In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If average marks of the complete class is 72, then what is the average marks of the girls?
65
Calculating Average Marks of Girls
This problem involves calculating the average marks of a specific group (girls) within a larger group (the class), given the total class size, the average marks of the entire class, and the count and average marks of another subgroup (boys).
The key concept here is the relationship between total marks, the number of students, and the average marks. The formula for average is:
\[\text{Average Marks} = \frac{\text{Total Marks}}{\text{Number of Students}}\]
From this, we can derive:
\[\text{Total Marks} = \text{Average Marks} \times \text{Number of Students}\]
Step 1: Find the Total Marks of the Boys
We are given the number of boys and their average marks.
- Number of boys = 70
- Average marks of boys = 75
Using the formula for total marks:
\[\text{Total Marks of Boys} = \text{Average Marks of Boys} \times \text{Number of Boys}\]
\[\text{Total Marks of Boys} = 75 \times 70\]
\[\text{Total Marks of Boys} = 5250\]
So, the total marks scored by all the boys in the class is 5250.
Step 2: Find the Total Marks of the Complete Class
We are given the total number of students in the class and the average marks of the complete class.
- Total number of students in class = 100
- Average marks of the complete class = 72
Using the formula for total marks:
\[\text{Total Marks of Class} = \text{Average Marks of Class} \times \text{Total Number of Students}\]
\[\text{Total Marks of Class} = 72 \times 100\]
\[\text{Total Marks of Class} = 7200\]
The total marks scored by all 100 students in the class is 7200.
Step 3: Find the Total Marks of the Girls
The total marks of the class is the sum of the total marks of the boys and the total marks of the girls.
\[\text{Total Marks of Class} = \text{Total Marks of Boys} + \text{Total Marks of Girls}\]
We can rearrange this to find the total marks of the girls:
\[\text{Total Marks of Girls} = \text{Total Marks of Class} - \text{Total Marks of Boys}\]
Using the values we calculated in the previous steps:
\[\text{Total Marks of Girls} = 7200 - 5250\]
\[\text{Total Marks of Girls} = 1950\]
The total marks scored by all the girls is 1950.
Step 4: Find the Number of Girls
The total number of students in the class is the sum of the number of boys and the number of girls.
\[\text{Total Students} = \text{Number of Boys} + \text{Number of Girls}\]
We can rearrange this to find the number of girls:
\[\text{Number of Girls} = \text{Total Students} - \text{Number of Boys}\]
Using the given values:
\[\text{Number of Girls} = 100 - 70\]
\[\text{Number of Girls} = 30\]
There are 30 girls in the class.
Step 5: Calculate the Average Marks of the Girls
Now we have the total marks of the girls and the number of girls. We can use the average formula:
\[\text{Average Marks of Girls} = \frac{\text{Total Marks of Girls}}{\text{Number of Girls}}\]
Using the values calculated in Steps 3 and 4:
\[\text{Average Marks of Girls} = \frac{1950}{30}\]
\[\text{Average Marks of Girls} = 65\]
The average marks of the girls in the subject is 65.
| Group | Number of Students | Average Marks | Total Marks |
|---|---|---|---|
| Boys | 70 | 75 | \(70 \times 75 = 5250\) |
| Girls | \(100 - 70 = 30\) | ? | \(7200 - 5250 = 1950\) |
| Complete Class | 100 | 72 | \(100 \times 72 = 7200\) |
Using the values from the table for girls:
\[\text{Average Marks of Girls} = \frac{\text{Total Marks of Girls}}{\text{Number of Girls}} = \frac{1950}{30} = 65\]
Revision Table: Key Formulas for Average
| Concept | Formula |
|---|---|
| Average | \[\text{Average} = \frac{\text{Sum of all values}}{\text{Count of values}}\] |
| Sum of values (Total) | \[\text{Sum} = \text{Average} \times \text{Count}\] |
| Count | \[\text{Count} = \frac{\text{Sum of all values}}{\text{Average}}\] |
| Combined Average | \[\text{Combined Average} = \frac{(\text{Avg}_1 \times \text{Count}_1) + (\text{Avg}_2 \times \text{Count}_2) + ...}{\text{Count}_1 + \text{Count}_2 + ...}\] |
Additional Information on Calculating Averages
The average, also known as the mean, is a fundamental concept in statistics used to represent a typical value in a set of numbers. It is calculated by summing all the values in a set and then dividing by the number of values.
In this problem, we used the concept of weighted average implicitly. The class average is a weighted average of the boys' average and the girls' average, weighted by the number of boys and girls respectively. The formula for the class average could be written as:
\[\text{Class Average} = \frac{(\text{Boys Average} \times \text{Number of Boys}) + (\text{Girls Average} \times \text{Number of Girls})}{\text{Number of Boys} + \text{Number of Girls}}\]
Plugging in the known values:
\[72 = \frac{(75 \times 70) + (\text{Girls Average} \times 30)}{100}\]
\[72 \times 100 = (75 \times 70) + (\text{Girls Average} \times 30)\]
\[7200 = 5250 + (\text{Girls Average} \times 30)\]
\[7200 - 5250 = \text{Girls Average} \times 30\]
\[1950 = \text{Girls Average} \times 30\]
\[\text{Girls Average} = \frac{1950}{30} = 65\]
This confirms the result obtained through the step-by-step calculation of total marks. Understanding how to calculate total sums from averages and vice versa is crucial for solving such problems efficiently.
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