In a class of 60 students, there are 35 boys. The average weight of boys is 42 kg and the average weight of the full class is 46.5 kg. Find the average weight of girls in the class.
52.8 kg
Calculating Average Weight of Girls in a Class
This problem involves calculating the average weight of girls in a class, given the total number of students, the number of boys, the average weight of boys, and the average weight of the entire class.
Understanding the Given Information
We are provided with the following details:
- Total number of students in the class: 60
- Number of boys in the class: 35
- Average weight of boys: 42 kg
- Average weight of the full class: 46.5 kg
Finding the Number of Girls
The first step is to find the number of girls in the class. The total number of students is the sum of the number of boys and the number of girls.
Number of girls = Total number of students - Number of boys
Number of girls \( = 60 - 35 = 25 \)
So, there are 25 girls in the class.
Calculating Total Weights
The average weight is defined as the total weight divided by the number of individuals. We can use this formula to find the total weight of boys and the total weight of the class.
Average weight \( = \frac{\text{Total Weight}}{\text{Number of Individuals}} \)
This means, Total Weight \( = \) Average weight \( \times \) Number of Individuals.
Total Weight of Boys:
Using the formula:
Total weight of boys \( = \) Average weight of boys \( \times \) Number of boys
Total weight of boys \( = 42 \, \text{kg} \times 35 \)
Total weight of boys \( = 1470 \, \text{kg} \)
Total Weight of the Class:
Using the formula:
Total weight of class \( = \) Average weight of class \( \times \) Total number of students
Total weight of class \( = 46.5 \, \text{kg} \times 60 \)
Total weight of class \( = 2790 \, \text{kg} \)
Determining the Total Weight of Girls
The total weight of the class is the sum of the total weight of boys and the total weight of girls.
Total weight of class \( = \) Total weight of boys \( + \) Total weight of girls
We can rearrange this to find the total weight of girls:
Total weight of girls \( = \) Total weight of class \( - \) Total weight of boys
Total weight of girls \( = 2790 \, \text{kg} - 1470 \, \text{kg} \)
Total weight of girls \( = 1320 \, \text{kg} \)
Calculating the Average Weight of Girls
Now that we have the total weight of girls and the number of girls, we can calculate the average weight of girls using the average weight formula:
Average weight of girls \( = \frac{\text{Total weight of girls}}{\text{Number of girls}} \)
Average weight of girls \( = \frac{1320 \, \text{kg}}{25} \)
Average weight of girls \( = 52.8 \, \text{kg} \)
Therefore, the average weight of girls in the class is 52.8 kg.
Revision Table: Class Weight Calculation
| Item | Number | Average Weight (kg) | Total Weight (kg) |
|---|---|---|---|
| Total Class | 60 | 46.5 | \(60 \times 46.5 = 2790\) |
| Boys | 35 | 42 | \(35 \times 42 = 1470\) |
| Girls | \(60 - 35 = 25\) | \(1320 / 25 = 52.8\) | \(2790 - 1470 = 1320\) |
Additional Information on Averages and Weighted Averages
The concept used here is related to averages and can be extended to weighted averages. The average is a measure of central tendency. When combining groups with different averages, the overall average is a weighted average.
- Average (Arithmetic Mean): Sum of all values divided by the number of values.
- Total Sum: The total sum of values in a group can be found by multiplying the average by the number of items in the group.
- Weighted Average: When calculating the average of combined groups, where each group has a different number of members and a different average, the overall average is influenced more by the group with more members. The formula for the weighted average of two groups (Group 1 and Group 2) is:
\[ \text{Overall Average} = \frac{(\text{Number in Group 1} \times \text{Average of Group 1}) + (\text{Number in Group 2} \times \text{Average of Group 2})}{\text{Total Number of Individuals}} \]
In this problem, if we let the average weight of girls be \(A_g\), we could set up the equation:
\[ 46.5 = \frac{(35 \times 42) + (25 \times A_g)}{60} \]
\[ 46.5 \times 60 = (35 \times 42) + (25 \times A_g) \]
\[ 2790 = 1470 + 25 A_g \]
\[ 2790 - 1470 = 25 A_g \]
\[ 1320 = 25 A_g \]
\[ A_g = \frac{1320}{25} = 52.8 \]
This confirms our step-by-step calculation results.
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