Solved Exercises on Remainders in Division for Grade 4

Master remainders in division: understanding remainders, interpreting division with remainders, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic remainder concept
Exercise 1
What is the remainder when 17 is divided by 5? Show your work.
Definition:

Remainder: The amount left over when one number cannot be divided evenly by another. In division: Dividend = Divisor × Quotient + Remainder.

Remainder calculation method:
  1. Divide the dividend by the divisor
  2. Find the largest multiple of the divisor that fits in the dividend
  3. Subtract that multiple from the dividend
  4. The result is the remainder
Division
17 ÷ 5
Quotient
3
Remainder
2
Step 1: Divide 17 by 5

How many times does 5 go into 17?

5 × 3 = 15 (fits in 17)

5 × 4 = 20 (too big for 17)

Step 2: Calculate the largest multiple

5 × 3 = 15 (this is the largest multiple of 5 that fits in 17)

Step 3: Find the remainder

17 - 15 = 2

This is what's left over after dividing 17 by 5

Step 4: Verify the answer

Dividend = Divisor × Quotient + Remainder

17 = 5 × 3 + 2

17 = 15 + 2 ✓

17 ÷ 5 = 3 R 2
Final answer:

The remainder when 17 is divided by 5 is 2.

Applied rules:

• Remainder: The leftover amount after division

• Verification: Dividend = Divisor × Quotient + Remainder

• Range: Remainder is always less than the divisor

2 Visual remainder representation
Exercise 2
Use visual grouping to show 23 ÷ 4 and identify the remainder.
Definition:

Visual grouping: Using physical objects or drawings to represent division and see the remainder clearly. Groups are formed with the divisor, and leftover items are the remainder.

Division
23 ÷ 4
Groups
5 groups of 4
Remainder
3
Step 1: Create groups of 4

We have 23 items and want to group them in sets of 4

Group 1: 4 items

Group 2: 4 items

Group 3: 4 items

Group 4: 4 items

Group 5: 4 items

Step 2: Count total items in groups

5 groups × 4 items = 20 items

Step 3: Find remaining items

23 total items - 20 items in groups = 3 items left over

Step 4: Express the result

23 ÷ 4 = 5 R 3

5 complete groups with 3 items remaining

23 ÷ 4 = 5 R 3
Final answer:

When 23 is divided by 4, we get 5 complete groups with 3 items remaining.

Applied rules:

• Visual representation: Helps understand division conceptually

• Grouping: Form groups equal to the divisor

• Remainder: Items that don't fit into complete groups

3 Interpreting remainders
Exercise 3
If 31 cookies are shared equally among 6 children, how many cookies does each child get and how many are left over?
Definition:

Interpreting remainders: Understanding what the remainder means in real-world contexts. Sometimes it's ignored, sometimes it needs to be accounted for.

Division
31 ÷ 6
Quotient
5
Remainder
1
Step 1: Set up the division

We have 31 cookies to share among 6 children

This is 31 ÷ 6

Step 2: Calculate how many per child

How many times does 6 go into 31?

6 × 5 = 30 (fits in 31)

6 × 6 = 36 (too big for 31)

Step 3: Find the remainder

31 - 30 = 1

This is the number of cookies left over

Step 4: Interpret the result

Each child gets 5 cookies

There is 1 cookie left over

Each child gets 5 cookies, 1 cookie remains
Final answer:

Each child gets 5 cookies and there is 1 cookie left over.

Applied rules:

• Real-world interpretation: Remainders have practical meaning

• Distribution: Quotient tells how many each person gets

• Leftover: Remainder tells what's left after equal distribution

Remainders in Division Summary
\(Dividend = Divisor × Quotient + Remainder\)
Division Algorithm
Definition
Remainder
Leftover after division
Formula
D = d × q + r
Where r < d
Range
0 ≤ r < d
Remainder is always smaller than divisor
Solution: Exercises 4 to 5
4 Word problem with remainders
Exercise 4
A teacher has 43 pencils to distribute equally among 7 students. How many pencils will each student get? How many pencils will be left over?
Definition:

Word problems with remainders: Real-world scenarios where division with remainders provides meaningful answers about equal distribution and leftovers.

Division
43 ÷ 7
Quotient
6
Remainder
1
Step 1: Set up the problem

43 pencils ÷ 7 students

We need to find how many pencils each student gets

Step 2: Calculate how many per student

How many times does 7 go into 43?

7 × 6 = 42 (fits in 43)

7 × 7 = 49 (too big for 43)

Step 3: Find the remainder

43 - 42 = 1

This is the number of pencils left over

Step 4: Interpret the result

Each student gets 6 pencils

There is 1 pencil left over

Each student gets 6 pencils, 1 pencil remains
Final answer:

Each student gets 6 pencils and there is 1 pencil left over.

