Solved Exercises on the Relationship Between Multiplication and Division in Grade 3

Master the relationship between multiplication and division: inverse operations, fact families, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Inverse Operations
Exercise 1
Show that multiplication and division are inverse operations using 4 × 3 = 12.
Definition:

Inverse Operations: Operations that undo each other. Multiplication and division are inverse operations because multiplying by a number and then dividing by the same number returns you to the original number.

Method:
  1. Start with a multiplication fact
  2. Use the result to create a division fact
  3. Verify that the operations undo each other
  4. Show that you return to the original number
×3
4 × 3 = 12
→
÷3
12 ÷ 3 = 4

Inverse Relationship: Starting with 4, multiplying by 3 gives 12, then dividing by 3 returns to 4.

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Start
4
Multiply by 3
4 × 3 = 12
Divide by 3
12 ÷ 3 = 4
Step 1: Begin with multiplication

4 × 3 = 12 (multiply 4 by 3 to get 12)

Step 2: Apply inverse division

12 ÷ 3 = 4 (divide 12 by 3 to return to 4)

Step 3: Verify the inverse relationship

We started with 4 and ended with 4, proving the operations are inverses

Step 4: Complete the fact family

Also: 3 × 4 = 12 and 12 ÷ 4 = 3

4 × 3 = 12 and 12 ÷ 3 = 4
Final answer:

Multiplication and division are inverse operations as shown: 4 × 3 = 12 and 12 ÷ 3 = 4

Applied rules:

• Inverse Operations: Multiplication and division undo each other

• Fact Family: Four related facts connect the same three numbers

• Reversibility: Operations can be reversed to return to original values

2 Fact Families
Exercise 2
Create the complete fact family for 5, 6, and 30 showing the relationship between multiplication and division.
Definition:

Fact Family: A set of related multiplication and division facts that use the same three numbers. The relationship shows how multiplication and division are connected.

5 × 6 = 30
6 × 5 = 30
30 ÷ 5 = 6
30 ÷ 6 = 5
Multiplication Facts
5 × 6 = 30, 6 × 5 = 30
Division Facts
30 ÷ 5 = 6, 30 ÷ 6 = 5
Total Facts
4 facts
Step 1: Identify the three numbers

We have 5, 6, and 30. Note that 5 × 6 = 30

Step 2: Write multiplication facts

5 × 6 = 30 and 6 × 5 = 30 (using commutative property)

Step 3: Write division facts

30 ÷ 5 = 6 and 30 ÷ 6 = 5 (inverse operations)

Step 4: Verify all facts

All four equations are mathematically correct and use only the numbers 5, 6, and 30

5 × 6 = 30, 6 × 5 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5
Final answer:

The fact family for 5, 6, and 30 shows the relationship: 5 × 6 = 30, 6 × 5 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5

Applied rules:

• Fact Family Rule: Three numbers create four related multiplication and division facts

• Commutative Property: a × b = b × a (creates two multiplication facts)

• Inverse Operations: Multiplication and division are opposite operations

3 Real-World Application
Exercise 3
If 4 friends share 20 apples equally, how many apples does each friend get? Show how this relates to 4 × 5 = 20.
Definition:

Real-World Application: Using the relationship between multiplication and division to solve practical problems involving equal sharing or grouping.

Friend 1
Friend 2
Friend 3
Friend 4
×
4 × 5 = 20
🔄
÷
20 ÷ 4 = 5
Total apples
20
Number of friends
4
Apples per friend
5
Step 1: Understand the problem

20 apples shared equally among 4 friends

Step 2: Set up the division

20 ÷ 4 = ? (how many apples per friend)

Step 3: Use the multiplication relationship

If 4 × ? = 20, then ? = 5, so 20 ÷ 4 = 5

Step 4: Verify the answer

4 friends × 5 apples each = 20 apples total ✓

20 ÷ 4 = 5 apples per friend
Final answer:

Each friend gets 5 apples because 20 ÷ 4 = 5, which is related to 4 × 5 = 20

Applied rules:

• Division as Sharing: Total ÷ Number of groups = Amount per group

• Multiplication-Division Relationship: If a × b = c, then c ÷ a = b

• Verification: Use multiplication to check division answers

Multiplication and Division Relationship
If a × b = c, then c ÷ a = b and c ÷ b = a
Inverse Relationship
Multiplication Division Relationship 3 × 4 = 12 12 ÷ 3 = 4 Start with 3, multiply by 4, divide by 4 to return to 3 5 × 6 = 30 30 ÷ 5 = 6 Start with 5, multiply by 6, divide by 6 to return to 5 7 × 8 = 56 56 ÷ 7 = 8 Start with 7, multiply by 8, divide by 8 to return to 7 9 × 2 = 18 18 ÷ 9 = 2 Start with 9, multiply by 2, divide by 2 to return to 9
Key definitions:

Multiplication: An operation that combines equal groups to find a total

Division: An operation that splits a number into equal groups

Inverse Operations: Operations that undo each other

Fact Family: A group of related facts using the same numbers

Understanding the Relationship:
  1. Identify the connection: Multiplication and division are opposite operations
  2. Recognize patterns: The same three numbers appear in related facts
  3. Use for problem-solving: If you know one operation, you can find the other
  4. Check answers: Use the inverse operation to verify results
Key Pattern: The same three numbers are used in both operations.
Inverse Relationship: Multiplication and division undo each other.
Tip 1: If you know 6 × 7 = 42, you automatically know 42 ÷ 7 = 6.
Tip 2: Use multiplication to check division answers.
Tip 3: Think of division as "undoing" multiplication.
Solution: Exercises 4 to 5
4 Verification Practice
Exercise 4
Verify that 8 × 3 = 24 and 24 ÷ 3 = 8 are related facts by showing they are inverse operations.
Definition:

Verification: Proving that two operations are inverses by showing that one undoes the effect of the other, returning to the original number.

