Solved Exercises on Using Multiplication to Solve Division in Grade 4

Master using multiplication to solve division: division strategies, fact families, mental math, and inverse operations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Using Multiplication to Solve Division
Exercise 1
Use multiplication to solve:
36 ÷ 6, 42 ÷ 7, 54 ÷ 9
Definition:

Using multiplication to solve division: Finding how many times the divisor goes into the dividend by using known multiplication facts

Division method using multiplication:
  1. Think of the multiplication fact that matches the division problem
  2. Use the fact that division is the inverse of multiplication
  3. Apply the known multiple to solve the division
Multiplication
6 × 6 = 36
Division
36 ÷ 6 = 6
Results
6, 6, 6
Step 1: Think of the multiplication fact for 36 ÷ 6

What number times 6 equals 36?

6 × 6 = 36, so 36 ÷ 6 = 6

Step 2: Think of the multiplication fact for 42 ÷ 7

What number times 7 equals 42?

6 × 7 = 42, so 42 ÷ 7 = 6

Step 3: Think of the multiplication fact for 54 ÷ 9

What number times 9 equals 54?

6 × 9 = 54, so 54 ÷ 9 = 6

Step 4: Verify the results

6 × 6 = 36 ✓, 6 × 7 = 42 ✓, 6 × 9 = 54 ✓

36 ÷ 6 = 6
42 ÷ 7 = 6
54 ÷ 9 = 6
Final answer:

36 ÷ 6 = 6, 42 ÷ 7 = 6, 54 ÷ 9 = 6

Applied rules:

• Multiplication and division are inverse operations

• If a × b = c, then c ÷ a = b

• Know your multiplication facts to solve division quickly

2 Fact Family Division
Exercise 2
Use fact families to solve:
48 ÷ 8, 56 ÷ 7, 72 ÷ 9
Definition:

Fact family: A group of related multiplication and division facts using the same numbers

Fact Family
6 × 8 = 48
Division
48 ÷ 8 = 6
Results
6, 8, 8
Step 1: Identify the fact family for 48 ÷ 8

6 × 8 = 48, so 48 ÷ 8 = 6

Step 2: Identify the fact family for 56 ÷ 7

8 × 7 = 56, so 56 ÷ 7 = 8

Step 3: Identify the fact family for 72 ÷ 9

8 × 9 = 72, so 72 ÷ 9 = 8

Step 4: Verify all results

6 × 8 = 48 ✓, 8 × 7 = 56 ✓, 8 × 9 = 72 ✓

48 ÷ 8 = 6
56 ÷ 7 = 8
72 ÷ 9 = 8
Final answer:

48 ÷ 8 = 6, 56 ÷ 7 = 8, 72 ÷ 9 = 8

Applied rules:

• Fact families connect multiplication and division

• If a × b = c, then c ÷ a = b and c ÷ b = a

• Use known multiplication facts to solve division problems

3 Mental Math Division
Exercise 3
Use mental math to solve:
64 ÷ 8, 72 ÷ 8, 81 ÷ 9
Definition:

Mental math division: Using known multiplication facts to solve division problems in your head

Mental Strategy
Use multiples
Division
64 ÷ 8 = 8
Results
8, 9, 9
Step 1: Think of multiples for 64 ÷ 8

How many groups of 8 make 64?

8 × 8 = 64, so 64 ÷ 8 = 8

Step 2: Think of multiples for 72 ÷ 8

How many groups of 8 make 72?

9 × 8 = 72, so 72 ÷ 8 = 9

Step 3: Think of multiples for 81 ÷ 9

How many groups of 9 make 81?

9 × 9 = 81, so 81 ÷ 9 = 9

Step 4: Verify all results

8 × 8 = 64 ✓, 9 × 8 = 72 ✓, 9 × 9 = 81 ✓

64 ÷ 8 = 8
72 ÷ 8 = 9
81 ÷ 9 = 9
Final answer:

64 ÷ 8 = 8, 72 ÷ 8 = 9, 81 ÷ 9 = 9

Applied rules:

• Division is finding how many equal groups

• Use known multiplication facts to solve division mentally

• Check your answer by multiplying back

Rules and methods, laws,...
\(a \div b = c \text{ means } b \times c = a\)
Division Relationship
Fact Family
8 × 7 = 56
56 ÷ 8 = 7, 56 ÷ 7 = 8
Multiplication Division
72 ÷ 8 = 9
Because 8 × 9 = 72
Mental Math
49 ÷ 7 = 7
Think: 7 × 7 = 49
Key definitions:

