Tim and James had a total of 432 game cards. Tim gave 2/5 of his game cards to James. Then, James gave 1/4 of the total number of game cards he had to Tim. Both of them had the same number of game cards in the end. How many game cards did Tim have first?

Let the number of game cards Tim initially had be \( t \), and the number James initially had be \( j \). We are given the following information:

1. The total number of game cards they had together is 432:
   \[
   t + j = 432
   \]

2. Tim gave \( \frac{2}{5} \) of his cards to James. After this, Tim has \( t - \frac{2}{5}t = \frac{3}{5}t \), and James has \( j + \frac{2}{5}t \).

3. After that, James gave \( \frac{1}{4} \) of the total number of cards he had at that point (i.e., \( j + \frac{2}{5}t \)) back to Tim. The number of cards James has left is:
   \[
   \left( j + \frac{2}{5}t \right) - \frac{1}{4}\left( j + \frac{2}{5}t \right) = \frac{3}{4} \left( j + \frac{2}{5}t \right)
   \]
   And the number of cards Tim has after receiving \( \frac{1}{4} \) of James' cards is:
   \[
   \frac{3}{5}t + \frac{1}{4}\left( j + \frac{2}{5}t \right)
   \]

4. At the end, both Tim and James have the same number of cards:
   \[
   \frac{3}{5}t + \frac{1}{4}\left( j + \frac{2}{5}t \right) = \frac{3}{4}\left( j + \frac{2}{5}t \right)
   \]

We can now solve this system of equations.

### Step 1: Express \( j \) in terms of \( t \)

From the total number of cards:
\[
j = 432 - t
\]

### Step 2: Substitute \( j = 432 - t \) into the equation

Substitute this into both sides of the equation for the number of cards they have at the end:
\[
\frac{3}{5}t + \frac{1}{4}(432 - t + \frac{2}{5}t) = \frac{3}{4}(432 - t + \frac{2}{5}t)
\]

Let's simplify and solve for \( t \).Tim initially had 240 game cards.

Would you like a detailed breakdown of how the solution was derived? 

Here are 5 related questions to explore further:

1. How many game cards did James initially have?
2. How many cards did Tim give to James in the first exchange?
3. How many cards did James have before giving 1/4 of his total to Tim?
4. How many cards did James end up with after the final exchange?
5. Can this type of problem be solved using a different method?

**Tip:** When solving word problems, always break down the information into clear equations before attempting to solve!

Learn how to solve a game card exchange problem involving Tim and James, with 432 total game cards. Tim gives away 2/5 of his cards, and James returns 1/4 of his. Both end up with the same number of cards. Suitable for Grades 7-9 students, the solution involves algebra and fractions.

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