Solve the fraction division maze to find the correct path from start to end.

Let's solve this fraction maze step by step.

🧩 Goal:

We need to start at "START" and reach "END" by moving through circles connected by paths.
Each circle contains a fraction division problem, and each path between circles has a value (a fraction or mixed number).

To move from one circle to another, the result of the division in the current circle must equal the value on the connecting path.

So we:
1. Solve each fraction division in every circle.
2. Check which paths match the result.
3. Trace a valid path from START to END where each step matches.

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🔢 Step 1: Solve Each Division Problem

We’ll compute each fraction division using the rule:

> $$
> \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
> $$

Let’s go through all the circles:

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#### ✔ Start: $ \frac{3}{8} \div \frac{3}{4} $

$$
\frac{3}{8} \times \frac{4}{3} = \frac{12}{24} = \frac{1}{2}
$$

✔ Result: $ \frac{1}{2} $

This matches the right path to the next green circle → $ \frac{4}{5} \div \frac{3}{9} $

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#### 🟨 $ \frac{4}{5} \div \frac{3}{9} $

First simplify $ \frac{3}{9} = \frac{1}{3} $

$$
\frac{4}{5} \div \frac{1}{3} = \frac{4}{5} \times 3 = \frac{12}{5} = 2\frac{2}{5}
$$

Wait — but the path from here is labeled $ \frac{12}{45} $, which is $ \frac{4}{15} $. That doesn’t match.

Wait — maybe I made a mistake?

Wait! Let’s double-check.

Actually: $ \frac{3}{9} = \frac{1}{3} $, so:

$$
\frac{4}{5} \div \frac{1}{3} = \frac{4}{5} \times 3 = \frac{12}{5} = 2.4
$$

But the path to the next orange circle says $ \frac{12}{45} = \frac{4}{15} \approx 0.267 $ — not matching.

So this path is not valid.

Wait — perhaps I misread the maze.

Let’s look again.

The START circle: $ \frac{3}{8} \div \frac{3}{4} = \frac{1}{2} $ — correct.

From there, the only path with label $ \frac{1}{2} $ goes to the next circle: $ \frac{4}{5} \div \frac{3}{9} $

So we must take that path if the result matches.

But earlier calculation gave $ \frac{12}{5} $, which is not $ \frac{1}{2} $.

Wait — contradiction!

That means either:
- We made a mistake in solving,
- Or the path value is incorrect,
- Or we're misunderstanding how the maze works.

Wait — let’s recheck.

Maybe the value on the path is the answer to the division problem in the circle.

So for each circle, we compute the division, and only if it equals the label on the path, we can traverse.

So for example:
- The START circle: $ \frac{3}{8} \div \frac{3}{4} = \frac{1}{2} $
- There is a path labeled $ \frac{1}{2} $ going right to $ \frac{4}{5} \div \frac{3}{9} $
- So we can go from START to that yellow circle, only if the result of $ \frac{4}{5} \div \frac{3}{9} $ is $ \frac{1}{2} $? No — that doesn't make sense.

Wait — actually, the path label is the value you get when you divide, so:

> You are at a circle, you solve its division, and then you can go along a path if the result matches the path's label.

So:
- At START: solve $ \frac{3}{8} \div \frac{3}{4} = \frac{1}{2} $
- Then, from START, you can go to any neighbor whose connecting path has label $ \frac{1}{2} $

Looking at START (top-left), it connects to:
- Right: path labeled $ \frac{1}{2} $ → leads to $ \frac{4}{5} \div \frac{3}{9} $
- Down: path labeled $ \frac{9}{32} $ → leads to $ \frac{1}{5} \div \frac{3}{9} $

Since we got $ \frac{1}{2} $, we can only go via the right path (labeled $ \frac{1}{2} $).

So we go to the yellow circle: $ \frac{4}{5} \div \frac{3}{9} $

Now compute that:

$ \frac{3}{9} = \frac{1}{3} $

So:
$$
\frac{4}{5} \div \frac{1}{3} = \frac{4}{5} \times 3 = \frac{12}{5} = 2\frac{2}{5}
$$

Now, what paths come out of this yellow circle?

It connects to:
- Right: path labeled $ \frac{12}{45} = \frac{4}{15} $
- Down: path labeled $ 2\frac{6}{15} = 2 + \frac{6}{15} = 2\frac{2}{5} = \frac{12}{5} $
- Diagonally down-right: path labeled $ \frac{2}{27} $

Our result is $ \frac{12}{5} = 2\frac{2}{5} $, which matches the down path labeled $ 2\frac{6}{15} $

Check: $ 2\frac{6}{15} = 2 + \frac{6}{15} = 2 + \frac{2}{5} = 2\frac{2}{5} $ → yes!

So we can go down to the purple circle: $ \frac{1}{3} \div \frac{3}{9} $

Compute that:
$ \frac{3}{9} = \frac{1}{3} $

So:
$$
\frac{1}{3} \div \frac{1}{3} = 1
$$

Now, check outgoing paths from this purple circle:
- Left: path labeled $ 1 $ → connects to $ \frac{1}{5} \div \frac{3}{9} $
- Right: path labeled $ \frac{3}{27} = \frac{1}{9} $
- Down: path labeled $ \frac{2}{32} = \frac{1}{16} $
- Diagonal right-down: path labeled $ \frac{2}{27} $

We got $ 1 $, so only the left path labeled 1 is valid.