Division: The operation of splitting a number into equal parts

Quotient: The result of a division problem

Dividend: The number being divided

Divisor: The number that divides the dividend

Fact family: A set of related multiplication and division facts

Complete methodology:
  1. Identify the dividend and divisor
  2. Think of the related multiplication fact
  3. Use the known multiple to find the quotient
  4. Verify by multiplying the quotient by the divisor
Tip 1: Always start by thinking of the related multiplication fact.
Tip 2: Use fact families to connect multiplication and division.
Tip 3: Practice your multiplication tables to improve division speed.
Tip 4: Always verify your division by multiplying back.
Common errors: Confusing dividend and divisor, forgetting to verify answers, not knowing multiplication facts.
Mental math strategies: Use known facts, break apart numbers, recognize patterns.
Formulas to know by heart:

• Basic division facts: Know all facts up to 12 ÷ 12

• Fact families: If a × b = c, then c ÷ a = b and c ÷ b = a

• Verification: Quotient × Divisor = Dividend

• Division and multiplication are inverse operations

Solution: Exercises 4 to 5
4 Using Multiples of 10
Exercise 4
Use multiplication to solve:
120 ÷ 12, 140 ÷ 14, 160 ÷ 16
Definition:

Division with multiples of 10: Using basic division facts and place value to solve division problems

Basic Fact
12 ÷ 12 = 1
With Zero
120 ÷ 12 = 10
Results
10, 10, 10
Step 1: Identify the basic division fact for 120 ÷ 12

12 ÷ 12 = 1, so 120 ÷ 12 = 10 (add one zero)

Step 2: Identify the basic division fact for 140 ÷ 14

14 ÷ 14 = 1, so 140 ÷ 14 = 10 (add one zero)

Step 3: Identify the basic division fact for 160 ÷ 16

16 ÷ 16 = 1, so 160 ÷ 16 = 10 (add one zero)

Step 4: Verify all results

10 × 12 = 120 ✓, 10 × 14 = 140 ✓, 10 × 16 = 160 ✓

120 ÷ 12 = 10
140 ÷ 14 = 10
160 ÷ 16 = 10
Final answer:

120 ÷ 12 = 10, 140 ÷ 14 = 10, 160 ÷ 16 = 10

Applied rules:

• Division with multiples of 10: divide the basic fact, add zeros

• Place value: each zero represents a power of 10

• Same basic fact applies to all multiples of 10

5 Pattern-Based Division
Exercise 5
Use patterns to solve:
45 ÷ 5, 54 ÷ 6, 63 ÷ 7
Definition:

Pattern-based division: Recognizing and extending division patterns to solve related problems

Pattern
45, 54, 63
Divide by
5, 6, 7
Results
9, 9, 9
Step 1: Recognize the pattern in dividends

45, 54, 63 - each number increases by 9

Step 2: Apply division for 45 ÷ 5

5 × 9 = 45, so 45 ÷ 5 = 9

Step 3: Apply division for 54 ÷ 6

6 × 9 = 54, so 54 ÷ 6 = 9

Step 4: Apply division for 63 ÷ 7

7 × 9 = 63, so 63 ÷ 7 = 9

Step 5: Notice the pattern

As the dividend increases by 9, and the divisor increases by 1, the quotient stays the same!

45 ÷ 5 = 9
54 ÷ 6 = 9
63 ÷ 7 = 9
Final answer:

45 ÷ 5 = 9, 54 ÷ 6 = 9, 63 ÷ 7 = 9

Applied rules:

• Pattern recognition: same quotient when dividend and divisor change proportionally

• Consistent relationship: when dividend increases by divisor's multiple, quotient may stay the same

• Verification: always check with multiplication

Key Concepts: Using Multiplication to Solve Division
\(a \div b = c \text{ means } b \times c = a\)
Division Relationship
Key definitions:

