₹ 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got $₹ 30$ less. Find the original number of persons. (M.A.T., 2005)
Answer
Answer (as printed):
Explanation
Let the original number of persons be $x$. Then, $\frac{6500}{x}-\frac{6500}{(x+15)}=30 \Leftrightarrow 6500\left[\frac{x+15-x}{x(x+15)}\right]=30 \Leftrightarrow 30 x(x+15)=6500 \times 15$ $$\begin{aligned} & \Leftrightarrow x^{2}+15 x=3250 \Leftrightarrow x^{2}+15 x-3250=0 \\ & \Leftrightarrow x^{2}+65 x-50 x-3250=0 \\ & \Leftrightarrow x(x+65)-50(x+65)=0 \\ & \Leftrightarrow(x+65)(x-50)=0 \Leftrightarrow x=50 . \end{aligned}$$ Hence, original number of persons = 50.
Explanation as extracted from the printed page; notation may be imperfect.