Question

Ranjana wants to distribute 540 oranges among some students. If 30 students were more each would get 3 oranges less. Find the number of students.

Answer 1

Total oranges = 540
Let the total number of students be x.
Orange each student gets will be $\frac{540}{x}$
If 30 more students are there so, the total number of students will be x + 30
Number of oranges per person will be $\frac{540}{x+30}$.
If 30 students were more each would get 3 oranges less = $\frac{540}{x}-3$
$\frac{540}{x+30}$ = $\frac{540}{x}-3$
$$\begin{gathered} \Rightarrow 3=540\left(\frac{1}{x}-\frac{1}{x+30}\right) \\ \Rightarrow 1=180\left(\frac{x+30-x}{x\left(x+30\right)}\right) \\ \Rightarrow 1=180\left(\frac{30}{x^2+30x}\right) \\ \Rightarrow x^2+30x=5400 \\ \Rightarrow x^2+30x-5400=0 \\ \Rightarrow x^2+90x-60x-5400=0 \\ \Rightarrow x\left(x+90\right)-60\left(x+90\right)=0 \\ \Rightarrow \left(x-60\right)\left(x+90\right)=0 \\ \Rightarrow x=60,-90 \end{gathered}$$
But number of students cannot be negative so, the number of students is 60.