Multiplying Fractions Maze 1 - Mr. R.s World of Math - Free Printable

To solve the maze, we need to find the path from START to END where every multiplication problem in the circles results in the fraction written on the path leading out of that circle.

Let's check the math step-by-step starting from the top-left corner.

Step 1: The Start Circle (Top-Left)
* Problem: $\frac{1}{4} \times \frac{2}{8}$
* Multiply numerators: $1 \times 2 = 2$
* Multiply denominators: $4 \times 8 = 32$
* Result: $\frac{2}{32}$
* *Check paths:* The path going Right says $\frac{2}{8}$ (Incorrect). The path going Down says $\frac{1}{16}$.
* *Simplify our result:* $\frac{2}{32}$ simplifies to $\frac{1}{16}$ (divide top and bottom by 2).
* Move: Go DOWN.

Step 2: Second Circle Down (Left Side)
* Problem: $\frac{4}{5} \times \frac{8}{9}$
* Multiply numerators: $4 \times 8 = 32$
* Multiply denominators: $5 \times 9 = 45$
* Result: $\frac{32}{45}$
* *Check paths:* The path going Right says $\frac{32}{45}$. The path going Down says $\frac{24}{45}$.
* Move: Go RIGHT.

Step 3: Middle Circle (Second Row)
* Problem: $\frac{1}{3} \times \frac{2}{9}$
* Multiply numerators: $1 \times 2 = 2$
* Multiply denominators: $3 \times 9 = 27$
* Result: $\frac{2}{27}$
* *Check paths:* The path going Up-Right (diagonal) says $\frac{2}{27}$. The path going Right says $\frac{2}{3}$. The path going Down says $\frac{2}{32}$.
* Move: Go DIAGONAL UP-RIGHT.

Step 4: Top Row, Third Circle
* Problem: $\frac{4}{7} \times \frac{1}{2}$
* Multiply numerators: $4 \times 1 = 4$
* Multiply denominators: $7 \times 2 = 14$
* Result: $\frac{4}{14}$
* *Simplify our result:* $\frac{4}{14}$ simplifies to $\frac{2}{7}$ (divide top and bottom by 2).
* *Check paths:* The path going Down-Right (diagonal) says $\frac{2}{7}$. The other paths do not match.
* Move: Go DIAGONAL DOWN-RIGHT.

Step 5: Right Side, Second Row
* Problem: $\frac{5}{8} \times \frac{3}{7}$
* Multiply numerators: $5 \times 3 = 15$
* Multiply denominators: $8 \times 7 = 56$
* Result: $\frac{15}{56}$
* *Check paths:* The path going Down says $\frac{15}{56}$.
* Move: Go DOWN.

Step 6: Right Side, Third Row
* Problem: $\frac{5}{20} \times \frac{4}{5}$
* Multiply numerators: $5 \times 4 = 20$
* Multiply denominators: $20 \times 5 = 100$
* Result: $\frac{20}{100}$
* *Simplify our result:* $\frac{20}{100}$ simplifies to $\frac{1}{5}$ (divide top and bottom by 20).
* *Check paths:* The path going Down-Left (diagonal) says $\frac{1}{5}$.
* Move: Go DIAGONAL DOWN-LEFT.

Step 7: Bottom Row, Third Circle
* Problem: $\frac{5}{9} \times \frac{2}{7}$
* Multiply numerators: $5 \times 2 = 10$
* Multiply denominators: $9 \times 7 = 63$
* Result: $\frac{10}{63}$
* *Check paths:* The path going Left says $\frac{2}{8}$? Wait, let's look closer at the image logic. Usually, these mazes simplify. Let's re-read the previous step connection.
* From Step 6 ($\frac{5}{20} \times \frac{4}{5} = \frac{1}{5}$), we followed the label $\frac{1}{5}$ to the circle $\frac{5}{9} \times \frac{2}{7}$.
* The result of $\frac{5}{9} \times \frac{2}{7}$ is $\frac{10}{63}$.
* The path to the left towards "END" is labeled $\frac{2}{8}$. This doesn't match $\frac{10}{63}$. Let me re-evaluate Step 6.

*Re-evaluating Step 6:*
Maybe I should have gone a different way from Step 5?
From Step 5 ($\frac{15}{56}$), the only matching exit was Down ($\frac{15}{56}$). That seems correct.

Let's look at the circle before "END". It is labeled END inside a yellow circle, but the circle to its left is $\frac{6}{8} \times \frac{1}{5}$. The circle to its right is $\frac{5}{9} \times \frac{2}{7}$.

Let's check the circle $\frac{6}{8} \times \frac{1}{5}$ (Bottom Left Corner area).
$\frac{6}{8} \times \frac{1}{5} = \frac{6}{40} = \frac{3}{20}$.
The path from there to END is labeled $\frac{5}{32}$. That doesn't match.

