In a class of 100 students, the average weight is 30 kg. If the average weight of the girls is 24 kg and that of the boys is 32 kg, then what is the number of girls in the class?
25
Solving Class Average Weight Problems
This problem involves calculating the number of girls in a class given the total number of students, the overall average weight, and the average weights of girls and boys separately. This is a classic example of a weighted average problem that can be solved using linear equations.
Understanding the Given Information
Let's break down the information provided in the question about the class of 100 students:
- Total number of students in the class = 100
- Overall average weight of all students = 30 kg
- Average weight of the girls = 24 kg
- Average weight of the boys = 32 kg
We need to find the number of girls in the class.
Setting Up the Equations for Class Weight
Let's use variables to represent the unknown quantities:
- Let \(g\) be the number of girls in the class.
- Let \(b\) be the number of boys in the class.
Based on the total number of students, we have our first equation:
Total students = Number of girls + Number of boys
\(g + b = 100\) (Equation 1)
Now, let's consider the total weight of the students. The total weight is the overall average weight multiplied by the total number of students.
Total weight of all students = Overall average weight \(\times\) Total number of students
Total weight = \(30 \text{ kg/student} \times 100 \text{ students} = 3000 \text{ kg}\)
The total weight of the students can also be calculated as the sum of the total weight of the girls and the total weight of the boys.
Total weight of girls = Average weight of girls \(\times\) Number of girls
Total weight of girls = \(24g\)
Total weight of boys = Average weight of boys \(\times\) Number of boys
Total weight of boys = \(32b\)
So, we can set up our second equation based on the total weight:
Total weight = Total weight of girls + Total weight of boys
\(24g + 32b = 3000\) (Equation 2)
Solving the System of Equations to Find Number of Girls
We now have a system of two linear equations with two variables:
- \(g + b = 100\)
- \(24g + 32b = 3000\)
We want to find the value of \(g\). We can solve this system using substitution or elimination. Let's use substitution.
From Equation 1, we can express \(b\) in terms of \(g\):
\(b = 100 - g\)
Now substitute this expression for \(b\) into Equation 2:
\(24g + 32(100 - g) = 3000\)
Distribute the 32:
\(24g + 3200 - 32g = 3000\)
Combine the \(g\) terms:
\((24g - 32g) + 3200 = 3000\)
\(-8g + 3200 = 3000\)
Subtract 3200 from both sides:
\(-8g = 3000 - 3200\)
\(-8g = -200\)
Divide by -8 to solve for \(g\):
\(g = \frac{-200}{-8}\)
\(g = 25\)
So, the number of girls in the class is 25.
We can also find the number of boys if needed: \(b = 100 - g = 100 - 25 = 75\). Let's check if these numbers satisfy Equation 2:
\(24(25) + 32(75) = 600 + 2400 = 3000\)
This matches the total weight calculated earlier, so our values for \(g\) and \(b\) are correct.
Summary of Calculation for Number of Girls
Here is a quick look at the key values:
| Category | Average Weight (kg) | Number | Total Weight (kg) |
|---|---|---|---|
| Girls | 24 | \(g\) (to find) | \(24g\) |
| Boys | 32 | \(b\) | \(32b\) |
| Total Class | 30 | 100 (\(g+b\)) | 3000 (\(24g+32b\)) |
Using the equations \(g+b=100\) and \(24g+32b=3000\), we found \(g=25\).
Conclusion on Number of Girls
Based on the calculations using the average weights and the total number of students, the number of girls in the class is 25.
Revision Table: Class Average Weight
| Concept | Description | Formula/Relation |
|---|---|---|
| Average Weight | Total weight divided by the number of items (students). | \(\text{Average Weight} = \frac{\text{Total Weight}}{\text{Number of Items}}\) |
| Total Weight | Sum of the weights of all individual items. | \(\text{Total Weight} = \text{Average Weight} \times \text{Number of Items}\) |
| Weighted Average | An average where each item's value is multiplied by a weight before summing. In this problem, the 'weights' are the number of students in each group (girls/boys). | \(\text{Overall Avg} = \frac{(\text{Avg}_1 \times \text{Count}_1) + (\text{Avg}_2 \times \text{Count}_2)}{\text{Count}_1 + \text{Count}_2}\) |
Understanding these basic concepts is crucial for solving problems involving averages and mixtures.
Additional Information on Average Problems
Problems involving averages are common in quantitative aptitude sections of exams. They often require setting up equations based on the total sum or total value. In problems like this, where you have subgroups with different averages contributing to an overall average, you can use the concept of weighted average.
Alternative approach (Mixture and Alligation): This problem can also be solved using the rule of alligation, which is a shortcut method for problems involving mixtures of two components with different prices or averages to form a mixture with a specific average price or average value. The ratio of the number of girls to boys can be found using the differences in average weights:
- Difference between boy's average and overall average: \(32 - 30 = 2\)
- Difference between overall average and girl's average: \(30 - 24 = 6\)
The ratio of the number of girls to boys is the inverse ratio of these differences, i.e., ratio of girls to boys = \(2 : 6\), which simplifies to \(1 : 3\). This means for every 1 girl, there are 3 boys. The total parts are \(1 + 3 = 4\).
Number of girls = \(\frac{\text{Ratio part for girls}}{\text{Total ratio parts}} \times \text{Total students} = \frac{1}{4} \times 100 = 25\).
Number of boys = \(\frac{\text{Ratio part for boys}}{\text{Total ratio parts}} \times \text{Total students} = \frac{3}{4} \times 100 = 75\).
Both the equation method and the alligation method yield the same result, confirming the number of girls is 25.
Important Questions from Average
The average height of 20 students of class 8 is 152 cm and the average height of 15 students of class 9 is 168 cm. What is the average height (to the nearest cm) of the students of both classes?
The average of 4, 6, 8, 12 and x is 7 and the average of x, 9, 13, 15 and y is 9. What is the value of 2x - 3y?
The average weight of 20 girls in a school was 52 kg. Two new students of weight 54 kg and 50 kg were admitted. The ratio of this new average to the old one is:
If the average of two numbers is 13 and the square root of their product is 12, then the difference between the numbers is:
If the average of 5 consecutive odd integers in increasing order is 11 , then the average of the last 3 of them is:
