Rupees 9000 were divided equally among a certain number of persons. Had therebeen 20 more persons each would have got Rupees 160 less. Find the originalnumber of persons.

Let's solve this step by step:

Step 1: Let's assume the original number of persons is 'x'. So, each person gets Rs. 9000/x.

Step 2: If there were 20 more persons, the number of persons becomes 'x+20'. Now, each person gets Rs. 9000/(x+20).

Step 3: According to the problem, Rs. 9000/x - Rs. 9000/(x+20) = Rs. 160.

Step 4: Simplify the equation to solve for 'x'.

Step 5: Multiply the whole equation by 'x*(x+20)' to get rid of the denominators. You will get 9000*(x+20) - 9000x = 160x*(x+20).

Step 6: Simplify further to get 180000 - 9000x = 160x^2 + 3200x.

Step 7: Rearrange the terms to form a quadratic equation 160x^2 + 3200x - 180000 = 0.

Step 8: Divide the whole equation by 160 to simplify it. You will get x^2 + 20x - 1125 = 0.

Step 9: This is a quadratic equation in the form of ax^2 + bx + c = 0. Solve for 'x' using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a.

Step 10: Substituting the values, we get x = [-20 ± sqrt((20)^2 - 4*(-1125))] / 2*1.

Step 11: Simplifying further, we get x = [-20 ± sqrt(400 + 4500)] / 2.

Step 12: Simplifying further, we get x = [-20 ± sqrt(4900)] / 2.

Step 13: Simplifying further, we get x = [-20 ± 70] / 2.

Step 14: We get two solutions x = 25 and x = -45.

Step 15: Since the number of persons cannot be negative, the original number of persons is 25.