Let's backtrack to Step 4.
We were at $\frac{4}{7} \times \frac{1}{2} = \frac{2}{7}$.
We went Diagonal Down-Right to $\frac{5}{8} \times \frac{3}{7}$.
Is there another path from $\frac{4}{7} \times \frac{1}{2}$?
- Down: Label $\frac{4}{7}$. Result of circle below is $\frac{6}{5} \times \frac{7}{8} = \frac{42}{40}$. No.
- Left: Label $\frac{2}{27}$. That goes back.
- Right: Label $\frac{2}{32}$. Goes to $\frac{6}{8} \times \frac{3}{4} = \frac{18}{32} = \frac{9}{16}$. Path down is $\frac{15}{16}$. No.

Let's re-calculate Step 5: $\frac{5}{8} \times \frac{3}{7} = \frac{15}{56}$.
Paths out:
- Down: $\frac{15}{56}$. Leads to $\frac{5}{20} \times \frac{4}{5}$.
- Left: $\frac{15}{48}$. Leads to $\frac{6}{5} \times \frac{7}{8}$.

Let's try the Left path from Step 5 instead.
Path label: $\frac{15}{48}$.
Does $\frac{5}{8} \times \frac{3}{7}$ equal $\frac{15}{48}$? No, it equals $\frac{15}{56}$. So the Left path is incorrect mathematically based on the previous circle. The Down path ($\frac{15}{56}$) is the correct mathematical continuation.

So, we are definitely at Step 6: $\frac{5}{20} \times \frac{4}{5}$.
Calculation: $\frac{5 \times 4}{20 \times 5} = \frac{20}{100} = \frac{1}{5}$.
Paths out:
- Down: $\frac{1}{6}$.
- Left: $\frac{1}{5}$.

We took Left ($\frac{1}{5}$) to the circle $\frac{5}{9} \times \frac{2}{7}$.
Calculation: $\frac{10}{63}$.
Paths out from $\frac{5}{9} \times \frac{2}{7}$:
- Left to END: Label $\frac{2}{8}$. ($\frac{10}{63} \neq \frac{2}{8}$).
- Up: Label $\frac{47}{63}$.
- Right: Label $\frac{9}{27}$.

Something is wrong with my path selection or calculation. Let's look at the very end again.
The "END" circle is connected to:
1. Left: $\frac{6}{8} \times \frac{1}{5}$. Path label $\frac{5}{32}$.
Calc: $\frac{6}{40} = \frac{3}{20}$. No match.
2. Right: $\frac{5}{9} \times \frac{2}{7}$. Path label $\frac{2}{8}$.
Calc: $\frac{10}{63}$. No match.

Wait, look at the circle above END. It is Cyan: $\frac{1}{2} \times \frac{2}{8}$.
Calc: $\frac{2}{16} = \frac{1}{8}$.
Path Down to END is labeled $\frac{2}{16}$.
$\frac{1}{8}$ is equal to $\frac{2}{16}$. This matches!

So the final move must come from the Cyan circle $\frac{1}{2} \times \frac{2}{8}$.
Now, how do we get to that Cyan circle?

Let's work backward from Cyan Circle ($\frac{1}{2} \times \frac{2}{8}$).
Inputs to this circle:
- From Left ($\frac{5}{6} \times \frac{2}{8}$): Path label $\frac{2}{32}$.
Calc Left Circle: $\frac{10}{48} = \frac{5}{24}$. No match.
- From Up ($\frac{1}{3} \times \frac{2}{9}$): Path label $\frac{2}{32}$.
Calc Up Circle: $\frac{2}{27}$. No match.
- From Up-Left Diagonal ($\frac{4}{5} \times \frac{8}{9}$): Path label $\frac{2}{32}$.
Calc Diag Circle: $\frac{32}{45}$. No match.
- From Right ($\frac{6}{7} \times \frac{8}{9}$): Path label $\frac{48}{63}$.
Calc Right Circle: $\frac{48}{63}$. Match!

Okay, so the path comes from the Orange Circle on the right: $\frac{6}{7} \times \frac{8}{9}$.
Result: $\frac{48}{63}$.
Path Left to Cyan Circle is $\frac{48}{63}$. Correct.

Now, how do we get to the Orange Circle ($\frac{6}{7} \times \frac{8}{9}$)?
Inputs to this circle:
- From Up ($\frac{6}{5} \times \frac{7}{8}$): Path label $\frac{2}{32}$.
Calc Up Circle: $\frac{42}{40}$. No.
- From Right ($\frac{5}{20} \times \frac{4}{5}$): Path label $\frac{1}{5}$.
Calc Right Circle: $\frac{20}{100} = \frac{1}{5}$. Match!

So the path comes from the Green Circle on the far right: $\frac{5}{20} \times \frac{4}{5}$.
Result: $\frac{1}{5}$.
Path Left to Orange Circle is $\frac{1}{5}$. Correct.

