - 15 N_G = 20 N_B - 18 N_B $

$ 3 N_G = 2 N_B $

This gives us the ratio of boys to girls:

$ \frac{N_B}{N_G} = \frac{3}{2} $

This means the ratio of boys to girls in the group is 3:2.

Calculating the Percentage of Boys

The ratio $N_B : N_G = 3:2$ indicates that for every 3 boys, there are 2 girls. Therefore, the total number of 'parts' in the ratio is $3 + 2 = 5$.

To find the percentage of boys, we calculate the proportion of boys relative to the total number of people in the group:

$ \text{Percentage of Boys} = \left( \frac{\text{Number of Boys}}{\text{Total Number of People}} \right) \times 100 $

Using the ratio parts:

$ \text{Percentage of Boys} = \left( \frac{N_B}{N_B + N_G} \right) \times 100 $

$ \text{Percentage of Boys} = \left( \frac{3}{3 + 2} \right) \times 100 $

$ \text{Percentage of Boys} = \left( \frac{3}{5} \right) \times 100 $

$ \text{Percentage of Boys} = 0.6 \times 100 $

$ \text{Percentage of Boys} = 60\% $

Based on the calculations, the percentage of boys in the group is 60%.