Question

In a mathematics test, the average marks of boy is x% and the average marks of girls is y% with $x \ne y$. If the average marks of all students is z%, the ratio of the number of girls to the total number of students is?
$\left( A \right)\dfrac{{z - x}}{{y - x}}$
$\left( B \right)\dfrac{{z - y}}{{y - x}}$
$\left( C \right)\dfrac{{z + y}}{{y - x}}$
$\left( D \right)\dfrac{{z + x}}{{y - x}}$

Answer 1

Hint – In this particular question use the concept that the percentage of average marks of the boys is the ratio sum of individual percentage of marks obtained by the boys to the total number of boys, and the percentage of average marks of the girls is the ratio sum of individual percentage of marks obtained by the girls to the total number of girls so use these concepts to reach the solution of the question.

Complete step by step solution:
Le the total number of students in the class = P.
Let there be M number of boys and F number of girls in the class.
Therefore, total number of students = number of boys + number of girls
Therefore, P = M + F
Now it is given that in a mathematics test, the average mark of a boy is x%.
As we all know that the percentage of average marks of the boys is the ratio sum of individual percentage of marks obtained by the boys to the total number of boys.
Let sum of individual percentage of marks obtained by the boys = b%
Now construct the linear equation according to this information we have,
Therefore, x% = \[\dfrac{{{\text{b% }}}}{{{\text{total number of boys}}}}\]
Therefore, x% = \[\dfrac{{{\text{b% }}}}{{\text{M}}}\]
Therefore, x = (b/M)
Therefore, b = M (x)....... (1)
Now it is also given that the average marks of girls is y%.
As we all know that the percentage of average marks of the girls is the ratio sum of individual percentage of marks obtained by the girls to the total number of girls.
Let sum of individual percentage of marks obtained by the girls = g%
Now construct the linear equation according to this information we have,
Therefore, y% = \[\dfrac{{{\text{g% }}}}{{{\text{total number of girls}}}}\]
Therefore, y% = \[\dfrac{{{\text{g% }}}}{F}\]
Therefore, g = F(y)....... (2)
Now add these two equations we have,
$ \Rightarrow b + g = Mx + Fy$................. (3)
Now it is also given that the total average percentage of marks obtained is z%
$ \Rightarrow {\text{z% }} = \dfrac{{{\text{b% + g% }}}}{{M + F}}$
\[ \Rightarrow z = \dfrac{{b + g}}{{M + F}}\]
Now from equation (3) we have,
\[ \Rightarrow z = \dfrac{{Mx + Fy}}{{M + F}}\]
Now we know that P = M + F
Therefore, M = P – F
\[ \Rightarrow z = \dfrac{{\left( {P - F} \right)x + Fy}}{P}\]
Now simplify this we have,
$ \Rightarrow zP = Px - Fx + Fy$
$ \Rightarrow P\left( {z - x} \right) = F\left( {y - x} \right)$
So the ratio of the total number of girls (F) to the total number of students (P) is
$ \Rightarrow \dfrac{F}{P} = \dfrac{{z - x}}{{y - x}}$
So this is the required ratio.
Hence option (A) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that the total percentage of average marks obtained in the class is the ratio of the sum of individual percentage marks obtained in the class to the total number of students in the class (${\text{z% }} = \dfrac{{{\text{b% + g% }}}}{{M + F}}$), so first find out the sum of individual total percentage of marks in terms of given variable as above then use this formula and simplify as above, we will get the required answer.