The total expenses of a trip for certain number of people is ₹ 18,000. If three more people join them, then the share of each reduces by ₹ 3,000. Take x to be the original number of people, form a quadratic equation in x and solve it to find the value of x.
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Let no. of people be x.
Total expense = ₹ 18,000
Expense per person = ₹ $\frac{18000}{x}$
Given,
If three more people join them, then the share of each reduces by ₹ 3,000.
No, of people now = x + 3
Expense per person = ₹ $\frac{18000}{x + 3}$
According to question,
$$\begin{aligned}\Rightarrow \frac{18000}{x} - \frac{18000}{x + 3} = 3000 \\[1em] \Rightarrow \frac{18000(x + 3) - 18000x}{x(x + 3)} = 3000 \\[1em] \Rightarrow \frac{18000x + 54000 - 18000x}{x^2 + 3x} = 3000 \\[1em] \Rightarrow \frac{54000}{x^2 + 3x} = 3000 \\[1em] \Rightarrow x^2 + 3x = \frac{54000}{3000} \\[1em] \Rightarrow x^2 + 3x = 18 \\[1em] \Rightarrow x^2 + 3x - 18 = 0 \\[1em] \Rightarrow x^2 + 6x - 3x - 18 = 0 \\[1em] \Rightarrow x(x + 6) - 3(x + 6) = 0 \\[1em] \Rightarrow (x - 3)(x + 6) = 0 \\[1em] \Rightarrow x - 3 = 0 \text{ or } x + 6 = 0 \\[1em] \Rightarrow x = 3 \text{ or } x = -6.\end{aligned}$$
Since, no. of people cannot be negative.
∴ x = 3.
Hence, original number of people = 3.