In a class of 60 students, 20 are girls. The average weight of the boys in the class is 40 kg, while that of all the girls is 35 kg. What is the average weight (in kg) of the entire class (correct to two decimal places)?
38.33
Let's break down how to calculate the average weight of the entire class. We are given the total number of students, the number of girls, and the average weights for boys and girls separately. To find the average weight of the entire class, we need to calculate the total weight of all students and divide it by the total number of students.
Understanding the Class Composition
We are given the following information about the class of 60 students:
- Total number of students = 60
- Number of girls = 20
From this, we can find the number of boys in the class.
Number of boys = Total number of students - Number of girls
Number of boys = \(60 - 20 = 40\)
So, there are 40 boys and 20 girls in the class.
Calculating Total Weight
We are given the average weight for boys and girls:
- Average weight of boys = 40 kg
- Average weight of girls = 35 kg
The total weight of a group is calculated by multiplying the number of individuals in the group by their average weight.
Total weight of boys = Number of boys \(\times\) Average weight of boys
Total weight of boys = \(40 \times 40 \, \text{kg} = 1600 \, \text{kg}\)
Total weight of girls = Number of girls \(\times\) Average weight of girls
Total weight of girls = \(20 \times 35 \, \text{kg} = 700 \, \text{kg}\)
Now, we find the total weight of the entire class by adding the total weight of boys and the total weight of girls.
Total weight of the class = Total weight of boys + Total weight of girls
Total weight of the class = \(1600 \, \text{kg} + 700 \, \text{kg} = 2300 \, \text{kg}\)
Finding the Average Weight of the Entire Class
The average weight of the entire class is the total weight of the class divided by the total number of students in the class.
Average weight of the class = \(\frac{\text{Total weight of the class}}{\text{Total number of students}}\)
Average weight of the class = \(\frac{2300 \, \text{kg}}{60}\)
Let's perform the division:
\[ \frac{2300}{60} = \frac{230}{6} = \frac{115}{3} \]Now, we calculate the decimal value of \(\frac{115}{3}\):
\[ 115 \div 3 \approx 38.3333... \]We are asked to provide the average weight correct to two decimal places. Rounding 38.3333... to two decimal places gives 38.33 kg.
So, the average weight of the entire class is approximately 38.33 kg.
Summary of Calculations
| Detail | Value |
|---|---|
| Total Students | 60 |
| Number of Girls | 20 |
| Number of Boys | \(60 - 20 = 40\) |
| Average Weight of Boys | 40 kg |
| Average Weight of Girls | 35 kg |
| Total Weight of Boys | \(40 \times 40 = 1600\) kg |
| Total Weight of Girls | \(20 \times 35 = 700\) kg |
| Total Weight of Class | \(1600 + 700 = 2300\) kg |
| Average Weight of Class | \(\frac{2300}{60} \approx 38.33\) kg |
Final Answer on Class Average Weight
The average weight of the entire class, correct to two decimal places, is 38.33 kg.
Revision Table: Calculating Class Average Weight
| Step | Description | Formula / Calculation |
|---|---|---|
| 1 | Find the number of boys. | Total Students - Number of Girls |
| 2 | Calculate total weight of boys. | Number of Boys \(\times\) Average Weight of Boys |
| 3 | Calculate total weight of girls. | Number of Girls \(\times\) Average Weight of Girls |
| 4 | Calculate total weight of the class. | Total Weight of Boys + Total Weight of Girls |
| 5 | Calculate the average weight of the class. | Total Weight of Class / Total Students |
Additional Information: Weighted Average
This problem is an example of calculating a weighted average. When combining groups with different average values, the overall average is influenced more by the group with more members. The formula for a weighted average is:
\[ \text{Weighted Average} = \frac{\sum (\text{weight}_i \times \text{value}_i)}{\sum \text{weight}_i} \]In this case, the 'weights' are the number of students in each group (boys and girls), and the 'values' are their average weights.
Let \(N_b\) be the number of boys, \(Avg_b\) be the average weight of boys, \(N_g\) be the number of girls, and \(Avg_g\) be the average weight of girls. The total number of students is \(N_b + N_g\).
The total weight is \(N_b \times Avg_b + N_g \times Avg_g\).
The average weight of the class is:
\[ \text{Average Weight}_{\text{Class}} = \frac{(N_b \times Avg_b) + (N_g \times Avg_g)}{N_b + N_g} \]Plugging in the values:
\[ \text{Average Weight}_{\text{Class}} = \frac{(40 \times 40) + (20 \times 35)}{40 + 20} = \frac{1600 + 700}{60} = \frac{2300}{60} \approx 38.33 \]This confirms our calculation using the weighted average concept.
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