Go left to: $ \frac{1}{5} \div \frac{3}{9} $

Simplify $ \frac{3}{9} = \frac{1}{3} $

$$
\frac{1}{5} \div \frac{1}{3} = \frac{1}{5} \times 3 = \frac{3}{5}
$$

Now, check connections from this light blue circle:
- Up: already came from there
- Down: path labeled $ \frac{3}{45} = \frac{1}{15} $
- Diagonal down-right: path labeled $ \frac{9}{15} = \frac{3}{5} $
- Right: path labeled $ 1 $ → but we just came from right?

Wait — let’s map:

From $ \frac{1}{5} \div \frac{3}{9} $, possible paths:
- Down: $ \frac{3}{45} = \frac{1}{15} $
- Diagonal down-right: $ \frac{9}{15} = \frac{3}{5} $
- Right: $ 1 $ — but we came from right? Wait no — we came from left.

Wait: we were at $ \frac{1}{3} \div \frac{3}{9} $, went left to $ \frac{1}{5} \div \frac{3}{9} $

So from $ \frac{1}{5} \div \frac{3}{9} $, we can go:
- Down: $ \frac{3}{45} $
- Diagonal down-right: $ \frac{9}{15} $
- Right: $ 1 $

But our result is $ \frac{3}{5} $, so only diagonal down-right labeled $ \frac{9}{15} = \frac{3}{5} $ matches.

So go to cyan circle: $ \frac{1}{2} \div \frac{2}{8} $

Compute:
$ \frac{2}{8} = \frac{1}{4} $

$$
\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2
$$

Now, from this cyan circle ($ \frac{1}{2} \div \frac{2}{8} $), outgoing paths:
- Left: $ \frac{2}{16} = \frac{1}{8} $
- Right: $ 2 $
- Down: $ \frac{2}{16} = \frac{1}{8} $
- Diagonal down-right: $ \frac{2}{24} = \frac{1}{12} $

We got $ 2 $, so only right path labeled 2 is valid.

Go to orange circle: $ \frac{6}{7} \div \frac{8}{9} $

Compute:
$$
\frac{6}{7} \div \frac{8}{9} = \frac{6}{7} \times \frac{9}{8} = \frac{54}{56} = \frac{27}{28}
$$

Now, from this orange circle, paths:
- Right: $ \frac{48}{63} $
- Down: $ \frac{47}{63} $
- Diagonal down-right: $ \frac{27}{28} $

We got $ \frac{27}{28} $, so only diagonal down-right path matches.

Go to light green circle: $ \frac{5}{20} \div \frac{4}{5} $

Simplify $ \frac{5}{20} = \frac{1}{4} $

$$
\frac{1}{4} \div \frac{4}{5} = \frac{1}{4} \times \frac{5}{4} = \frac{5}{16}
$$

Now, from this circle, paths:
- Down: $ \frac{5}{16} $ → matches!
- Right: $ \frac{9}{27} = \frac{1}{3} $

So go down to END!

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✔ Final Path:

Let’s trace it:

1. Start: $ \frac{3}{8} \div \frac{3}{4} = \frac{1}{2} $ → use path $ \frac{1}{2} $ → go right
2. $ \frac{4}{5} \div \frac{3}{9} = \frac{12}{5} = 2\frac{2}{5} $ → use path $ 2\frac{6}{15} = 2\frac{2}{5} $ → go down
3. $ \frac{1}{3} \div \frac{3}{9} = 1 $ → use path $ 1 $ → go left
4. $ \frac{1}{5} \div \frac{3}{9} = \frac{3}{5} $ → use path $ \frac{9}{15} = \frac{3}{5} $ → go diagonal down-right
5. $ \frac{1}{2} \div \frac{2}{8} = 2 $ → use path $ 2 $ → go right
6. $ \frac{6}{7} \div \frac{8}{9} = \frac{27}{28} $ → use path $ \frac{27}{28} $ → go diagonal down-right
7. $ \frac{5}{20} \div \frac{4}{5} = \frac{5}{16} $ → use path $ \frac{5}{16} $ → go down to END

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✔ Answer: The correct path is:

> START → $ \frac{4}{5} \div \frac{3}{9} $ → $ \frac{1}{3} \div \frac{3}{9} $ → $ \frac{1}{5} \div \frac{3}{9} $ → $ \frac{1}{2} \div \frac{2}{8} $ → $ \frac{6}{7} \div \frac{8}{9} $ → $ \frac{5}{20} \div \frac{4}{5} $ → END

All divisions match the path labels.

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✔ Final Solution Summary:

| Circle | Division | Result | Path Taken |
|--------|---------|--------|------------|
| Start | $ \frac{3}{8} \div \frac{3}{4} $ | $ \frac{1}{2} $ | → Right |
| $ \frac{4}{5} \div \frac{3}{9} $ | $ \frac{12}{5} $ | → Down |
| $ \frac{1}{3} \div \frac{3}{9} $ | $ 1 $ | → Left |
| $ \frac{1}{5} \div \frac{3}{9} $ | $ \frac{3}{5} $ | → Diag ↓→ |
| $ \frac{1}{2} \div \frac{2}{8} $ | $ 2 $ | → Right |
| $ \frac{6}{7} \div \frac{8}{9} $ | $ \frac{27}{28} $ | → Diag ↓→ |
| $ \frac{5}{20} \div \frac{4}{5} $ | $ \frac{5}{16} $ | → Down → END |

✔ Maze solved!

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Let me know if you'd like a visual diagram of the path!