Applied rules:

• Equal distribution: Quotient shows equal shares

• Leftover items: Remainder shows what's left after equal distribution

• Verification: 7 × 6 + 1 = 42 + 1 = 43 ✓

5 Remainder applications
Exercise 5
A baker has 58 cupcakes and wants to pack them in boxes that hold 8 cupcakes each. How many full boxes can be filled? How many cupcakes will be left over?
Definition:

Remainder applications: Practical problems where remainders help determine capacity, scheduling, and resource allocation.

Division
58 ÷ 8
Quotient
7
Remainder
2
Step 1: Identify the division

58 cupcakes ÷ 8 cupcakes per box

How many complete boxes can be filled?

Step 2: Calculate complete boxes

How many times does 8 go into 58?

8 × 7 = 56 (fits in 58)

8 × 8 = 64 (too big for 58)

Step 3: Find remaining cupcakes

58 - 56 = 2

This is the number of cupcakes that won't fit in a full box

Step 4: State the complete answer

7 full boxes can be filled

2 cupcakes will be left over

7 full boxes, 2 cupcakes remain
Final answer:

7 full boxes can be filled and 2 cupcakes will be left over.

Applied rules:

• Capacity problems: Quotient shows how many full containers

• Leftover items: Remainder shows excess that doesn't fill a container

• Verification: 8 × 7 + 2 = 56 + 2 = 58 ✓

Remainders in Division Guide
\(Dividend = Divisor × Quotient + Remainder\)
Division Algorithm
Key definitions:

Division: The process of splitting a number into equal groups

Dividend: The number being divided

Divisor: The number we divide by

Quotient: The result of division (how many equal groups)

Remainder: The amount left over when division is not exact

Finding remainders method:
  1. Divide: How many times does the divisor go into the dividend?
  2. Multiply: Multiply the divisor by the quotient
  3. Subtract: Subtract this product from the dividend
  4. Check: The result is the remainder (should be less than divisor)
Tip 1: The remainder is always smaller than the divisor.
Tip 2: If remainder is 0, the division is exact (no remainder).
Tip 3: Use multiplication facts to find how many times the divisor goes into the dividend.
Tip 4: Always verify your answer: Divisor × Quotient + Remainder = Dividend.
Common error: Having a remainder larger than the divisor (remainder must always be smaller).
Key insight: Remainders help us understand incomplete divisions in real-world contexts.
Important rules to remember:

• Remainder range: 0 ≤ remainder < divisor

• Exact division: When remainder = 0

• Verification: Dividend = Divisor × Quotient + Remainder

• Remainder meaning: The amount left over after equal distribution

Remainders in Division: Understanding Patterns
Exercise 6: Remainder Patterns
Explore how remainders change when dividing consecutive numbers by the same divisor:
Division with remainders and their patterns

Analysis: The chart shows how remainders cycle when dividing consecutive numbers by the same divisor.

  • When dividing by 4: remainders cycle 0, 1, 2, 3, 0, 1, 2, 3...
  • When dividing by 5: remainders cycle 0, 1, 2, 3, 4, 0, 1, 2, 3, 4...
  • The remainder pattern repeats every divisor number of terms
  • Remainder is always less than the divisor

Questions & Answers

Question: Why do we have remainders in division? Can't we always divide evenly?

Answer: Great question! We have remainders because not all numbers divide evenly. Here's why:

  • Not all numbers are multiples: For example, 17 is not a multiple of 5
  • Real-world situations: Sometimes we can't divide things perfectly (like sharing 17 cookies among 5 people)
  • Mathematical reality: Division shows how many complete groups we can make and what's left over
  • Remainders are useful: They tell us what's left after equal distribution

Remainders help us understand the complete picture of division, not just the equal parts.

Question: Can the remainder ever be bigger than the divisor?

Answer: No, the remainder can never be bigger than the divisor. Here's why:

  • Definition: The remainder is what's left AFTER we've made as many complete groups as possible
  • Logic: If remainder was bigger than divisor, we could make another complete group
  • Example: If we had 17 ÷ 5 = 2 R 7, this would be wrong because 7 > 5
  • Correction: We'd use 7 ÷ 5 = 1 R 2, so 17 ÷ 5 = 3 R 2

The rule is: 0 ≤ remainder < divisor

Question: How can I check if my division with remainder is correct?

Answer: Use the division algorithm to verify: Dividend = Divisor × Quotient + Remainder

  • Example: 23 ÷ 4 = 5 R 3
  • Check: 4 × 5 + 3 = 20 + 3 = 23 ✓
  • Another check: Make sure remainder < divisor (3 < 4 ✓)
  • Verification: If the original number comes back, your answer is correct

This verification works for every division problem with remainders!