A certain amount of money is equally divided between some persons .If there were 8 persons more each would have got one rupee less. The number of the person and the amount are?

let the amount of money be m

let the number of people sharing be x
let each persons share be y

then xy = m

second case:
number of people = x+8
share of each person = y-1
then m = (x+8)(y-1)
= xy + 8y - x - 8

thus:
xy+8y-x-8 = xy
x = 8y - 8 , where y > 0 , and y is a natural number

form the following table:
y x m
1 0 0 ---- not likely
2 8 16
3 16 48
4 24 96
5 32 160
... etc

let's test 2 8 16
number of people is 8
amount of money = 16
share for each person = 2
had there been 8 more people or 16 people, each would get 1, which is 1 peso less

testing 5 32 160
number of people = 32
amount of money = 160
share per person = 5
had there been 8 more people or 40 people, each would get 4, which is 1 peso less

There will be an infinite number of solutions as you can see

Thankyou!

Let's assume that the initial amount of money divided among the persons is x rupees.

If there were 8 more persons, the new number of persons would be (n + 8), where n is the initial number of persons.

In the initial scenario, each person would receive x/n rupees.

In the new scenario, each person would receive x/(n + 8) rupees.

According to the given information, if there were 8 more persons, each person would receive one rupee less.

So, we can set up the equation:

x/n - 1 = x/(n + 8)

Multiplying both sides of the equation by n(n + 8) to eliminate the denominators:

(x(n + 8))/n - n(n + 8) = x

Expanding and simplifying the equation:

(xn + 8x) - n^2 - 8n = xn

Rearranging the equation:

8x - n^2 - 8n = 0

We can solve this quadratic equation for x by factoring or using the quadratic formula. However, we need more information to determine the exact values of x, n, and the amount of money.

To solve this problem, let's break it down step by step.

Let's assume the original amount of money divided among the persons is x rupees. We are told that if there were 8 more persons, each person would receive one rupee less.

So, if there were (number of persons + 8) persons, each person would receive (x - 1) rupees.

Now, let's set up an equation using the information given:

x / n = (x - 1) / (n + 8)

Where:
x = total amount of money
n = number of persons

Now, let's solve this equation to find the values of x and n.

Cross-multiplying the equation, we get:

x(n + 8) = (x - 1)n

Expanding both sides:

xn + 8x = xn - n

Simplifying:

8x = -n

We know that the number of persons cannot be negative, so we can ignore the negative sign. Let's take the absolute value of n and substitute it back into the equation:

8x = |n|

Now, let's analyze the equation further. The right side of the equation represents the absolute value of n. Since n represents the number of persons, it cannot be negative. Therefore, the right side of the equation simplifies to just n.

So the equation becomes:

8x = n

Now we need to find a pair of values (x, n) that satisfies this equation.

Since the number of persons (n) cannot be fractional, we need to find a multiple of 8 that satisfies the equation. By trial and error, we find that when n = 8, the equation is satisfied.

Plugging in n = 8 into the equation:

8x = 8

Dividing both sides by 8, we get:

x = 1

Therefore, the total amount of money (x) is 1 rupee, and the number of persons (n) is 8.