Question

The average marks of the girls in a class is equal to the number of boys and the average marks of boys is equal to the number of girls. If the class average is 4 less than the average of both the boys' and the girls'average marks, what will be the number of students in the class?
1) 24
2) 48
3) 50
4) 64
5) None of these​

Answer 1

Concept:

The outcome of adding two or more numbers together and dividing the result by the number of numbers you added together is an average.

Given:

The average marks of girls in a class is equal to the number of boys.

The average marks of boys is equal to the number of girls.

The class average is 4 less than the average of both boys' and girls' average marks.

Find:

The number of students in class.

Solution:

Let x be the average marks of boys and hence the number of girls will be also x.

Let y be the average marks of girls and hence the number of boys will also be y.

Therefore,

The class average is $\frac{x+y}{2}$ .

The average marks of both the boys and girls is:

$=\frac{xy+xy}{x+y}\\=\frac{2xy}{x+y}$

Now, the class average is 4 less than the average marks of both boys and girls.

So,

$\frac{x+y}{2}-4 =\frac{2xy}{x+y}$

$\frac{x+y-8}{2}=\frac{2xy}{x+y}$

$(x+y-8)(x+y)=4xy\\x^2+xy+xy+y^2-8x-8y=4xy\\x^2+y^2-2xy-8x-8y=0\\(x-y)^2=8(x+y)$

Now, x-y is an integer so 8(x+y) should also be a perfect square.

Therefore, when x+y=50 we get 8(50) = 400, which makes the perfect square.

Therefore, the number of students in the class is 50.

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