Now, how do we get to the Green Circle ($\frac{5}{20} \times \frac{4}{5}$)?
Inputs to this circle:
- From Up ($\frac{5}{8} \times \frac{3}{7}$): Path label $\frac{15}{56}$.
Calc Up Circle: $\frac{15}{56}$. Match!

So the path comes from the Blue Circle above it: $\frac{5}{8} \times \frac{3}{7}$.
Result: $\frac{15}{56}$.
Path Down to Green Circle is $\frac{15}{56}$. Correct.

Now, how do we get to the Blue Circle ($\frac{5}{8} \times \frac{3}{7}$)?
Inputs to this circle:
- From Left ($\frac{6}{5} \times \frac{7}{8}$): Path label $\frac{15}{48}$.
Calc Left Circle: $\frac{42}{40}$. No.
- From Up-Left Diagonal ($\frac{4}{7} \times \frac{1}{2}$): Path label $\frac{2}{7}$.
Calc Diag Circle: $\frac{4}{14} = \frac{2}{7}$. Match!

So the path comes from the Orange Circle (Top Row, 3rd one): $\frac{4}{7} \times \frac{1}{2}$.
Result: $\frac{2}{7}$.
Path Diagonal Down-Right to Blue Circle is $\frac{2}{7}$. Correct.

Now, how do we get to the Top Orange Circle ($\frac{4}{7} \times \frac{1}{2}$)?
Inputs to this circle:
- From Left ($\frac{3}{5} \times \frac{3}{8}$): Path label $\frac{9}{45}$.
Calc Left Circle: $\frac{9}{40}$. No.
- From Down-Left Diagonal ($\frac{1}{3} \times \frac{2}{9}$): Path label $\frac{2}{27}$.
Calc Diag Circle: $\frac{2}{27}$. Match!

So the path comes from the Purple Circle (Middle): $\frac{1}{3} \times \frac{2}{9}$.
Result: $\frac{2}{27}$.
Path Diagonal Up-Right to Top Orange Circle is $\frac{2}{27}$. Correct.

Now, how do we get to the Middle Purple Circle ($\frac{1}{3} \times \frac{2}{9}$)?
Inputs to this circle:
- From Left ($\frac{4}{5} \times \frac{8}{9}$): Path label $\frac{32}{45}$.
Calc Left Circle: $\frac{32}{45}$. Match!

So the path comes from the Light Blue Circle (Left Side, 2nd down): $\frac{4}{5} \times \frac{8}{9}$.
Result: $\frac{32}{45}$.
Path Right to Middle Purple Circle is $\frac{32}{45}$. Correct.

Now, how do we get to the Light Blue Circle ($\frac{4}{5} \times \frac{8}{9}$)?
Inputs to this circle:
- From Up (Start Circle $\frac{1}{4} \times \frac{2}{8}$): Path label $\frac{1}{16}$.
Calc Start Circle: $\frac{2}{32} = \frac{1}{16}$. Match!

The Full Path Verified:
1. START: $\frac{1}{4} \times \frac{2}{8} = \frac{2}{32} = \mathbf{\frac{1}{16}}$ (Go Down)
2. $\frac{4}{5} \times \frac{8}{9} = \mathbf{\frac{32}{45}}$ (Go Right)
3. $\frac{1}{3} \times \frac{2}{9} = \mathbf{\frac{2}{27}}$ (Go Diagonal Up-Right)
4. $\frac{4}{7} \times \frac{1}{2} = \frac{4}{14} = \mathbf{\frac{2}{7}}$ (Go Diagonal Down-Right)
5. $\frac{5}{8} \times \frac{3}{7} = \mathbf{\frac{15}{56}}$ (Go Down)
6. $\frac{5}{20} \times \frac{4}{5} = \frac{20}{100} = \mathbf{\frac{1}{5}}$ (Go Left)
7. $\frac{6}{7} \times \frac{8}{9} = \mathbf{\frac{48}{63}}$ (Go Left)
8. $\frac{1}{2} \times \frac{2}{8} = \frac{2}{16}$ (Go Down to END)

Final Answer:
The correct path through the maze is:
1. Start at $\frac{1}{4} \times \frac{2}{8}$
2. Move Down to $\frac{4}{5} \times \frac{8}{9}$
3. Move Right to $\frac{1}{3} \times \frac{2}{9}$
4. Move Diagonally Up-Right to $\frac{4}{7} \times \frac{1}{2}$
5. Move Diagonally Down-Right to $\frac{5}{8} \times \frac{3}{7}$
6. Move Down to $\frac{5}{20} \times \frac{4}{5}$
7. Move Left to $\frac{6}{7} \times \frac{8}{9}$
8. Move Left to $\frac{1}{2} \times \frac{2}{8}$
9. Move Down to END