Question

In a class of 60 students, each boy contributed rupees equal to the number of girls and each girl contributed rupees equal to the number of boys. If the total money then collected was Rs.1600. How many boys were there in the class?

Answer 1

Hint: In this question total number of students are given and also their contributions are given so by taking the number of boys and girls as unknown variables we will get two equations and then by further solving those equations we will get the number of boys and the girls.

Complete step-by-step answer:
Let,
The number of boys in the class be x
Number of girls in the class be y
Now since there are a total of 60 students in the class so we can write
\[x + y = 60 - - (i)\]
Now it is said that each boy contributed sum of money which is equal to the number of girls hence the contribution from the boys will be \[ = x \times y - - (ii)\]
Also each girl contributed money which is equal to the number of boys in the class hence the contribution from the girls will be \[ = y \times x - - (iii)\]
Now since the total contribution from the class is of Rs.1600 hence we can write
\[x \times y + y \times x = 1600 - - (iv)\]
By further solving this equation we can write
\[
\Rightarrow xy + yx = 1600 \\
\Rightarrow 2xy = 1600 - - (v) \;
 \]
Now since \[x + y = 60\]from equation (i), hence we can write equation (v) as
\[2x\left( {60 - x} \right) = 1600\]
Hence by further solving the equation we get
\[
\Rightarrow 120x - 2{x^2} = 1600 \\
\Rightarrow 2{x^2} - 120x + 1600 = 0 \\
\Rightarrow {x^2} - 60x + 800 = 0 \;
 \]
Now solve the quadratic equation to find the value of x, we get
\[
\Rightarrow {x^2} - 60x + 800 = 0 \\
  {x^2} - 40x - 20x + 800 = 0 \\
  x\left( {x - 40} \right) - 20\left( {x - 40} \right) = 0 \\
  \left( {x - 20} \right)\left( {x - 40} \right) = 0 \;
 \]
Hence we get the number of boys 20 or 40 and number of girls 40 or 20

Note: Students must note that in this question if the number of boys will be 20 then the number of girls will be 40 and if the number of boys is 40 then the number of girls will be 20 since the total number of students in the class is 60.