Understanding the Relationship Between Multiplication and Division

📖 Understanding How Multiplication and Division Work Together

**What Are Inverse Operations?**

Multiplication and division are called "inverse operations" because they undo each other.
• If you multiply a number by something, you can divide to get back to the original number
• If you divide a number by something, you can multiply to get back to the original number

**The Relationship Rule:**
• If a × b = c, then a = c/b and b = c/a
• Example: 3 × 8 = 24, so 3 = 24/8 and 8 = 24/3

**Using Models to Show Operations:**

When you group items (like cherries on plates), you can use multiplication:
• 6 plates with 9 cherries each = 6 × 9 = 54 cherries
• You can break this into smaller groups: (4 plates + 2 plates) × 9 cherries = (4 + 2) × 9

**KEY TAKEAWAY:**
The number of cherries on EACH plate stays the same. You add the number of PLATES, not the cherries!

**Testing If a Number Works:**

To check if a number makes an equation true, substitute it in and solve:
• Does 8 work in 36 ÷ ? = 6?
• Try it: 36 ÷ 8 = 4.5 (not 6!)
• So the answer is NO, 8 does not work
• What does work? 36 ÷ 6 = 6 ✓

💡 Worked Examples

Study these examples carefully before starting the practice questions.

Example 1

Problem: What value makes both equations true? 20 ÷ 4 = ? and 4 × ? = 20

Division and multiplication are inverse operations. Since 20 ÷ 4 = 5, we know that 4 × 5 = 20.

Answer:

Example 2

Problem: Maria has 5 bags. She puts 7 candies in each bag. She finds the total using (3 + 2) × 7. Why does this work?

When grouping, you add the number of GROUPS (bags), then multiply by what's in EACH group (candies per bag).

Answer:

Example 3

Problem: Does putting 7 in the box make this equation true? 42 ÷ ? = 6

Always substitute the number and calculate. If both sides equal the same value, it's true.

Answer:

✏️ Practice Questions

Solve each problem. Show your work in the space provided.

1. What value makes both equations true? 24 ÷ 3 = ? and 3 × ? = 24

A) 6 B) 8 C) 7 D) 9

2. What value makes both equations true? 35 ÷ 5 = ? and 5 × ? = 35

A) 8 B) 7 C) 5 D) 9

3. What value makes both equations true? 48 ÷ 6 = ? and 6 × ? = 48

A) 8 B) 6 C) 7 D) 10

4. What value makes both equations true? 54 ÷ 9 = ? and 9 × ? = 54

A) 5 B) 8 C) 6 D) 4

5. Lyle has 6 plates. He puts 9 cherries on each plate. He finds the total number of cherries by splitting the plates into 4 and 2. Which expression shows this?

A) (4 + 2) × 9 B) (9 + 6) × 2 C) (4 + 9) × (2 + 9)

6. Amy has 7 boxes. She puts 8 toys on each boxe. She finds the total number of toys by splitting the boxes into 3 and 4. Which expression shows this?

A) (3 + 4) × 8 B) (3 + 8) × (4 + 8) C) (8 + 7) × 4

7. Tom has 5 bags. He puts 6 apples on each bag. He finds the total number of apples by splitting the bags into 2 and 3. Which expression shows this?

A) (2 + 6) × (3 + 6) B) (6 + 5) × 3 C) (2 + 3) × 6

8. Sarah has 8 trays. She puts 5 cookies on each tray. She finds the total number of cookies by splitting the trays into 5 and 3. Which expression shows this?

A) (5 + 8) × 3 B) (5 + 3) × 5 C) (5 + 5) × (3 + 5)

9. Decide if putting the number 8 in the box makes the equation true: 36 ÷ ? = 6

A) Yes B) No

10. Decide if putting the number 6 in the box makes the equation true: 42 ÷ ? = 7

A) Yes B) No

11. Decide if putting the number 7 in the box makes the equation true: 56 ÷ ? = 8

A) Yes B) No

12. Decide if putting the number 8 in the box makes the equation true: 45 ÷ ? = 5

A) Yes B) No

13. What number goes in the box? 7 × ? = 42

14. What number goes in the box? 8 × ? = 56

15. What number goes in the box? 9 × ? = 63

16. What number goes in the box? 6 × ? = 48

17. What number goes in the box? 32 ÷ ? = 8

18. What number goes in the box? 49 ÷ ? = 7

19. What number goes in the box? 30 ÷ ? = 5

20. What number goes in the box? 28 ÷ ? = 4