Integer Maze Multiply Divide #1 - Mr. R.s World of Math - Free Printable

Let's solve this step-by-step by following the flowchart from START to END, filling in the missing values in each circle based on the operations given.

We'll go through each column one by one.

---

🟩 Left Column (Start at -8)

Start: -8

1. $-8 \times (-2) = 16$
→ First blank: 16

2. $16 \times (+3) = 48$
→ Second blank: 48

3. $48 \times (-3) = -144$
→ Third blank: -144

4. $-144 \div (+2) = -72$
→ Fourth blank: -72

5. $-72 \times (+5) = -360$
→ Fifth blank: -360

Now, we have:
> -360 → ÷ (-9) → +40 ✔ (matches checkpoint)

Check: $-360 \div (-9) = 40$ ✔️

✔ Left column verified.

---

🔵 Middle Column (Checkpoint at +40)

We are told that the bottom of the middle column is +40, and we need to work upward using reverse operations.

But let’s go from bottom to top, reversing the operations.

Start: +40

We’re going upward, so we reverse each operation:

1. Operation before +40: $x (-5)$ → So reverse: divide by (-5)
- $40 \div (-5) = -8$
→ Next blank: -8

2. Before that: $÷ (+2)$ → Reverse: multiply by (+2)
- $-8 \times (+2) = -16$
→ Next blank: -16

3. Before that: $x (-4)$ → Reverse: divide by (-4)
- $-16 \div (-4) = 4$
→ Next blank: 4

4. Before that: $x (+3)$ → Reverse: divide by (+3)
- $4 \div 3 = \frac{4}{3}$ → Wait, this is not an integer. Let's double-check.

Wait! Let's retrace carefully.

We're going upward, so we must reverse the order of operations.

The sequence from bottom to top (reverse):

- Start: +40
- Previous operation: $x (-5)$ → So reverse: $÷ (-5)$
- $40 ÷ (-5) = -8$
- Previous: $÷ (+2)$ → Reverse: $× (+2)$
- $-8 × 2 = -16$
- Previous: $x (-4)$ → Reverse: $÷ (-4)$
- $-16 ÷ (-4) = 4$
- Previous: $x (+3)$ → Reverse: $÷ (+3)$
- $4 ÷ 3 ≈ 1.333...$ → Not nice.

But wait — this suggests a problem. Let's check if the top value connects correctly to the right column.

Wait — there's another checkpoint: the top of the middle column goes to -150 via $÷ (+2)$.

So let's use that clue.

---

🔵 Top of Middle Column

We know:

- The top circle → $÷ (+2)$ → -150
- So: $ \text{Top} \div (+2) = -150 $
- Therefore: $ \text{Top} = -150 \times (+2) = -300 $

So the top of the middle column is -300

Now we can fill the middle column from top to bottom.

---

#### Middle Column (From Top to Bottom)

1. Top: -300
↓ $÷ (-4)$ → $-300 ÷ (-4) = 75$

2. $75 × (+3) = 225$

3. $225 × (-4) = -900$

4. $-900 ÷ (+2) = -450$

5. $-450 × (-5) = 2250$

Wait — but earlier we said the bottom should be +40, but here we get 2250? That doesn’t match.

Contradiction!

Wait — let’s re-express.

We have two checkpoints:

- Top of middle column: some number → $÷ (+2)$ → -150 ⇒ So top = $-150 × 2 = -300$ ✔️

- Bottom of middle column: some number → $÷ (-9)$ → +40 ⇒ So bottom = $40 × (-9) = -360$ ✘

Wait — no: the arrow is from left column to middle column: -360 → ÷ (-9) → +40

So the bottom of the left column is -360, then $-360 ÷ (-9) = 40$, which is the start of the middle column.

So the bottom of the middle column is +40.

So the middle column goes:

- Start: +40 (bottom)
- Then ↑: $x (-5)$ → so previous value = $40 ÷ (-5) = -8$
- Then ↑: $÷ (+2)$ → reverse: $× (+2)$ → $-8 × 2 = -16$
- Then ↑: $x (-4)$ → reverse: $÷ (-4)$ → $-16 ÷ (-4) = 4$
- Then ↑: $x (+3)$ → reverse: $÷ (+3)$ → $4 ÷ 3 ≈ 1.333$ → not good

But we also know the top of the middle column is -300, because it goes to -150 via $÷ (+2)$

So let’s go from top to bottom with known top: -300

- Top: -300
- ↓ $÷ (-4)$ → $-300 ÷ (-4) = 75$
- ↓ $x (+3)$ → $75 × 3 = 225$
- ↓ $x (-4)$ → $225 × (-4) = -900$
- ↓ $÷ (+2)$ → $-900 ÷ 2 = -450$
- ↓ $x (-5)$ → $-450 × (-5) = 2250$

So bottom of middle column = 2250

But earlier we said it should be +40?

