In a class of 39 students, there are 26 girls with average weight 42 kg. If the average weight of the whole class is 48 kg, then what is the average weight of the boys in the class?
(c) 60 kg
Finding the Average Weight of Boys in a Class
This problem asks us to calculate the average weight of the boys in a class, given the total number of students, the number of girls, the average weight of the girls, and the average weight of the entire class. We need to use the concept of averages and total values to solve this.
The average weight is calculated by dividing the total weight by the number of individuals.
\(\text{Average Weight} = \frac{\text{Total Weight}}{\text{Number of Individuals}}\)
From this, we can find the Total Weight:
\(\text{Total Weight} = \text{Average Weight} \times \text{Number of Individuals}\)
Step-by-Step Calculation
Here's how we can solve the problem:
- Identify the given information.
- Calculate the number of boys in the class.
- Calculate the total weight of the girls.
- Calculate the total weight of the entire class.
- Calculate the total weight of the boys.
- Calculate the average weight of the boys.
Given Information:
- Total number of students in the class = 39
- Number of girls = 26
- Average weight of girls = 42 kg
- Average weight of the whole class = 48 kg
1. Calculate the Number of Boys
The number of boys is the total number of students minus the number of girls.
\(\text{Number of Boys} = \text{Total Students} - \text{Number of Girls}\)
\(\text{Number of Boys} = 39 - 26 = 13\)
So, there are 13 boys in the class.
2. Calculate the Total Weight of Girls
Using the formula \(\text{Total Weight} = \text{Average Weight} \times \text{Number of Individuals}\):
\(\text{Total Weight of Girls} = \text{Average Weight of Girls} \times \text{Number of Girls}\)
\(\text{Total Weight of Girls} = 42 \, \text{kg/girl} \times 26 \, \text{girls}\)
\(\text{Total Weight of Girls} = 1092 \, \text{kg}\)
3. Calculate the Total Weight of the Whole Class
Using the formula \(\text{Total Weight} = \text{Average Weight} \times \text{Number of Individuals}\):
\(\text{Total Weight of Class} = \text{Average Weight of Class} \times \text{Total Students}\)
\(\text{Total Weight of Class} = 48 \, \text{kg/student} \times 39 \, \text{students}\)
\(\text{Total Weight of Class} = 1872 \, \text{kg}\)
4. Calculate the Total Weight of Boys
The total weight of the class is the sum of the total weight of girls and the total weight of boys.
\(\text{Total Weight of Class} = \text{Total Weight of Girls} + \text{Total Weight of Boys}\)
So, the total weight of boys is the total weight of the class minus the total weight of girls.
\(\text{Total Weight of Boys} = \text{Total Weight of Class} - \text{Total Weight of Girls}\)
\(\text{Total Weight of Boys} = 1872 \, \text{kg} - 1092 \, \text{kg}\)
\(\text{Total Weight of Boys} = 780 \, \text{kg}\)
5. Calculate the Average Weight of Boys
Now, we use the formula \(\text{Average Weight} = \frac{\text{Total Weight}}{\text{Number of Individuals}}\) for the boys.
\(\text{Average Weight of Boys} = \frac{\text{Total Weight of Boys}}{\text{Number of Boys}}\)
\(\text{Average Weight of Boys} = \frac{780 \, \text{kg}}{13}\)
\(\text{Average Weight of Boys} = 60 \, \text{kg}\)
Thus, the average weight of the boys in the class is 60 kg.
| Item | Number/Average | Total Weight (kg) |
|---|---|---|
| Girls | 26 (Number), 42 kg (Average) | \(26 \times 42 = 1092\) |
| Class | 39 (Number), 48 kg (Average) | \(39 \times 48 = 1872\) |
| Boys | \(39 - 26 = 13\) (Number), ? kg (Average) | \(1872 - 1092 = 780\) |
| Boys (Average) | 13 (Number) | \(780 \div 13 = 60\) |
The calculated average weight of the boys is 60 kg.
Average Weight Problem Revision Table
Reviewing the key information and calculation steps helps reinforce understanding of average weight problems.
- Average = Sum of values / Number of values
- Total Sum = Average \(\times\) Number of values
- In this problem, we treated the class as composed of two groups: girls and boys.
- The total weight of the class is the sum of the total weight of girls and total weight of boys.
- We used the total weight and number of boys to find the average weight of boys.
Additional Information on Averages
The concept of average, or arithmetic mean, is widely used in statistics and daily life to represent a typical value in a set of numbers. For grouped data, like in this problem where we have subgroups (girls and boys) within a larger group (the class), the overall average is a weighted average of the subgroup averages.
Let \(n_g\) be the number of girls, \(A_g\) their average weight, \(n_b\) the number of boys, \(A_b\) their average weight, \(N\) the total number of students, and \(A_c\) the class average weight.
Total weight of girls = \(n_g \times A_g\)
Total weight of boys = \(n_b \times A_b\)
Total weight of class = \(N \times A_c\)
We know \(N = n_g + n_b\).
Also, \(N \times A_c = (n_g \times A_g) + (n_b \times A_b)\).
In our case, \(N=39\), \(n_g=26\), \(A_g=42\), \(A_c=48\). We found \(n_b = 13\). We need to find \(A_b\).
\(39 \times 48 = (26 \times 42) + (13 \times A_b)\)
\(1872 = 1092 + 13 \times A_b\)
\(1872 - 1092 = 13 \times A_b\)
\(780 = 13 \times A_b\)
\(A_b = \frac{780}{13} = 60\)
This confirms our step-by-step calculation using the definition of average and total weight.
Important Questions from Average
The average salary of employees of a factory is ₹15,000. The average salary of 250 of the employees is ₹16,000 and that of the remaining employees is ₹13,750. Total number of employees in the factory is:
Find the mean of prime numbers between 1 and 30.
If mode and mean of a data are 18 and 21 respectively, then median of the data is
The average of the numbers 5, 3, 9, 11, 29 and (p+1) is 12. The average increases by 2 when the numbers (q – 5) and (q +11) are also included. Find the value of q.
In a particular week, the average earning per day of a plumber from Monday to Wednesday remained ₹580 and from Thursday to Saturday it was ₹612. If the average earning for the whole week was ₹675, then how much did he earn on Sunday?
