The mean weight of 100 students in a class is 46 kg. The mean weight of boys is 50 kg and that of girls is 40 kg. The number of boys exceeds the number of girls by
20
Calculating the Difference in Number of Boys and Girls by Mean Weight
The problem provides us with the total number of students, the overall mean weight, and the mean weights specifically for boys and girls. We need to find the difference between the number of boys and the number of girls in the class.
Let's break down the information given:
- Total number of students = 100
- Overall mean weight of 100 students = 46 kg
- Mean weight of boys = 50 kg
- Mean weight of girls = 40 kg
We can use this information to set up equations and solve for the number of boys and girls.
Setting Up the Equations
Let:
- \(B\) = Number of boys
- \(G\) = Number of girls
From the total number of students, we get the first equation:
\(\qquad B + G = 100 \quad (1)\)
The total weight of all students is the sum of the total weight of boys and the total weight of girls.
Total weight of all students = Overall mean weight \(\times\) Total number of students
Total weight of all students = \(46 \times 100 = 4600\) kg
Total weight of boys = Mean weight of boys \(\times\) Number of boys
Total weight of boys = \(50 \times B = 50B\) kg
Total weight of girls = Mean weight of girls \(\times\) Number of girls
Total weight of girls = \(40 \times G = 40G\) kg
From the total weight information, we get the second equation:
\(\qquad 50B + 40G = 4600 \quad (2)\)
We now have a system of two linear equations with two variables (\(B\) and \(G\)).
Solving the System of Equations
We can use substitution or elimination method to solve these equations. Let's use the substitution method.
From equation (1), we can express \(B\) in terms of \(G\):
\(\qquad B = 100 - G\)
Now, substitute this expression for \(B\) into equation (2):
\(\qquad 50(100 - G) + 40G = 4600\)
Distribute the 50:
\(\qquad 5000 - 50G + 40G = 4600\)
Combine the terms with \(G\):
\(\qquad 5000 - 10G = 4600\)
Subtract 5000 from both sides:
\(\qquad -10G = 4600 - 5000\)
\(\qquad -10G = -400\)
Divide by -10 to find \(G\):
\(\qquad G = \frac{-400}{-10}\)
\(\qquad G = 40\)
So, the number of girls is 40.
Now, substitute the value of \(G\) back into equation (1) to find \(B\):
\(\qquad B = 100 - G\)
\(\qquad B = 100 - 40\)
\(\qquad B = 60\)
So, the number of boys is 60.
Finding the Difference
The question asks for the number by which the number of boys exceeds the number of girls. This is the difference between the number of boys and the number of girls (\(B - G\)).
Difference = Number of boys - Number of girls
Difference = \(B - G\)
Difference = \(60 - 40\)
Difference = 20
The number of boys exceeds the number of girls by 20.
| Category | Number | Mean Weight (kg) | Total Weight (kg) |
|---|---|---|---|
| Boys | \(B=60\) | 50 | \(60 \times 50 = 3000\) |
| Girls | \(G=40\) | 40 | \(40 \times 40 = 1600\) |
| Total | \(100\) | \(3000 + 1600 = 4600\) |
The overall mean weight check: Total weight / Total students = \(4600 / 100 = 46\) kg, which matches the given information.
Summary of Steps
- Define variables for the number of boys and girls.
- Formulate two equations based on the total number of students and the total weight.
- Solve the system of equations to find the number of boys and girls.
- Calculate the difference between the number of boys and girls.
Revision Table: Mean Weight Calculation
| Concept | Formula | Application in this problem |
|---|---|---|
| Mean (Average) | Mean = \(\frac{\text{Sum of all values}}{\text{Number of values}}\) | Mean weight = \(\frac{\text{Total weight}}{\text{Total number of students}}\) |
| Total Sum from Mean | Sum = Mean \(\times\) Number of values | Total weight = Mean weight \(\times\) Number of students |
Additional Information: Weighted Average Concept
This problem can also be viewed through the lens of a weighted average. The overall mean weight is a weighted average of the mean weight of boys and girls, weighted by the number of students in each group.
Overall Mean Weight = \(\frac{(\text{Mean weight of boys} \times \text{Number of boys}) + (\text{Mean weight of girls} \times \text{Number of girls})}{\text{Total number of students}}\)
\(46 = \frac{(50 \times B) + (40 \times G)}{B + G}\)
Since \(B + G = 100\), we get:
\(46 = \frac{50B + 40G}{100}\)
\(4600 = 50B + 40G\)
This is the same second equation we derived earlier, confirming the relationship. Solving this with \(B+G=100\) gives the same result for the number of boys and girls.
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