Division: The operation of splitting a number into equal parts

Quotient: The result of a division problem

Dividend: The number being divided

Divisor: The number that divides the dividend

Fact family: A set of related multiplication and division facts

Complete methodology:
  1. Identify the dividend and divisor
  2. Think of the related multiplication fact
  3. Use the known multiple to find the quotient
  4. Verify by multiplying the quotient by the divisor
Tip 1: Always start by thinking of the related multiplication fact.
Tip 2: Use fact families to connect multiplication and division.
Tip 3: Practice your multiplication tables to improve division speed.
Tip 4: Always verify your division by multiplying back.
Tip 5: Look for patterns when dividing by the same number repeatedly.
Common errors: Confusing dividend and divisor, forgetting to verify answers, not knowing multiplication facts.
Mental math strategies: Use known facts, break apart numbers, recognize patterns.
Pattern recognition: Look for relationships between related division problems.
Formulas to know by heart:

• Basic division facts: Know all facts up to 12 ÷ 12

• Fact families: If a × b = c, then c ÷ a = b and c ÷ b = a

• Verification: Quotient × Divisor = Dividend

• Division and multiplication are inverse operations

Questions & Answers

Question: I don't understand how using multiplication helps me solve division faster. Can you explain?

Answer: Using multiplication makes division faster because division and multiplication are inverse operations! Here's how:

  • Instead of dividing 36 ÷ 6 directly, think: "What number times 6 equals 36?"
  • You know 6 × 6 = 36, so 36 ÷ 6 = 6
  • This uses your knowledge of multiplication facts to solve division problems

The more multiplication facts you know, the faster you can solve division!

Question: What is a fact family and how does it help with division?

Answer: A fact family is a group of related multiplication and division facts using the same numbers!

For example, with 6, 7, and 42:

  • 6 × 7 = 42
  • 7 × 6 = 42
  • 42 ÷ 6 = 7
  • 42 ÷ 7 = 6

If you know the multiplication facts, you automatically know the division facts!

Question: What if I forget my multiplication facts? How can I still use multiplication to solve division?

Answer: If you forget multiplication facts, you can still use multiplication to solve division by counting up!

For example, for 48 ÷ 8:

  • Count by 8s: 8, 16, 24, 32, 40, 48
  • How many 8s did you count? 6
  • So 48 ÷ 8 = 6

This takes longer but still uses the concept of multiplication!

Complete Summary: Using Multiplication to Solve Division
Key Definitions:

Division: The operation of splitting a number into equal parts. It's the opposite of multiplication.

Quotient: The result you get when you divide one number by another.

Dividend: The number you're dividing up (the big number).

Divisor: The number you're dividing by (the number outside).

Fact Family: A group of related multiplication and division facts using the same numbers.

Core Rules and Principles:

• Division and multiplication are inverse operations: If a × b = c, then c ÷ a = b

• Fact families connect multiplication and division: Same numbers create related facts.

• Use known multiples: Think of multiplication facts to solve division problems.

• Verification: Always check your division by multiplying the quotient by the divisor.

Step-by-Step Method:
  1. Identify the dividend and divisor: Find the number being divided and the number dividing it.
  2. Think of the related multiplication fact: What number times the divisor equals the dividend?
  3. Use the known multiple: Apply the multiplication fact to find the quotient.
  4. Verify: Multiply the quotient by the divisor to check your answer.
Simple Examples

Using multiples: 28 ÷ 4 = 7 because 4 × 7 = 28

Fact family: 7 × 8 = 56, so 56 ÷ 7 = 8 and 56 ÷ 8 = 7

Mental math: 45 ÷ 9 = 5 because you know 9 × 5 = 45

Advanced Examples

Multiples of 10: 90 ÷ 9 = 10 (because 9 × 10 = 90)

Pattern recognition: 40 ÷ 5 = 8, 48 ÷ 6 = 8, 56 ÷ 7 = 8

Multiple steps: 84 ÷ 7 = (70 ÷ 7) + (14 ÷ 7) = 10 + 2 = 12

Tips: Always start by thinking of the related multiplication fact. Practice your times tables regularly!
Common Pitfalls: Don't confuse the dividend and divisor. Always verify your answer by multiplying back.
Memorization Tips: Practice fact families together. If you know 9 × 8 = 72, you also know 72 ÷ 8 = 9.
Key Takeaways:

• Using multiplication makes division faster and more accurate.

• Division and multiplication are inverse operations.

• Fact families connect multiplication and division facts.

• Always verify your division by multiplying back.

• Practice multiplication tables to improve division speed.