Solved Exercises on Division with Remainders in Grade 4

Master division with remainders: basic division, word problems, and interpreting remainders through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Basic division with remainder
Exercise 1
What is 23 divided by 4? Write the answer with a remainder.
Definition:

Division with remainder: When one number doesn't divide evenly into another, the leftover amount is called the remainder. The result is expressed as: Dividend ÷ Divisor = Quotient R Remainder.

Division Method:
  1. Find how many times the divisor goes into the dividend evenly
  2. Multiply that number by the divisor
  3. Subtract that product from the dividend
  4. The result is the remainder
  5. Express as: Quotient R Remainder
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23 ÷ 4 = 5 R 3
Step 1: Find how many times 4 goes into 23

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

4 × 5 = 20 (fits in 23), but 4 × 6 = 24 (too big)

Step 2: Multiply and subtract

5 × 4 = 20, then 23 - 20 = 3

Step 3: Write the answer

23 ÷ 4 = 5 R 3 (5 groups of 4 with 3 left over)

Step 4: Verify the answer

5 × 4 + 3 = 20 + 3 = 23 ✓

23 ÷ 4 = 5 R 3
Final answer:

23 divided by 4 equals 5 with a remainder of 3.

Applied rules:

• Division equation: Dividend = Divisor × Quotient + Remainder

• Remainder rule: Remainder must be less than the divisor

• Verification: Always check by multiplying and adding remainder

2 Word problem with remainder
Exercise 2
Sarah has 37 stickers. She wants to put them equally on 5 pages of her sticker book. How many stickers will go on each page, and how many will be left over?
Definition:

Word problem interpretation: In word problems, the quotient tells us how many items go to each group, and the remainder tells us how many items are left over after equal distribution.

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37 ÷ 5 = 7 R 2
Step 1: Set up the division

Total stickers = 37, Number of pages = 5, So 37 ÷ 5

Step 2: Find how many times 5 goes into 37

5 × 7 = 35 (fits in 37), but 5 × 8 = 40 (too big)

Step 3: Calculate the remainder

37 - 35 = 2

Step 4: Interpret the answer

Each page gets 7 stickers, with 2 stickers left over

Each page gets 7 stickers, with 2 stickers remaining.
Final answer:

Each page will have 7 stickers, and 2 stickers will be left over.

Applied rules:

• Quotient meaning: How many items per group

• Remainder meaning: Items left after equal distribution

• Context interpretation: Answer must make sense in the problem context

3 Interpreting remainders
Exercise 3
A teacher has 29 students and wants to arrange them in groups of 6. How many complete groups can be made? Will any students be left out?
Definition:

Interpreting remainders: The remainder often has practical significance in real-world problems. Sometimes you need to round up, sometimes ignore it, depending on the context.

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29 ÷ 6 = 4 R 5
Step 1: Set up the division

Total students = 29, Group size = 6, So 29 ÷ 6

Step 2: Find how many times 6 goes into 29

6 × 4 = 24 (fits in 29), but 6 × 5 = 30 (too big)

Step 3: Calculate the remainder

29 - 24 = 5

Step 4: Interpret the answer in context

4 complete groups can be made, with 5 students left over who won't form a complete group

4 complete groups can be made, with 5 students left over.
Final answer:

4 complete groups can be made, and 5 students will be left out of complete groups.

Applied rules:

• Context matters: Remainder interpretation depends on the problem

• Complete groups: Only count full sets when asked for complete units

• Leftover meaning: Remainder represents incomplete or extra items

Rules and methods, laws,...
\(\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\)
Division Relationship
Division
Dividend ÷ Divisor = Quotient R Remainder
Standard form of division with remainder
Verification
Divisor × Quotient + Remainder = Dividend
Check if your answer is correct
Remainder
Remainder < Divisor
Remainder must be smaller than divisor

Division Components

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Dividend

Definition: The number being divided

Example: In 23 ÷ 4, 23 is the dividend

Role: Total amount to be distributed

Divisor

Definition: The number that divides the dividend

Example: In 23 ÷ 4, 4 is the divisor

Role: Size of each group

Quotient

Definition: The result of division (how many groups)

Example: In 23 ÷ 4 = 5 R 3, 5 is the quotient

Role: Number of complete groups

Remainder

Definition: Leftover amount after division

Example: In 23 ÷ 4 = 5 R 3, 3 is the remainder

Rule: Must be less than the divisor

Key Concepts for Division with Remainders
\(\text{23} ÷ \text{4} = \text{5 R 3}\)
Division Example
Key definitions:

Division: The process of sharing or grouping a number into equal parts

Quotient: The result of division; how many equal groups can be made

Remainder: The amount left over when a number doesn't divide evenly

Dividend: The number being divided

Divisor: The number by which the dividend is divided

Complete division method:
  1. Estimate: Think about how many times the divisor fits in the dividend
  2. Multiply: Multiply your estimate by the divisor
  3. Subtract: Subtract from the dividend to find remainder
  4. Check: Verify remainder is less than divisor
  5. Verify: Multiply quotient by divisor and add remainder
Tip 1: Always check that your remainder is smaller than the divisor.
Tip 2: Use multiplication facts to help with division.
Tip 3: Draw pictures or use objects to visualize division.
Tip 4: In word problems, consider what the remainder means in context.
Common errors: Forgetting to include remainder, remainder larger than divisor, misinterpreting word problems.
Learning outcomes: Develop number sense, understand division concept, solve real-world problems.
Essential rules:

• Remainder rule: Remainder must always be less than the divisor

• Verification rule: Divisor × Quotient + Remainder = Dividend

• Context rule: Interpret remainder based on the problem situation

• Zero remainder: When remainder is 0, division is exact

Comprehensive Summary
Complete Guide to Division with Remainders
Definitions of Key Concepts

Division: The mathematical operation that splits a number into equal groups. When division doesn't result in equal groups, there's a remainder.

Dividend: The number being divided. It's the total amount you start with.

Divisor: The number by which you divide. It tells you the size of each group.

Quotient: The result of division. It tells you how many complete groups you can make.

Remainder: The amount left over after division. It's what remains after making as many complete groups as possible.

Core Rules and Principles

Division Relationship: Dividend = Divisor × Quotient + Remainder

Remainder Constraint: The remainder must always be smaller than the divisor.

Verification Rule: To check your answer, multiply the divisor by the quotient and add the remainder. You should get the original dividend.

Exact Division: When the remainder is 0, the division is said to be exact.

Step-by-Step Methods

Basic Division Steps:

  1. Estimate how many times the divisor fits into the dividend
  2. Multiply your estimate by the divisor
  3. Subtract this product from the dividend
  4. The result is your remainder
  5. Express as: Quotient R Remainder
  6. Verify: Divisor × Quotient + Remainder = Dividend

Word Problem Steps:

  1. Identify what needs to be divided and into how many groups
  2. Set up the division problem
  3. Solve using division with remainder
  4. Interpret the quotient and remainder in context
  5. Answer the specific question asked
Examples from Simple to Advanced

Simple Example: 15 ÷ 3 = 5 R 0 (exact division, no remainder)

Intermediate Example: 17 ÷ 4 = 4 R 1 (4 groups of 4, with 1 left over)

Advanced Example: 100 ÷ 7 = 14 R 2 (14 complete groups of 7, with 2 remaining)

Tips, Tricks, and Common Pitfalls

Tips:

  • Use multiplication facts to help estimate quotients
  • Always check that remainder is less than divisor
  • Draw pictures or use objects for visualization
  • Verify your answer by multiplying and adding remainder
  • In word problems, think about what remainder means in context

Common Mistakes:

  • Having a remainder that's larger than the divisor
  • Forgetting to include the remainder in the answer
  • Misinterpreting what the remainder means in word problems
  • Not checking work by verifying the answer
  • Confusing which number is the dividend vs. divisor
Key Notes for Memorization

Memory Aid: "Divide, Multiply, Subtract, Bring Down" (for long division) and remember "Remainder must be smaller than Divisor."

Verification Check: Always use "Divisor × Quotient + Remainder = Dividend" to check your work.

Context Clue: In word problems, the question often indicates how to interpret the remainder (round up, ignore, or report as is).

Quick Check: If remainder ≥ divisor, you need to increase your quotient.

Questions & Answers

Question: Why can't the remainder be bigger than the divisor? What happens if it is?

Answer: Great question! The remainder cannot be bigger than the divisor because if it were, you could still divide more. Think of it this way:

If you have 17 ÷ 4 and somehow got a remainder of 5, that would mean you still had 5 left over. But since 5 is bigger than 4 (the divisor), you could still make another group of 4! So 5 ÷ 4 = 1 R 1, which means you can make one more group and have 1 left over.

The correct answer would be: Instead of 4 R 5, it would be 5 R 1 (4 + 1 = 5 groups, with 1 remaining).

So whenever your remainder is equal to or larger than the divisor, you need to divide again!

Question: My child struggles with knowing what to do with the remainder in word problems. Sometimes they should ignore it, sometimes round up. How do I help them decide?

Answer: The key is to focus on what the question is asking:

  • Ignore remainder: When the question asks for complete groups only ("How many full boxes can be filled?")
  • Round up: When everything needs to be accounted for ("How many buses are needed?")
  • Report remainder: When the question specifically asks about leftovers ("How many are left over?")

Help your child identify key phrases: "complete," "full," "whole" usually mean to ignore the remainder. "Total needed," "required" often mean to round up. "Remaining," "left over" mean to report the remainder.

Always encourage them to reread the question to see exactly what is being asked!

Question: How can I help my students visualize division with remainders?

Answer: Visual representations are very effective! Here are some strategies:

  • Manipulatives: Use counters, cubes, or other objects to physically group and show remainders
  • Arrays: Draw rectangular arrays and circle groups to show complete sets
  • Number lines: Show jumps of the divisor size until you can't make another complete jump
  • Story contexts: Use real scenarios like sharing cookies or arranging chairs

Emphasize that the remainder is what's "left over" after making as many complete groups as possible. Students often grasp the concept better when they can see the physical separation of complete groups versus leftovers.