Division with Remainders with Verified Answers for 4th Grade Students

Master division with remainders: understanding remainders, long division steps, word problems, and practical applications through these 5 detailed exercises with verified answers.

Solution: Exercises 1 to 3
1 Basic Division with Remainder
Exercise 1
Solve: 23 ÷ 4 and verify your answer
Definition:

Division with remainder: When one number doesn't divide evenly into another, the leftover amount is called the remainder.

Division method:
  1. Find how many times the divisor goes into the dividend completely
  2. Multiply the divisor by that number
  3. Subtract to find the remainder
  4. Check: Dividend = (Divisor × Quotient) + Remainder
Total Items: 23
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Groups of 4
Group 1: 1,2,3,4
Group 2: 5,6,7,8
Group 3: 9,10,11,12
Group 4: 13,14,15,16
Group 5: 17,18,19,20
Leftover: 21,22,23
Problem
23 ÷ 4
Quotient
5
Remainder
3
Step 1: Determine how many groups of 4 fit in 23

4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, 4 × 4 = 16, 4 × 5 = 20, 4 × 6 = 24

Since 24 > 23, the largest multiple of 4 that fits in 23 is 20 (4 × 5)

Step 2: Calculate the remainder

23 - 20 = 3

So, 23 ÷ 4 = 5 remainder 3

Answer Verification:

Formula: Dividend = (Divisor × Quotient) + Remainder

23 = (4 × 5) + 3 = 20 + 3 = 23 ✓

Remainder (3) < Divisor (4) ✓

23 ÷ 4 = 5 R 3
Final answer:

23 ÷ 4 = 5 remainder 3

Applied rules:

• Division: How many times does the divisor fit into the dividend?

• Remainder: The amount left over after division

• Verification: Dividend = (Divisor × Quotient) + Remainder

2 Long Division Method
Exercise 2
Solve: 73 ÷ 8 using long division and verify your answer
Definition:

Long division: A systematic method for dividing larger numbers step by step.

______
8 ) 73
- 72
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1
Step 1: Set up the long division

Write 73 under the division symbol and 8 outside

Step 2: Divide 7 by 8

Since 7 < 8, we consider 73 instead

How many 8s are in 73? 8 × 9 = 72, 8 × 10 = 80

Since 80 > 73, the answer is 9

Step 3: Multiply and subtract

9 × 8 = 72

73 - 72 = 1

Step 4: Check the remainder

Since 1 < 8, we cannot divide further

So, 1 is the remainder

Answer Verification:

Formula: Dividend = (Divisor × Quotient) + Remainder

73 = (8 × 9) + 1 = 72 + 1 = 73 ✓

Remainder (1) < Divisor (8) ✓

73 ÷ 8 = 9 R 1
Final answer:

73 ÷ 8 = 9 remainder 1

Applied rules:

• Divide: How many times does the divisor go into the current part?

• Multiply: Multiply divisor by the quotient digit

• Subtract: Subtract the product from the current part

• Bring down: Bring down the next digit (if any)

3 Word Problem Application
Exercise 3
Tom has 38 marbles to share equally among 6 friends. How many marbles does each friend get and how many are left over? Verify your answer.
Definition:

Word problems: Real-life scenarios that require mathematical operations to solve.

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38 Marbles
→
👤👤👤👤👤👤
6 Friends
Problem
38 ÷ 6
Each gets
6 marbles
Left over
2 marbles
Step 1: Identify the operation needed

We need to divide 38 marbles among 6 friends

This means 38 ÷ 6

Step 2: Perform the division

6 × 6 = 36, 6 × 7 = 42

Since 42 > 38, each friend gets 6 marbles

38 - 36 = 2 marbles left over

Step 3: State the complete answer

Each friend gets 6 marbles, and 2 marbles are left over

Answer Verification:

Formula: Dividend = (Divisor × Quotient) + Remainder

38 = (6 × 6) + 2 = 36 + 2 = 38 ✓

Remainder (2) < Divisor (6) ✓

Each friend gets 6 marbles, remainder 2
Final answer:

Each friend gets 6 marbles and 2 marbles are left over

Applied rules:

• Identify key words: "share equally" indicates division

• Understand context: The remainder represents items that can't be distributed

• Complete answer: Include both quotient and remainder in context

Division with Remainders - Complete Summary
Dividend = (Divisor × Quotient) + Remainder
Fundamental Division Equation
Complete Summary: Division with Remainders
Core Definitions
  • Dividend: The number being divided (the top number)
  • Divisor: The number we divide by (the bottom number)
  • Quotient: The result of division (how many times it divides)
  • Remainder: What's left over after division
  • Division with remainder: When one number doesn't divide evenly into another
    • Core Rules and Laws
  • Remainder Rule: Remainder is always less than the divisor
  • Division Algorithm: Dividend = (Divisor × Quotient) + Remainder
  • Exact Division: When remainder is 0
  • Remainder Bounds: 0 ≤ Remainder < Divisor
  • Verification: Always check your answer using the division algorithm
    • Step-by-Step Methods
  • Step 1: Identify dividend and divisor
  • Step 2: Find how many times divisor fits in dividend
  • Step 3: Multiply divisor by quotient
  • Step 4: Subtract to find remainder
  • Step 5: Verify: (Divisor × Quotient) + Remainder = Dividend
    • Examples (Simple to Advanced)
  • Simple: 10 ÷ 3 = 3 R 1 (because 3×3=9, 10-9=1)
  • Medium: 25 ÷ 4 = 6 R 1 (because 4×6=24, 25-24=1)
  • Advanced: 87 ÷ 9 = 9 R 6 (because 9×9=81, 87-81=6)
  • Word Problem: 43 ÷ 7 = 6 R 1 (6 friends get 7 each, 1 left over)
    • Tips and Tricks
  • Tip 1: Always check that remainder < divisor
  • Tip 2: Use multiplication facts to help with division
  • Tip 3: Draw pictures for word problems
  • Tip 4: Always verify your answer using the division algorithm
  • Tip 5: Remember: remainder cannot equal or exceed the divisor
    • Common Mistakes to Avoid
  • Mistake 1: Having remainder ≥ divisor
  • Mistake 2: Not verifying the answer
  • Mistake 3: Incorrect subtraction in the division process
  • Mistake 4: Confusing quotient and remainder
  • Mistake 5: Forgetting to include remainder in the final answer
    • Key Takeaways for Memorization
  • Memory Aid: "Divide, Multiply, Subtract, Check"
  • Formula: Dividend = Divisor × Quotient + Remainder
  • Rule: Remainder < Divisor
  • Process: Always verify your answer
  • Concept: Remainder is what's left after equal sharing
    • Remainder Rules
  • Remainder is always less than the divisor
  • If remainder is 0, division is exact
  • Remainder cannot be equal to or greater than divisor
  • Always check: (Divisor × Quotient) + Remainder = Dividend
    • Long Division Steps
  • Divide: How many times does divisor go into current number?
  • Multiply: Multiply divisor by the quotient digit
  • Subtract: Subtract the product from current number
  • Bring Down: Bring down the next digit
  • Repeat: Continue until no digits remain
    • Real-World Applications
  • Sharing items among people
  • Grouping objects into equal sets
  • Distributing resources fairly
  • Calculating leftover materials
    • 20 apples
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    ÷
    6 baskets
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    =
    3 per basket
    2 left over
    20 ÷ 6 = 3 R 2
    Solution: Exercises 4 to 5
    4 Classroom Seating Problem
    Exercise 4
    There are 52 students in a class. If each table seats 7 students, how many tables are needed and how many students won't have a seat? Verify your answer.
    Definition:

    Application problem: Real-world scenarios requiring division with remainders to find practical solutions.

