Solved Exercises on Division as Repeated Subtraction in Grade 3

Master division as repeated subtraction: visual representations, step-by-step subtraction, and connection to multiplication through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Repeated Subtraction
Exercise 1
Use repeated subtraction to solve: 15 ÷ 3
Show each step and explain how many groups you make.
Definition:

Division as repeated subtraction: Dividing by subtracting the divisor from the dividend repeatedly until reaching zero.

Repeated subtraction method:
  1. Start with the dividend (the number being divided)
  2. Subtract the divisor (the number you're dividing by)
  3. Count how many times you subtract
  4. Continue until you reach zero
Dividend
15
Divisor
3
Groups
5
15 - 3 = 12
1
12 - 3 = 9
2
9 - 3 = 6
3
6 - 3 = 3
4
3 - 3 = 0
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Step 1: Start with 15

Subtract 3: 15 - 3 = 12 (1 group made)

Step 2: Continue subtracting 3

12 - 3 = 9 (2 groups), 9 - 3 = 6 (3 groups), 6 - 3 = 3 (4 groups)

Step 3: Final subtraction

3 - 3 = 0 (5 groups made)

Step 4: Count the groups

We subtracted 3 a total of 5 times to reach 0

15 ÷ 3 = 5
Final answer:

15 ÷ 3 = 5. We made 5 groups of 3 by repeatedly subtracting 3 from 15.

Applied rules:

• Repeated subtraction: Keep subtracting the divisor until reaching zero

• Group counting: Count how many times you subtract to find the quotient

• Division concept: How many equal groups can be made

2 Larger Numbers
Exercise 2
Use repeated subtraction to solve: 24 ÷ 4
Show how many groups of 4 you can make from 24.
Definition:

Division with larger numbers: Same principle applies, but may require more subtraction steps.

24 - 4 = 20
1
20 - 4 = 16
2
16 - 4 = 12
3
12 - 4 = 8
4
8 - 4 = 4
5
4 - 4 = 0
6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Step 1: Start with 24

Subtract 4 repeatedly: 24 - 4 = 20 (1 group)

Step 2: Continue the process

20 - 4 = 16 (2 groups), 16 - 4 = 12 (3 groups), 12 - 4 = 8 (4 groups)

Step 3: Continue until zero

8 - 4 = 4 (5 groups), 4 - 4 = 0 (6 groups)

Step 4: Count the total groups

We subtracted 4 a total of 6 times to reach 0

24 ÷ 4 = 6
Final answer:

24 ÷ 4 = 6. We can make 6 groups of 4 from 24.

Applied rules:

• Consistent subtraction: Keep subtracting the same number each time

• Progress tracking: Keep track of how many subtractions you perform

• Zero termination: Stop when you reach zero

3 Connection to Multiplication
Exercise 3
Use repeated subtraction to solve: 18 ÷ 6
Then show how this connects to multiplication.
Definition:

Multiplication-division connection: Division as repeated subtraction shows the inverse relationship with multiplication.

18 - 6 = 12
1
12 - 6 = 6
2
6 - 6 = 0
3
18 ÷ 6 = 3
↔
3 × 6 = 18
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Step 1: Start with 18

Subtract 6: 18 - 6 = 12 (1 group)

Step 2: Continue subtracting 6

12 - 6 = 6 (2 groups)

Step 3: Final subtraction

6 - 6 = 0 (3 groups)

Step 4: Connect to multiplication

If 18 ÷ 6 = 3, then 3 × 6 = 18 (inverse operations)

3 × 6 = 18
↔
18 ÷ 6 = 3
Final answer:

18 ÷ 6 = 3. This connects to multiplication: 3 × 6 = 18.

Applied rules:

• Inverse operations: Division and multiplication are opposite operations

• Verification: Multiply the quotient by the divisor to get the dividend

• Relationship understanding: Shows how division undoes multiplication

Division as Repeated Subtraction Rules
a ÷ b = n means a - b - b - ... - b = 0 (n times)
Definition
Division
12 ÷ 4 = 3
3 groups of 4
Subtraction
12-4-4-4=0
3 times
Multiplication
3 × 4 = 12
Inverse connection
Key Rule: Division as repeated subtraction shows how many equal groups can be made.
Connection Logic: Division and multiplication are inverse operations.
Tip 1: Always subtract the same number (the divisor) each time.
Tip 2: Count each subtraction to find the quotient.
Tip 3: Stop when you reach zero.
Key Rules to Remember:

• Repeated subtraction: Keep subtracting the divisor until reaching zero

• Quotient counting: Count how many times you subtract to find the answer

• Inverse relationship: Division and multiplication are opposite operations

• Verification: Multiply quotient by divisor to get the original dividend

a ÷ b = c ⟺ a = b × c
Inverse Relationship
Division Repeated Subtraction Groups Made Multiplication Check 12 ÷ 3 12-3-3-3-3=0 4 times 4×3=12 20 ÷ 5 20-5-5-5-5=0 4 times 4×5=20 18 ÷ 6 18-6-6-6=0 3 times 3×6=18 15 ÷ 5 15-5-5-5=0 3 times 3×5=15
Additional Practice
4 Mixed Practice
Exercise 4
Use repeated subtraction to solve:
a) 16 ÷ 4
b) 21 ÷ 7
c) 30 ÷ 5
Definition:

Mixed practice: Applying repeated subtraction to various division problems.

