Tommy, Shawn and Ricky had a total of 192 picture cards. In a game, Tommy lost some picture cars to Shawn and Shawn’s picture cards doubled. Shawn in turn lost some picture cards to Ricky and Ricky’s picture cards doubled. Then they each had the same number of picture cards in the end. How many picture cards did each of them have at first?
64
To solve this problem, we can use a system of equations. Let's break down the information given:
1. Tommy, Shawn, and Ricky had a total of 192 picture cards.
2. Tommy lost some picture cards to Shawn, and Shawn's picture cards doubled.
3. Shawn then lost some picture cards to Ricky, and Ricky's picture cards doubled.
Let's assign variables to represent the unknowns:
Let T represent the number of picture cards Tommy had initially.
Let S represent the number of picture cards Shawn had initially.
Let R represent the number of picture cards Ricky had initially.
Now we can set up the equations based on the given information:
Equation 1: T + S + R = 192 (Since the total number of picture cards they had is 192)
Equation 2: T - x = S + x (Tommy lost x cards to Shawn, resulting in Shawn's cards doubling)
Equation 3: S - y = R + y (Shawn lost y cards to Ricky, resulting in Ricky's cards doubling)
From Equation 2, we can rearrange it as T - S = 2x, since S + x - x = S and T - x = T - S.
Similarly, from Equation 3, we can rearrange it as S - R = 2y.
Now, we can substitute these rearranged equations into Equation 1:
(T - S) + S + (S - R) = 192
T - R + S = 192
Now, substitute T - S = 2x and S - R = 2y:
2x + 2y + S = 192
So, we have reduced the problem to a single equation: 2x + 2y + S = 192.
We need additional information to solve this problem completely. Please provide the values of x and y, or any other relevant information if available.