No — contradiction.

Wait — the arrow from left column to middle column says:

> (last of left column) → ÷ (-9) → +40

So the value before dividing by (-9) is $40 × (-9) = -360$

So the last value in left column is -360

Then: $-360 ÷ (-9) = 40$ → this is the first value in middle column (bottom)

So the middle column starts at +40 (bottom), and goes upward.

But we just calculated from top that the top is -300, and going down gives us 2250 at bottom.

But that contradicts +40 at bottom.

So either:

- We made a mistake in assuming the top of middle column is -300?
- Or we misunderstood the flow?

Let’s look again.

---

🔹 Key Clue: The top of the middle column has an arrow → $÷ (+2)$ → -150

So:

> (some value) ÷ (+2) = -150
⇒ that value = $-150 × 2 = -300$

So the top of the middle column is -300

And the bottom of the middle column is +40

So we need to go from -300 (top) → down to +40 (bottom), applying the operations.

Operations in middle column (from top to bottom):

1. ÷ (-4)
2. × (+3)
3. × (-4)
4. ÷ (+2)
5. × (-5)

Wait — direction matters.

Look at arrows:

- From top: ↓ $÷ (-4)$
- Then ↓ $× (+3)$
- Then ↓ $× (-4)$
- Then ↓ $÷ (+2)$
- Then ↓ $× (-5)$
- Then → to +40

Wait — but the last operation is × (-5), and result is +40

So:

Let’s call the value before × (-5) as X

Then: $X × (-5) = 40$ ⇒ $X = 40 ÷ (-5) = -8$

So the value before the last operation is -8

Now go back:

- Before $÷ (+2)$: let Y → $Y ÷ (+2) = -8$ ⇒ $Y = -16$

- Before $× (-4)$: Z → $Z × (-4) = -16$ ⇒ $Z = (-16) ÷ (-4) = 4$

- Before $× (+3)$: W → $W × (+3) = 4$ ⇒ $W = 4 ÷ 3 ≈ 1.333$ → not integer

Hmm. But earlier we had top = -300

But now we have:

- Top: ?
- ↓ ÷ (-4) → ?
- ↓ × (+3) → ?
- ↓ × (-4) → ?
- ↓ ÷ (+2) → ?
- ↓ × (-5) → 40

We found:

- × (-5) → 40 ⇒ previous = -8
- ÷ (+2) → -8 ⇒ previous = -16
- × (-4) → -16 ⇒ previous = 4
- × (+3) → 4 ⇒ previous = 4/3
- ÷ (-4) → 4/3 ⇒ previous = (4/3) × (-4) = -16/3 ≈ -5.333

But we were told that the top of the middle column is connected to -150 via $÷ (+2)$

So top value → $÷ (+2)$ → -150 ⇒ top = -300

So the top of the middle column must be -300

But our calculation from bottom gives top ≈ -5.333 — not matching.

So contradiction.

Therefore, the only way both checkpoints are satisfied is if the middle column path is consistent.

But we have two constraints:

1. Top of middle column → $÷ (+2)$ → -150 ⇒ top = -300
2. Bottom of middle column → $× (-5)$ → 40 ⇒ previous = -8

But we can't connect -300 to -8 using the operations in the middle column.

Let’s try to compute the full path from top to bottom in middle column, starting with -300

1. -300
2. ↓ $÷ (-4)$ → $-300 ÷ (-4) = 75$
3. ↓ $× (+3)$ → $75 × 3 = 225$
4. ↓ $× (-4)$ → $225 × (-4) = -900$
5. ↓ $÷ (+2)$ → $-900 ÷ 2 = -450$
6. ↓ $× (-5)$ → $-450 × (-5) = 2250$

So final value = 2250

But we are told that after this, it goes to +40? No — the arrow is from left column to middle column:

> (last of left) → ÷ (-9) → +40

So +40 is the start of the middle column, i.e., the bottom of the middle column.

But we just got 2250 at the bottom — not +40

So something is wrong.

Wait — perhaps I misread the diagram.

Let me re-read the diagram carefully.

There are three columns.

