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Let's break down the problem step by step:

- At the **start of the game**, the ratio of Isla's counters to Bob's counters to Marie's counters is 5:4:3.
- At the **end of the game**, the ratio becomes 6:5:7.
- The total number of counters stays constant throughout the game.
- Isla ends the game with 12 counters.
- We need to find how many counters Isla had at the start of the game.

### Step 1: Express ratios as variables
Let’s define the number of counters at the **start of the game** as:
- Isla's counters = \( 5x \)
- Bob's counters = \( 4x \)
- Marie's counters = \( 3x \)

At the **end of the game**, the number of counters is given by:
- Isla's counters = \( 6y \)
- Bob's counters = \( 5y \)
- Marie's counters = \( 7y \)

### Step 2: Use the fact that the total number of counters is constant
At the start, the total number of counters is:
\[
5x + 4x + 3x = 12x
\]

At the end, the total number of counters is:
\[
6y + 5y + 7y = 18y
\]

Since the total number of counters remains the same:
\[
12x = 18y
\]

Solving for \(x\) in terms of \(y\):
\[
x = \frac{3y}{2}
\]

### Step 3: Use the information that Isla ends with 12 counters
At the end of the game, Isla has \(6y\) counters. Since Isla ends with 12 counters:
\[
6y = 12
\]

Solving for \(y\):
\[
y = 2
\]

### Step 4: Find the number of counters Isla started with
We now know that \(y = 2\). Substituting \(y = 2\) into \(x = \frac{3y}{2}\):
\[
x = \frac{3(2)}{2} = 3
\]

At the start of the game, Isla had \(5x\) counters:
\[
5x = 5(3) = 15
\]

Thus, Isla started the game with **15 counters**.

Would you like further clarification or a deeper explanation on any part?

Here are 5 related questions for further practice:
1. If Bob ended the game with 10 counters, how many counters did he start with?
2. What was the total number of counters in the game?
3. If the ratio at the end was instead 7:6:8, how would that affect Isla's starting number of counters?
4. How many counters did Marie have at the start of the game?
5. What is the total number of counters that changed hands between the start and the end?

**Tip**: When dealing with ratios, expressing unknowns as multiples of a variable simplifies solving for one unknown at a time.

Isla, Bob, and Marie are playing a game involving counters. The total number of counters stays the same throughout the game. At the start of the game, the ratio of Isla's counters to Bob's counters to Marie's counters is 5:4:3. At the end of the game, this ratio is 6:5:7. Isla ends the game with 12 counters. How many counters did Isla start the game with?

Learn how to solve a ratio problem involving Isla, Bob, and Marie with counters. This problem involves determining how many counters Isla started with based on the ratio at the start and end of the game. The total number of counters stays constant. Suitable for students in Grades 7-9.

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