Let the original number of persons be $x$.

Step 1: Formulate the equation

• Total amount = $9000$
• Share per person with $x$ persons: $\frac{9000}{x}$
• If there were $x + 20$ persons, share per person: $\frac{9000}{x + 20}$
• Difference in share: $\frac{9000}{x} - \frac{9000}{x + 20} = 160$

Step 2: Simplify the equation

$\frac{9000}{x} - \frac{9000}{x + 20} = 160$ Factor out $9000$: $9000 \left( \frac{1}{x} - \frac{1}{x + 20} \right) = 160$ Simplify the fraction: $9000 \cdot \frac{(x + 20) - x}{x(x + 20)} = 160$ $9000 \cdot \frac{20}{x(x + 20)} = 160$

Step 3: Solve for $x$

Divide both sides by 20: $\frac{9000}{x(x + 20)} = 8$ Multiply through by $x(x + 20)$: $9000 = 8x(x + 20)$ Expand and simplify: $9000 = 8x^2 + 160x$ Divide through by 8: $1125 = x^2 + 20x$ Rearrange into standard quadratic form: $x^2 + 20x - 1125 = 0$

Step 4: Solve the quadratic equation

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 20$, $c = -1125$: $x = \frac{-20 \pm \sqrt{20^2 - 4(1)(-1125)}}{2(1)}$ $x = \frac{-20 \pm \sqrt{400 + 4500}}{2}$ $x = \frac{-20 \pm \sqrt{4900}}{2}$ $x = \frac{-20 \pm 70}{2}$ $x = \frac{50}{2} \quad \text{or} \quad x = \frac{-90}{2}$ $x = 25 \quad \text{or} \quad x = -45$

Since $x$ must be positive, $x = 25$.

Final Answer:

The original number of persons is 25.

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Do you want a detailed explanation or further clarification?
Here are 5 related questions to explore further:

1. What are the steps to solve a quadratic equation?
2. How does dividing Rs. 9000 among people affect individual shares?
3. Can we solve this problem graphically?
4. What would happen if the additional persons were 30 instead of 20?
5. How can quadratic equations be applied in other real-life scenarios?

Tip: Always check if your quadratic roots make sense in the context of the problem!