- Left column: starts at -8, ends at some number → then → ÷ (-9) → +40

- Middle column: starts at +40 (from left), then goes up with operations, and at the top, → ÷ (+2) → -150

- Right column: starts at -150, ends at 100

Ah! So the flow is not vertical only — there are horizontal connections.

Let’s map it clearly.

---

🔁 Flow Summary

1. Left column:
- Start: -8
- ↓ ×(-2) → ?
- ↓ ×(+3) → ?
- ↓ ×(-3) → ?
- ↓ ÷(+2) → ?
- ↓ ×(+5) → ?
- → ÷(-9) → +40 (this is the start of middle column)

2. Middle column:
- Start: +40
- ↑ ×(-5) → ?
- ↑ ÷(+2) → ?
- ↑ ×(-4) → ?
- ↑ ×(+3) → ?
- ↑ ÷(-4) → ?
- → ÷(+2) → -150 (this is the start of right column)

3. Right column:
- Start: -150
- ↓ ×(-5) → ?
- ↓ ÷(-5) → ?
- ↓ ÷(-2) → ?
- ↓ ×(-4) → ?
- ↓ ÷(-3) → 100 (END)

Note: The arrows in middle column go upward, so we go from bottom to top.

So let's solve each column in order.

---

## ✔ Step 1: Solve Left Column

Start: -8

1. $-8 × (-2) = 16$
2. $16 × (+3) = 48$
3. $48 × (-3) = -144$
4. $-144 ÷ (+2) = -72$
5. $-72 × (+5) = -360$

Now: $-360 ÷ (-9) = 40$ → matches checkpoint ✔

So left column filled:

- -8 → 16 → 48 → -144 → -72 → -360 → → 40

---

## ✔ Step 2: Middle Column (from bottom to top)

Start: +40

Operations are upward, so reverse the operations.

List of operations from bottom to top:

1. × (-5) ← from 40 → previous = $40 ÷ (-5) = -8$
2. ÷ (+2) ← from -8 → previous = $-8 × (+2) = -16$
3. × (-4) ← from -16 → previous = $-16 ÷ (-4) = 4$
4. × (+3) ← from 4 → previous = $4 ÷ (+3) = \frac{4}{3}$
5. ÷ (-4) ← from $\frac{4}{3}$ → previous = $\frac{4}{3} × (-4) = -\frac{16}{3}$

Now, this top value ($-\frac{16}{3}$) should go to -150 via $÷ (+2)$

Check: $-\frac{16}{3} ÷ (+2) = -\frac{16}{6} = -\frac{8}{3} ≈ -2.666$ ≠ -150

✘ Not matching.

But we are told: top of middle column → ÷ (+2) → -150

So the value before ÷(+2) is $-150 × 2 = -300$

So the top of the middle column is -300

But from bottom-up calculation, we get $-\frac{16}{3}$ — not -300

So inconsistency.

Unless the operations are applied in different order.

Wait — maybe the middle column operations are from top to bottom, but arrows are upward?

Look at the diagram:

- Arrows in middle column are upward.
- Operations are labeled between circles.
- So when we see:
Circle → ↑ ×(+3) → Circle
It means: lower circle ×(+3) = upper circle

So to go from bottom to top, we apply the operations as shown.

So from +40 at bottom, we go up:

- × (-5) → next: $40 × (-5) = -200$
- ÷ (+2) → $-200 ÷ (+2) = -100$
- × (-4) → $-100 × (-4) = 400$
- × (+3) → $400 × (+3) = 1200$
- ÷ (-4) → $1200 ÷ (-4) = -300$

Now, -300 → ÷ (+2) → $-300 ÷ 2 = -150$ ✔

Perfect!

So the top of the middle column is -300, and it goes to -150 via ÷(+2)

Now let's fill the middle column from bottom to top:

1. +40 (bottom)
2. ↑ × (-5) → $40 × (-5) = -200$
3. ↑ ÷ (+2) → $-200 ÷ 2 = -100$
4. ↑ × (-4) → $-100 × (-4) = 400$
5. ↑ × (+3) → $400 × 3 = 1200$
6. ↑ ÷ (-4) → $1200 ÷ (-4) = -300$ (top)

Then: $-300 ÷ (+2) = -150$ → matches checkpoint ✔

So middle column values (from bottom to top):

- +40
- -200
- -100
- 400
- 1200
- -300

---

## ✔ Step 3: Right Column

Start: -150

Operations downward:

1. × (-5) → $-150 × (-5) = 750$
2. ÷ (-5) → $750 ÷ (-5) = -150$
3. ÷ (-2) → $-150 ÷ (-2) = 75$
4. × (-4) → $75 × (-4) = -300$
5. ÷ (-3) → $-300 ÷ (-3) = 100$ ✔ END

Perfect!

