In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average

Q: Wait, why can't I just take the average of 75 and 72 to find the girls' average?

That is a classic trap! Averages are weighted by the number of students. Because there are more boys (70) than girls (30), the class average of 72 is pulled much closer to the boys' average of 75. You cannot simply average the averages; you must work with the 'Total Marks' (the sum) to account for the different group sizes.

Q: Is there a faster way to solve this without calculating such large numbers?

Yes, absolutely! You can use the 'Deviation Method' or 'Alligation'. You look at how far the boys' average is from the class average ($75 - 72 = +3$). Since there are 70 boys, the total 'surplus' is $70 \times 3 = 210$. This surplus must be balanced by the girls. If the girls' average is $x$, then $30 \times (72 - x) = 210$. Solving this gives $72 - x = 7$, so $x = 65$. It is a beautiful way to see the balance!

Q: Why do we subtract the boys' marks from the total class marks?

Think of the class as a container holding all the marks. If you remove the 'boys' portion' of the marks, what remains in the container must be the 'girls' portion'. It is the principle of conservation: the whole is equal to the sum of its parts.

Question

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Options:

(A) 73

(B) 65

(C) 68

(D) 74

Sigma Percentile · Accepted Answer

### The Classroom Ecosystem: A Balancing Act

My dear student, welcome to the fascinating world of averages. Often, when we look at a problem like this, we see numbers and immediately reach for a formula.

But I want you to pause. I want you to visualize this classroom. Imagine $100$ students sitting in a hall. Some are boys, some are girls.

They have all taken an exam, and their scores are like weights on a scale. The class average is the point where that scale perfectly balances.

### The Geometry of Averages

Let us start by defining our universe. We have $100$ students in total. We know $70$ are boys, which leaves us with $30$ girls. This is our foundation.

Now, how do we handle the 'average'? Think of the average not just as a number, but as the height of a rectangle. If the width of the rectangle is the number of students, then the area of that rectangle is the total marks scored by that group.

This is the core principle:

$$	ext{Total Marks} = 	ext{Average} 	imes 	ext{Number of Students}$$

### The Conservation of Marks

Consider the entire class as one giant rectangle. The average is $72$, and the width is $100$.

Therefore, the total area—the total marks scored by everyone—is:

$$100 	imes 72 = 7200$$

This is our 'Total Pool' of marks. Now, we know the boys' contribution. They are $70$ in number, with an average of $75$.

Their rectangle has an area of:

$$70 	imes 75 = 5250$$

This is the 'Blue Rectangle' of our classroom. The remaining marks, the 'Orange Rectangle', must belong to the girls. By the law of conservation, the girls' total marks must be the difference:

$$7200 - 5250 = 1950$$

### The Final Synthesis

We are almost there. We have the total marks for the girls ($1950$) and we know there are $30$ girls. To find their average, we simply redistribute those marks equally among them.

We calculate:

$$	ext{Average}_{	ext{girls}} = \frac{1950}{30} = 65$$

When you perform this division, you get **$65$**.

Notice the elegance here: the boys' average ($75$) is $3$ points above the class average ($72$), while the girls' average ($65$) is $7$ points below. Because there are more boys, their 'pull' on the average is stronger, which is why the class average sits closer to $75$ than to $65$.

You have just mastered the art of weighted averages. Keep this intuition, and no problem will ever be too complex for you.

Animated Solution for Mathematics - Statistics: In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Select Answer:

Visualized Solution

Visualizing the Class Split

The Area Model for Total Marks

Calculating Total Class Marks

Calculating Total Marks of Boys

Finding Total Marks of Girls

Computing Girls' Total Marks

Calculating the Girls' Average

Final Answer

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

The Classroom Ecosystem

The Geometry of Averages

The Conservation of Marks

The Final Synthesis

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