Key Concepts and Formulas

• Definition of Average (Arithmetic Mean): The average of a set of values is the sum of all values divided by the total number of values. $\text{Average} = \frac{\text{Sum of all values}}{\text{Total number of values}}$
• Calculating Total Sum from Average: This formula can be rearranged to find the total sum of values if the average and the number of values are known. $\text{Sum of all values} = \text{Average} \times \text{Total number of values}$
• Combined Averages (Weighted Average): When a larger group is composed of distinct subgroups, the total sum of the larger group is the sum of the total sums of its subgroups. If group 1 has $N_1$ members with average $A_1$, and group 2 has $N_2$ members with average $A_2$, then the combined average $A_{total}$ is given by: $A_{total} = \frac{N_1 A_1 + N_2 A_2}{N_1 + N_2}$

Step-by-Step Solution

Let's systematically break down the problem to find the average marks of the girls.

Step 1: Determine the number of girls in the class.

• Why this step? To calculate the average marks for the girls, we first need to know how many girls are in the class. The total class population is the sum of boys and girls.
• Given: Total number of students in the class = 100
• Given: Number of boys = 70
• We subtract the number of boys from the total number of students to find the number of girls: $\text{Number of girls} = \text{Total students} - \text{Number of boys}$ $\text{Number of girls} = 100 - 70 = 30$ So, there are 30 girls in the class.

Step 2: Calculate the total marks of the entire class.

• Why this step? We are given the average marks for the complete class. Using the formula $\text{Sum} = \text{Average} \times \text{Number}$, we can find the total marks obtained by all 100 students. This total represents the combined performance of both boys and girls.
• Given: Average marks of the complete class (100 students) = 72
• Using the formula: $\text{Total marks of 100 students} = \text{Average marks of class} \times \text{Total number of students}$ $\text{Total marks of 100 students} = 72 \times 100 = 7200$

Step 3: Calculate the total marks of the boys.

• Why this step? We are provided with the average marks for the boys. Similar to Step 2, we can use the formula $\text{Sum} = \text{Average} \times \text{Number}$ to find the total marks contributed by the boys. This value is essential because it allows us to isolate the girls' marks by subtracting the boys' contribution from the total class marks.
• Given: Average marks of boys (70 boys) = 75
• Using the formula: $\text{Total marks of 70 boys} = \text{Average marks of boys} \times \text{Number of boys}$ $\text{Total marks of 70 boys} = 75 \times 70 = 5250$

Step 4: Calculate the total marks of the girls.

• Why this step? We now have the total marks for the entire class (from Step 2) and the total marks for the boys (from Step 3). Since the class consists only of boys and girls, the total marks obtained by the girls can be found by subtracting the boys' total marks from the overall class's total marks.
• $\text{Total marks of 30 girls} = \text{Total marks of 100 students} - \text{Total marks of 70 boys}$ $\text{Total marks of 30 girls} = 7200 - 5250 = 1950$

Step 5: Calculate the average marks of the girls.

• Why this step? We have successfully determined the total marks obtained by the girls (from Step 4) and we know the number of girls (from Step 1). Now, we apply the original definition of average to find their average marks.
• Using the formula: $\text{Average marks of 30 girls} = \frac{\text{Total marks of 30 girls}}{\text{Number of girls}}$ $\text{Average marks of 30 girls} = \frac{1950}{30}$ $\text{Average marks of 30 girls} = 65$

Common Mistakes & Tips

• Master the Core Formula: Always remember the relationship: $\text{Sum} = \text{Average} \times \text{Number}$. This is the fundamental tool for solving most average problems, especially those involving combined groups or subgroups.
• Systematic Approach: Break down the problem into smaller, manageable steps. First, identify the size of each group. Then, calculate total sums for known groups. Finally, use these sums to find the unknown values.
• Weighted Average Shortcut: For competitive exams, proficiency with the weighted average formula can save time. Let $A_g$ be the average marks of the girls: $A_{total} = \frac{N_{boys} A_{boys} + N_{girls} A_{girls}}{N_{total}}$ $72 = \frac{70 \times 75 + 30 \times A_g}{100}$ $7200 = 5250 + 30 A_g$ $30 A_g = 7200 - 5250$ $30 A_g = 1950$ $A_g = \frac{1950}{30} = 65$ This method combines several steps into one equation.
• Careful Calculations: Arithmetic errors are a common pitfall. Double-check your multiplication, subtraction, and division to avoid losing marks on simple mistakes.

Summary

This problem is a classic application of the concept of averages, particularly involving subgroups within a larger group. The approach involves using the definition of average to calculate the total marks for the entire class and for the known subgroup (boys). By subtracting the boys' total marks from the total class marks, we find the total marks for the girls. Finally, dividing the girls' total marks by the number of girls gives their average marks. This systematic method, or the equivalent weighted average formula, ensures an accurate solution.

The final answer is $\boxed{\text{65}}$, which corresponds to option (B).