Animated Solution for Mathematics - Statistics: The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

Imagine you are standing in a classroom, looking at two distinct groups: the boys and the girls. We are given their average marks: $52$ for the boys and $42$ for the girls.

Now, visualize a number line. Place the boys' average at $52$ on the right and the girls' average at $42$ on the left. The combined average of the class is $50$.

Notice something fascinating? The value $50$ is sitting much closer to $52$ than it is to $42$.

In physics, this is exactly like a center of mass problem. If you were balancing a rod with weights at $42$ and $52$, and the balance point (the fulcrum) was at $50$, you would intuitively know that the weight at $52$ must be much heavier to pull the balance point toward itself.

This is our first, powerful clue: there must be more boys than girls in this class.

The Mathematical Engine

To turn this intuition into a rigorous proof, we need the weighted average formula. It is the bedrock of statistics:

Here, $n_1$ is the number of boys ($x$), ${\bar{x}}_1$ is their average ($52$), $n_2$ is the number of girls ($y$), and ${\bar{x}}_2$ is their average ($42$).

By substituting our known values, we get the equation:

This equation is the bridge between our word problem and the solution.

Solving for the Ratio

Now, let us perform the algebra with precision. We start by multiplying both sides by $(x+y)$ to clear the denominator:

Next, we expand the left side:

Now, we group the like terms. Let us bring all the $y$ terms to the left and all the $x$ terms to the right:

This simplifies beautifully to $8y=2x$. We are looking for the ratio of boys to girls, which is $\frac{x}{y}$.

Rearranging our equation, we find:

This ratio, $4:1$, tells us that for every $4$ boys, there is $1$ girl.

Calculating the Percentage

We have the ratio, but the question asks for the percentage of boys. The total number of parts in the class is $4+1=5$.

The boys represent $4$ of these $5$ parts. To find the percentage, we calculate:

We have arrived at our destination: the percentage of boys in the class is $80\%$.

This confirms our initial intuition from the number line—the boys are indeed the vast majority, pulling the average heavily toward their score of $52$. You have successfully navigated the logic of weighted averages, a concept that will serve you well in everything from chemistry mixtures to physics center-of-mass problems.