Beth & J had $2880. Beth gave 1/4 of her money to J. Then J gave 1/3 of the total amount she had to Beth. In the end, both girls had same amount of money. 1 - how much money did beth have at first? 2- how much money did J had at first?

Let's solve this step by step.

1. Let's assume the initial amount of money Beth had as "x".
After giving 1/4 of her money to J, Beth will have 3/4 of her initial amount: 3/4 * x = (3/4)x.

Therefore, after giving the money to J, Beth will have (3/4)x.

2. As per the given information, J initially received 1/4 of Beth's money.
So, J initially received (1/4)x.

Now, let's calculate the final amount of money J had after giving 1/3 of the total amount she had to Beth.
J will have 2/3 of her initial amount after giving money to Beth: 2/3 * (1/4)x = (2/12)x = (1/6)x.

From the given information, it is mentioned that both Beth and J ended up having the same amount of money, which means (3/4)x = (1/6)x.

Now, we can set up an equation and solve for "x":
(3/4)x = (1/6)x

Multiplying both sides by 24 to eliminate the fractions:
18x = 4x

Subtracting 4x from both sides:
18x - 4x = 4x - 4x
14x = 0

Dividing both sides by 14 to solve for "x":
x = 0

Therefore, the initial amount of money Beth had was $0.

To find the initial amount of money J had, we can substitute this value of x into the equation we used earlier:
J initially received (1/4) * 0 = 0.

So, the initial amount of money J had was also $0.

To find the answers to your questions, we can break down the problem step by step.

1. Let's assume Beth had X dollars at the beginning.
- Beth gave 1/4 of her money to J, which is (1/4) * X = X/4 dollars.
- After giving her 1/4 of the money, Beth has (X - X/4) = (3/4)X dollars left.

2. Now, let's figure out how much money J had at first.
- Beth gave (1/4) of her money, which is X/4 dollars, to J.
- So, J received X/4 dollars. Since both girls had the same amount of money at the end, J also had (3/4)X dollars.

According to the problem, J then gave 1/3 of the total amount she had to Beth. So, J gave (1/3) * [(3/4)X] = X/4 dollars to Beth.

Now, we know that both girls had the same amount of money in the end. Therefore, we can set up an equation:

(3/4)X + (X/4) = X

To solve this equation, let's simplify it:

(3/4)X + (1/4)X = X

Multiplying through by 4, we get:

3X + X = 4X

Combining like terms:

4X = 4X

Since the equation is satisfied for any value of X, we can conclude that there are infinite solutions. This means that we cannot determine the exact amount of money each girl had at the beginning based on the information provided.