Part a: 16 ÷ 4

16-4=12 (1), 12-4=8 (2), 8-4=4 (3), 4-4=0 (4)

16 ÷ 4 = 4

Part b: 21 ÷ 7

21-7=14 (1), 14-7=7 (2), 7-7=0 (3)

21 ÷ 7 = 3

Part c: 30 ÷ 5

30-5=25 (1), 25-5=20 (2), 20-5=15 (3), 15-5=10 (4), 10-5=5 (5), 5-5=0 (6)

30 ÷ 5 = 6

Final answers:

a) 16 ÷ 4 = 4

b) 21 ÷ 7 = 3

c) 30 ÷ 5 = 6

Applied rules:

• Consistency: Apply the same method to all problems

• Counting: Keep track of how many subtractions you perform

• Termination: Stop when you reach zero

5 Word Problem Application
Exercise 5
Sarah has 24 stickers. She wants to share them equally among her 6 friends. How many stickers will each friend get? Use repeated subtraction to solve.
Definition:

Word problem application: Using repeated subtraction to solve real-world division problems.

24 - 6 = 18
1
18 - 6 = 12
2
12 - 6 = 6
3
6 - 6 = 0
4
24 ÷ 6 = 4
↔
4 × 6 = 24
Step 1: Identify the problem

Sarah has 24 stickers to share among 6 friends

Step 2: Set up the division

24 ÷ 6 = ?

Step 3: Use repeated subtraction

24-6=18 (1), 18-6=12 (2), 12-6=6 (3), 6-6=0 (4)

Step 4: Interpret the result

Each friend gets 4 stickers

Final answer:

Each friend will get 4 stickers. 24 ÷ 6 = 4.

Applied rules:

• Problem translation: Convert word problems to mathematical operations

• Method application: Use repeated subtraction to solve division

• Result interpretation: Answer the original question

Comprehensive Summary: Division as Repeated Subtraction
a ÷ b = n ⟺ a - b - b - ... - b = 0 (n times)
Core Concept
Key Definitions:

Division as repeated subtraction: A method of dividing by subtracting the divisor from the dividend repeatedly until reaching zero.

Dividend: The number being divided (the starting number).

Divisor: The number being divided by (the number you subtract).

Quotient: The result of division (how many times you subtract).

Remainder: What's left over after division (zero when evenly divisible).

Inverse operations: Operations that undo each other (division and multiplication).

Complete Methodology:
  1. Identify dividend and divisor: Recognize the number being divided and the number dividing
  2. Start the process: Begin with the dividend
  3. Subtract repeatedly: Keep subtracting the divisor from the current number
  4. Count subtractions: Track how many times you subtract
  5. Stop at zero: Continue until you reach zero
  6. Record the quotient: The count is your answer
Tip 1: Always subtract the same number (the divisor) each time.
Tip 2: Count each subtraction to find the quotient.
Tip 3: Stop when you reach zero.
Tip 4: Verify by multiplying quotient by divisor to get dividend.
Common errors: Forgetting to count subtractions, subtracting wrong number, not stopping at zero.
Key insights: This method builds understanding of what division really means.
Core Rules and Principles:

• Repeated subtraction: Keep subtracting the divisor until reaching zero

• Quotient counting: The number of subtractions equals the quotient

• Inverse relationship: Division and multiplication are opposite operations

• Verification principle: Quotient × divisor = dividend

• Conceptual understanding: Shows how many equal groups can be made

Examples: 12÷3=4 (12-3-3-3-3=0), 20÷5=4 (20-5-5-5-5=0), 18÷6=3 (18-6-6-6=0)
Concrete Examples
Memory aid: "How many times can I take away the divisor?" - count each subtraction.
Success check: If 15÷3=5, then 5×3 should equal 15. ✓

Questions & Answers

Question: Why do we need to learn division as repeated subtraction? Isn't regular division faster?

Answer: Learning division as repeated subtraction is very important for several reasons:

Understanding the concept:

  • It helps you understand what division really means - making equal groups
  • You see that division is the opposite of multiplication
  • It builds a strong foundation for more advanced math

Problem-solving skills:

  • If you forget a division fact, you can figure it out using subtraction
  • It helps develop logical thinking and reasoning abilities

Connection to multiplication:

  • You see that 15 ÷ 3 = 5 means 3 × 5 = 15
  • This relationship is crucial for solving equations later

Once you understand the concept, you can use faster methods, but the understanding stays with you!

Question: My child is getting confused when doing repeated subtraction for larger numbers. How can I help them stay organized?

Answer: Here are effective strategies to help organize repeated subtraction:

Use a systematic approach:

  • Write each subtraction step on a new line
  • Circle or highlight the number you're subtracting each time
  • Keep a separate counter to track how many times you subtract

Visual organization:

  • Create a table with columns: "Current Number", "Subtract", "New Total", "Count"
  • Use tally marks to keep track of subtractions
  • Draw circles around each group as you form it

Chunking method:

  • For 30 ÷ 6, instead of 30-6-6-6-6-6, think "6+6=12, so two groups", then "12+12=24, so four groups", then "24+6=30, so five groups"
  • This reduces the number of steps for larger problems

Practice with smaller numbers first, then gradually work up to larger ones.

Question: What happens if I don't reach exactly zero when doing repeated subtraction? For example, with 17 ÷ 5?

Answer: Great question! When you don't reach exactly zero, you have a remainder:

Example with 17 ÷ 5:

  • 17 - 5 = 12 (1 group)
  • 12 - 5 = 7 (2 groups)
  • 7 - 5 = 2 (3 groups)

We stop here because 2 is less than 5 (our divisor).

Result: 17 ÷ 5 = 3 remainder 2, or 17 ÷ 5 = 3 R2

What this means:

  • We can make 3 complete groups of 5
  • We have 2 left over that aren't enough for another complete group
  • To check: 3 × 5 + 2 = 15 + 2 = 17 ✓

The remainder is always less than the divisor!