Division with Remainders: Comprehensive Examples for Grade 4

Complete collection of division with remainders examples ranging from basic to advanced, with step-by-step solutions and visual representations.

Basic Division Examples
\(Dividend = Divisor × Quotient + Remainder\)
Division Algorithm
Example 1: Basic Division
Problem: \(13 ÷ 4\)
Calculate the quotient and remainder
Components:

Dividend: 13 (the number being divided)

Divisor: 4 (the number we divide by)

Quotient: The result of division

Remainder: What's left over

Step 1: Estimate how many times 4 fits in 13

\(4 × 3 = 12\) and \(4 × 4 = 16\) (too big), so quotient = 3

Step 2: Calculate the product

\(4 × 3 = 12\)

Step 3: Find the remainder

\(13 - 12 = 1\)

Step 4: Verify remainder is smaller than divisor

\(1 < 4\) ✓

Step 5: Check with formula

\(13 = 4 × 3 + 1 = 12 + 1 = 13\) ✓

\(13 ÷ 4 = 3\) remainder \(1\)
Example 2: Another Basic Problem
Problem: \(21 ÷ 5\)
Calculate the quotient and remainder
Step 1: Estimate how many times 5 fits in 21

\(5 × 4 = 20\) and \(5 × 5 = 25\) (too big), so quotient = 4

Step 2: Calculate the product

\(5 × 4 = 20\)

Step 3: Find the remainder

\(21 - 20 = 1\)

Step 4: Verify remainder is smaller than divisor

\(1 < 5\) ✓

Step 5: Check with formula

\(21 = 5 × 4 + 1 = 20 + 1 = 21\) ✓

\(21 ÷ 5 = 4\) remainder \(1\)
Intermediate Division Examples
Example 3: Word Problem
Problem: Packing Bags
Sarah has 37 apples. She wants to pack them in bags of 6 apples each. How many full bags can she make and how many apples will remain?
Problem Analysis:

Dividend: 37 (total apples)

Divisor: 6 (apples per bag)

Quotient: Full bags that can be made

Remainder: Apples that won't fit in a full bag

Step 1: Set up the division

\(37 ÷ 6\)

Step 2: Estimate how many times 6 fits in 37

\(6 × 6 = 36\) and \(6 × 7 = 42\) (too big), so quotient = 6

Step 3: Calculate the product

\(6 × 6 = 36\)

Step 4: Find the remainder

\(37 - 36 = 1\)

Step 5: Interpret the result

Sarah can make 6 full bags with 1 apple remaining

6 full bags, 1 apple remaining
Example 4: Equal Distribution
Problem: Sharing Candies
A teacher has 45 candies to distribute equally among 7 students. How many candies does each student get and how many remain?
Step 1: Set up the division

Total candies (45) ÷ Number of students (7) = ?

Step 2: Estimate how many times 7 fits in 45

\(7 × 6 = 42\) and \(7 × 7 = 49\) (too big), so quotient = 6

Step 3: Calculate the product

\(7 × 6 = 42\)

Step 4: Find the remainder

\(45 - 42 = 3\)

Step 5: Interpret the result

Each student gets 6 candies with 3 candies remaining

Each student gets 6 candies, 3 candies remain
Advanced Division Examples
Example 5: Real-world Application
Problem: Packing Boxes
A store has 58 items to pack in boxes that hold 8 items each. How many full boxes can be filled and how many items will be left over?
Step 1: Set up the division

\(58 ÷ 8\)

Step 2: Estimate how many times 8 fits in 58

\(8 × 7 = 56\) and \(8 × 8 = 64\) (too big), so quotient = 7

Step 3: Calculate the product

\(8 × 7 = 56\)

Step 4: Find the remainder

\(58 - 56 = 2\)

Step 5: Interpret the result

7 full boxes can be filled with 2 items remaining

7 full boxes, 2 items remaining
Example 6: Transportation Scenario
Problem: Field Trip Planning
There are 89 students going on a field trip. Each bus can carry 12 students. How many buses are needed and how many seats will be empty in the last bus?
Step 1: Set up the division

\(89 ÷ 12\)

Step 2: Estimate how many times 12 fits in 89

\(12 × 7 = 84\) and \(12 × 8 = 96\) (too big), so quotient = 7

Step 3: Calculate the product

\(12 × 7 = 84\)

Step 4: Find the remainder

\(89 - 84 = 5\)

Step 5: Determine total buses needed

Since 5 students remain, we need 1 more bus for them, so 7 + 1 = 8 buses

Step 6: Calculate empty seats

Last bus has 5 students, so 12 - 5 = 7 empty seats

8 buses needed, 7 empty seats in the last bus
Key Concepts and Takeaways
Core Concepts:

Division Algorithm: \(Dividend = Divisor × Quotient + Remainder\)

Remainder Property: Remainder must always be smaller than the divisor

Verification: Always check your answer using the division algorithm

Real-world Meaning: Quotient = complete groups, Remainder = leftover items

Step-by-step Method:
  1. Identify components: Find dividend and divisor
  2. Estimate quotient: Find how many times divisor fits in dividend
  3. Multiply: Multiply divisor by estimated quotient
  4. Subtract: Subtract product from dividend to get remainder
  5. Verify: Ensure remainder is smaller than divisor
  6. Check: Verify using division algorithm
Tip 1: Use multiplication facts to quickly estimate quotients.
Tip 2: Always verify remainder is less than the divisor.
Tip 3: Check your answer by multiplying quotient by divisor and adding remainder.
Tip 4: In word problems, interpret quotient and remainder in context.
Common errors: Forgetting to check remainder is smaller than divisor, miscalculating multiplication facts, confusing quotient with remainder.
Key to success: Practice regularly, use visual aids, verify answers, understand real-world applications.

Questions & Answers

Question: How do I know if my remainder is correct?

Answer: You can verify your remainder is correct using two checks:

  1. Size check: Make sure remainder is smaller than the divisor
  2. Formula check: Use the division algorithm: Dividend = Divisor × Quotient + Remainder

Example: For 23 ÷ 5 = 4 remainder 3

  • Size check: 3 < 5 ✓
  • Formula check: 5 × 4 + 3 = 20 + 3 = 23 ✓

If both checks pass, your answer is correct!

Question: Why do we sometimes need to round up in word problems?

Answer: You need to round up when the remainder represents additional items that still need accommodation:

  • Transportation problems: If you have 23 people and buses hold 10 people each, 23÷10=2 remainder 3. You need 3 buses, not 2, because the 3 remaining people still need transportation.
  • Container problems: If you have 47 items and boxes hold 6 items each, 47÷6=7 remainder 5. You need 8 boxes, not 7, because the 5 remaining items need their own box.
  • Grouping problems: If you have 31 students and groups must have 4 students each, 31÷4=7 remainder 3. You need 8 groups because the 3 remaining students form another group.

Always consider the real-world context of the problem!

Question: What if I get confused between the quotient and remainder?

Answer: Here are some memory aids to remember the difference:

  • Quotient: Think "Q" for "Quantity" - it's the number of complete groups
  • Remainder: Think "R" for "Remains" - it's what's left over
  • Quotient: The bigger number in the answer (unless divisor is very large)
  • Remainder: Always smaller than the divisor

Example: In 23 ÷ 5 = 4 remainder 3:

  • Quotient (4) = number of complete groups of 5
  • Remainder (3) = number left over after making complete groups