    👥
    52 Students
    ÷
    🪑
    7 per table
    =
    📋
    ? Tables
    Problem
    52 ÷ 7
    Tables needed
    8 tables
    Students left
    4 students
    Step 1: Identify the division needed

    We need to divide 52 students by 7 students per table

    This means 52 ÷ 7

    Step 2: Perform the division

    7 × 7 = 49, 7 × 8 = 56

    Since 56 > 52, we can fully fill 7 tables

    But we still have 52 - 49 = 3 students left

    These 3 students need an additional table

    Wait, let me recalculate: 7 × 7 = 49, remainder is 52 - 49 = 3

    Actually, 7 × 7 = 49, so 3 students are left over

    Step 3: Interpret the result

    7 full tables + 1 partially filled table = 8 total tables

    3 students will sit at the last table

    Answer Verification:

    Formula: Dividend = (Divisor × Quotient) + Remainder

    52 = (7 × 7) + 3 = 49 + 3 = 52 ✓

    Remainder (3) < Divisor (7) ✓

    So we need 7 full tables + 1 table for 3 students = 8 tables total

    8 tables needed, 3 students in the last table
    Final answer:

    8 tables are needed, and 3 students will be seated at the last table

    Applied rules:

    • Context interpretation: Remainder may require additional unit

    • Practical rounding: Need 8 tables even though only 3 students remain

    • Verification: (7 × 7) + 3 = 49 + 3 = 52 ✓

    5 Gift Packaging Problem
    Exercise 5
    Alex has 91 stickers to put on gift bags. If he puts 8 stickers on each bag, how many full bags can he make and how many stickers will be left? Verify your answer.
    Definition:

    Distribution problem: Problems where items are distributed equally with possible leftovers.

    91 stickers
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    + 71 more...
    →
    Groups of 8
    8 per bag
    Problem
    91 ÷ 8
    Full bags
    11 bags
    Leftover
    3 stickers
    Step 1: Identify the division needed

    We need to divide 91 stickers by 8 stickers per bag

    This means 91 ÷ 8

    Step 2: Perform the division

    8 × 11 = 88, 8 × 12 = 96

    Since 96 > 91, Alex can make 11 full bags

    91 - 88 = 3 stickers left over

    Step 3: State the complete answer

    Alex can make 11 full bags with 3 stickers remaining

    Answer Verification:

    Formula: Dividend = (Divisor × Quotient) + Remainder

    91 = (8 × 11) + 3 = 88 + 3 = 91 ✓

    Remainder (3) < Divisor (8) ✓

    11 full bags, 3 stickers left
    Final answer:

    Alex can make 11 full bags and will have 3 stickers left over

    Applied rules:

    • Division: How many times does 8 go into 91?

    • Remainder: What's left after making full groups?

    • Verification: (8 × 11) + 3 = 88 + 3 = 91 ✓

    Questions & Answers

    Question: I don't understand why we sometimes have remainders in division. Why can't everything divide evenly?

    Answer: Great question! In real life, things don't always divide evenly. Think of it this way:

    • If you have 7 cookies and want to share them equally among 3 friends, each friend gets 2 cookies (3×2=6), and 1 cookie is left over
    • The remainder represents the "leftovers" that can't be shared equally
    • Mathematically, when the dividend isn't a multiple of the divisor, there will be a remainder

    Example: 7 ÷ 3 = 2 R 1 because 3 × 2 = 6, and 7 - 6 = 1 leftover

    Remainders help us deal with real-world situations where equal sharing isn't perfectly possible!

    Question: My child struggles with long division. How can I help them understand the steps better?

    Answer: Try these strategies to help your child with long division:

    • Use manipulatives: Physical objects like counters or base-ten blocks to model the division process
    • Remember the acronym: DMSB (Divide, Multiply, Subtract, Bring down)
    • Start simple: Begin with 2-digit dividends and single-digit divisors
    • Practice multiplication facts: Strong knowledge of multiplication helps with division

    Visual representation is key: Show how many groups of the divisor fit into the dividend step by step.

    Encourage your child to always check their work: (Divisor × Quotient) + Remainder should equal the original dividend.

    Question: How do I know if my remainder is correct? Sometimes I get confused about whether it's too big.

    Answer: There's an easy rule to check if your remainder is correct:

    • The remainder must always be smaller than the divisor
    • If your remainder is equal to or bigger than the divisor, you made a mistake
    • Always check: (Divisor × Quotient) + Remainder = Original Dividend

    Example: In 23 ÷ 5 = 4 R 3, check that 3 < 5 ✓

    Then verify: (5 × 4) + 3 = 20 + 3 = 23 ✓

    If you got 23 ÷ 5 = 4 R 7, you'd know it's wrong because 7 > 5, so you could still divide more!