×3
8 × 3 = 24
→
÷3
24 ÷ 3 = 8
8 × 3 = 24
3 × 8 = 24
24 ÷ 8 = 3
24 ÷ 3 = 8
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Original number
8
Multiply by 3
8 × 3 = 24
Divide by 3
24 ÷ 3 = 8
Step 1: Start with the original number

Begin with 8

Step 2: Apply multiplication

8 × 3 = 24 (multiply 8 by 3)

Step 3: Apply inverse division

24 ÷ 3 = 8 (divide 24 by 3)

Step 4: Verify the result

We returned to the original number 8, confirming the inverse relationship

8 × 3 = 24 and 24 ÷ 3 = 8
Final answer:

8 × 3 = 24 and 24 ÷ 3 = 8 are inverse operations because starting with 8, multiplying by 3 gives 24, and dividing by 3 returns to 8

Applied rules:

• Inverse Verification: Operations that undo each other return to original values

• Reversibility: Multiplication and division can reverse each other's effects

• Fact Family Completeness: All four related facts connect the same numbers

5 Problem Solving
Exercise 5
If 6 × 7 = 42, what are the other three facts in this fact family? How do they demonstrate the relationship between multiplication and division?
Definition:

Problem Solving: Using the known relationship between multiplication and division to find missing facts in a fact family and understand how the operations connect.

6 × 7 = 42
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6
×
6 × 7 = 42
🔄
÷
42 ÷ 7 = 6
Given fact
6 × 7 = 42
Find 3 more
7 × 6, 42 ÷ 6, 42 ÷ 7
Complete family
4 facts
Step 1: Use commutative property

6 × 7 = 42, so 7 × 6 = 42

Step 2: Find first division fact

42 ÷ 6 = 7 (what number times 6 equals 42?)

Step 3: Find second division fact

42 ÷ 7 = 6 (what number times 7 equals 42?)

Step 4: Verify all facts

All four facts use only the numbers 6, 7, and 42

6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6
Final answer:

The complete fact family is: 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6. These facts demonstrate that multiplication and division are inverse operations.

Applied rules:

• Fact Family Completion: Given one fact, find the other three related facts

• Commutative Property: a × b = b × a

• Inverse Relationship: If a × b = c, then c ÷ a = b and c ÷ b = a

Advanced Relationship Concepts
a × b = c ↔ c ÷ a = b ↔ c ÷ b = a
Bidirectional Relationship
Key definitions:

Factor: A number that divides another number evenly

Product: The result of multiplying two or more numbers

Dividend: The number being divided in a division problem

Divisor: The number that divides the dividend

Quotient: The result of a division problem

Problem-Solving Strategies:
  1. Identify the operation needed: Determine if multiplication or division is required
  2. Recognize the relationship: Use the inverse relationship to find unknowns
  3. Apply the appropriate fact: Use known multiplication or division facts
  4. Verify with inverse operation: Check answers using the opposite operation
  5. Connect to real-world contexts: Apply relationships to practical problems
Tip 1: If you forget a division fact, think of the related multiplication fact.
Tip 2: Always check division by multiplying your answer by the divisor.
Tip 3: Use arrays and groups to visualize the relationship between operations.
Tip 4: Understanding this relationship helps with all future math concepts.
Common Applications: Equal sharing, equal grouping, rate problems, and measurement conversions.
Foundation for Algebra: Understanding inverse operations is crucial for solving equations.
Advanced rules:

• Commutative Property: a × b = b × a (creates two multiplication facts)

• Inverse Operations: Multiplication and division undo each other

• Fact Family Rule: Three numbers create four related facts

• Verification Method: Use inverse operations to check answers

Example
Multiplication
Related Division
Basic
3 × 4 = 12
12 ÷ 4 = 3
Medium
6 × 7 = 42
42 ÷ 7 = 6
Advanced
8 × 9 = 72
72 ÷ 9 = 8
Challenge
12 × 5 = 60
60 ÷ 5 = 12

Questions & Answers

Question: Why are multiplication and division called inverse operations? What does "inverse" mean?

Answer: "Inverse" means "opposite" or "reverse." Multiplication and division are inverse operations because they undo each other:

  • Multiplication: Combines equal groups to make a larger number
  • Division: Separates a larger number into equal groups

Think of it like this:

  • Start with 5
  • Multiply by 3: 5 × 3 = 15
  • Now divide by 3: 15 ÷ 3 = 5
  • We're back to where we started!

It's like taking 3 steps forward and then 3 steps backward - you end up in the same place.

Question: How can I help my child understand this relationship better?

Answer: Here are some effective strategies:

  • Use physical objects: Counters, blocks, or toys to show grouping and sharing
  • Draw pictures: Arrays and groups help visualize the relationship
  • Practice fact families: Work with sets of related multiplication and division facts
  • Real-world examples: Share cookies equally among friends (division) or arrange chairs in rows (multiplication)
  • Check answers: Always verify division by multiplying the result

The key is to make the connection visible and tangible for your child.

Question: Does this relationship work with all numbers, even big ones?

Answer: Yes! The relationship between multiplication and division works for ALL numbers:

  • Small numbers: 3 × 4 = 12, so 12 ÷ 4 = 3
  • Medium numbers: 15 × 6 = 90, so 90 ÷ 6 = 15
  • Large numbers: 24 × 50 = 1200, so 1200 ÷ 50 = 24

The principle is universal: if a × b = c, then c ÷ a = b and c ÷ b = a. This relationship is fundamental to all of mathematics and doesn't depend on the size of the numbers involved.