So right column values:

- -150
- 750
- -150
- 75
- -300
- 100

---

✔ Final Answer: Fill in all blanks

#### Left Column (from top to bottom):
1. -8
2. $-8 × (-2) = 16$
3. $16 × (+3) = 48$
4. $48 × (-3) = -144$
5. $-144 ÷ (+2) = -72$
6. $-72 × (+5) = -360$

→ Then: $-360 ÷ (-9) = 40$ → connects to middle column

#### Middle Column (from bottom to top):
1. +40 (bottom)
2. $40 × (-5) = -200$
3. $-200 ÷ (+2) = -100$
4. $-100 × (-4) = 400$
5. $400 × (+3) = 1200$
6. $1200 ÷ (-4) = -300$ (top)

→ Then: $-300 ÷ (+2) = -150$ → connects to right column

#### Right Column (from top to bottom):
1. -150
2. $-150 × (-5) = 750$
3. $750 ÷ (-5) = -150$
4. $-150 ÷ (-2) = 75$
5. $75 × (-4) = -300$
6. $-300 ÷ (-3) = 100$ → END

---

✔ All Values Filled

#### Left Column:
- -8
- 16
- 48
- -144
- -72
- -360

#### Middle Column (from bottom to top):
- +40
- -200
- -100
- 400
- 1200
- -300

#### Right Column:
- -150
- 750
- -150
- 75
- -300
- 100

---

✔ Verification of Checkpoints

- Left end: $-360 ÷ (-9) = 40$ → ✓
- Middle top: $-300 ÷ (+2) = -150$ → ✓
- Right end: $-300 ÷ (-3) = 100$ → ✓

All correct!

---

🟩 Final Answer (Filling the Diagram)

#### Left Column:
- -8
- 16
- 48
- -144
- -72
- -360

#### Middle Column (top to bottom):
- -300
- -100
- 400
- 1200
- -200
- +40

Wait — the diagram shows upward arrows, so the top is first.

So middle column (from top to bottom):
- -300
- -100
- 400
- 1200
- -200
- +40

But the operations are labeled between them.

Let’s list them in order from top to bottom:

1. -300
2. ↓ ÷ (-4) → $-300 ÷ (-4) = 75$ ✘ Wait, we have -100?

Wait — mismatch.

Earlier we computed:

- Top: -300
- Then ↑ ÷ (-4) → meaning: below = above ÷ (-4)? No.

Wait — arrow is upward, so:

- Lower circle → ↑ ÷ (-4) → Upper circle

So: lower value ÷ (-4) = upper value

So to find lower value: upper × (-4)

So:

- Top: -300
- Below it: ? → ↑ ÷ (-4) → -300 → so ? ÷ (-4) = -300 → ? = (-300) × (-4) = 1200
- Below that: ? → ↑ × (+3) → 1200 → so ? × (+3) = 1200 → ? = 400
- Below that: ? → ↑ × (-4) → 400 → so ? × (-4) = 400 → ? = -100
- Below that: ? → ↑ ÷ (+2) → -100 → so ? ÷ (+2) = -100 → ? = -200
- Below that: ? → ↑ × (-5) → -200 → so ? × (-5) = -200 → ? = 40

So from top to bottom:

- -300
- 1200
- 400
- -100
- -200
- 40

Yes! This matches.

So the middle column (top to bottom):

1. -300
2. 1200
3. 400
4. -100
5. -200
6. 40

And operations:

- -300 ← ↑ ÷ (-4) from 1200: $1200 ÷ (-4) = -300$ ✔️
- 1200 ← ↑ × (+3) from 400: $400 × 3 = 1200$ ✔️
- 400 ← ↑ × (-4) from -100: $-100 × (-4) = 400$ ✔️
- -100 ← ↑ ÷ (+2) from -200: $-200 ÷ 2 = -100$ ✔️
- -200 ← ↑ × (-5) from 40: $40 × (-5) = -200$ ✔️

Perfect.

---

✔ Final Filled Diagram

#### Left Column (top to bottom):
- -8
- 16
- 48
- -144
- -72
- -360

#### Middle Column (top to bottom):
- -300
- 1200
- 400
- -100
- -200
- 40

#### Right Column (top to bottom):
- -150
- 750
- -150
- 75
- -300
- 100

---

✔ All checkpoints satisfied.

Answer is complete and verified.