To solve this problem, we need to set up equations based on the information provided:

Step 1: Define variables

Let:

• $I$ = the number of marbles Ibrahim had at the beginning
• $J$ = the number of marbles Jonathan had at the beginning

From the problem, we know:

1. $I + J = 3285$ (Equation 1: Total number of marbles at the beginning)
2. During the game, Jonathan lost 528 marbles to Ibrahim. Therefore, after the exchange:
   • Ibrahim has $I + 528$
   • Jonathan has $J - 528$
3. At the end of the game, Ibrahim has 4 times as many marbles as Jonathan: $I + 528 = 4(J - 528) \quad \text{(Equation 2)}$

Step 2: Solve the system of equations

From Equation 1:

$J = 3285 - I$

Substitute $J = 3285 - I$ into Equation 2:

$I + 528 = 4((3285 - I) - 528)$ Simplify: $I + 528 = 4(3285 - I - 528)$ $I + 528 = 4(2757 - I)$ $I + 528 = 11028 - 4I$ $I + 4I = 11028 - 528$ $5I = 10500$ $I = 2100$

Step 3: Verify the solution

• If $I = 2100$, then $J = 3285 - 2100 = 1185$.
• After the exchange:
  • Ibrahim has $2100 + 528 = 2628$,
  • Jonathan has $1185 - 528 = 657$.
• Check the final ratio: $2628 \div 657 = 4$ This is correct.

Final Answer:

Ibrahim had 2100 marbles at the beginning.

Do you want a step-by-step breakdown of any part, or have questions? Here are related questions to explore:

1. What if Ibrahim lost marbles instead of Jonathan?
2. How would the problem change if the total marbles were a different number?
3. Can we solve this using a graphical method?
4. How does the ratio of Ibrahim’s and Jonathan’s marbles change before and after the game?
5. Can we verify the solution using trial and error instead of equations?

Tip: Always double-check your equations when solving word problems involving multiple variables.