Division with Remainders Worksheet Free - 5 Detailed Exercises

Master division with remainders: step-by-step solutions, visual explanations, rules, and methods for 4th grade students.

Solution: Exercises 1 to 3
1 Basic division with remainder
Exercise 1
Divide 23 by 5:
\(23 ÷ 5 = ?\) remainder ?
Definition:

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

Division formula: Dividend = (Divisor × Quotient) + Remainder

Division method:
  1. Find how many times the divisor fits into the dividend
  2. Multiply divisor by quotient
  3. Subtract to find the remainder
  4. Check: Dividend = (Divisor × Quotient) + Remainder
Problem
\(23 ÷ 5\)
Quotient
\(4\)
Remainder
\(3\)
Step 1: Determine how many times 5 goes into 23

\(5 × 1 = 5\), \(5 × 2 = 10\), \(5 × 3 = 15\), \(5 × 4 = 20\), \(5 × 5 = 25\)

Since 25 > 23, the largest multiple of 5 that fits is 20 (which is 5 × 4)

Step 2: Calculate the quotient

5 goes into 23 exactly 4 times, so the quotient is 4

Step 3: Find the remainder

Subtract: \(23 - (5 × 4) = 23 - 20 = 3\)

Step 4: Verify the answer

Check: \(23 = (5 × 4) + 3 = 20 + 3 = 23\) ✓

\(23 ÷ 5 = 4\) remainder \(3\)
Final answer:

\(23 ÷ 5 = 4\) remainder \(3\)

Applied rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder rule: Remainder must be less than the divisor

• Verification: Always check your answer using the division algorithm

2 Division with visual representation
Exercise 2
Divide 31 by 6:
\(31 ÷ 6 = ?\) remainder ?
Definition:

Visual division: Grouping items into equal sets helps understand remainders.

Remainder concept: Leftover items that can't form a complete group.

Problem
\(31 ÷ 6\)
Quotient
\(5\)
Remainder
\(1\)
Step 1: Determine how many times 6 goes into 31

\(6 × 1 = 6\), \(6 × 2 = 12\), \(6 × 3 = 18\), \(6 × 4 = 24\), \(6 × 5 = 30\), \(6 × 6 = 36\)

Since 36 > 31, the largest multiple of 6 that fits is 30 (which is 6 × 5)

Step 2: Calculate the quotient

6 goes into 31 exactly 5 times, so the quotient is 5

Step 3: Find the remainder

Subtract: \(31 - (6 × 5) = 31 - 30 = 1\)

Step 4: Verify the answer

Check: \(31 = (6 × 5) + 1 = 30 + 1 = 31\) ✓

\(31 ÷ 6 = 5\) remainder \(1\)
Final answer:

\(31 ÷ 6 = 5\) remainder \(1\)

Applied rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder rule: Remainder must be less than the divisor

• Verification: Always check your answer using the division algorithm

3 Word problem with remainders
Exercise 3
Sarah has 47 candies. She wants to put them into bags of 7 candies each. How many full bags can she make and how many candies will be left over?
Definition:

Word problems: Real-life situations that require division with remainders.

Context interpretation: The quotient represents complete groups, the remainder represents leftovers.

Problem
\(47 ÷ 7\)
Quotient
\(6\)
Remainder
\(5\)
Step 1: Set up the division problem

Total candies (47) ÷ Candies per bag (7) = Number of full bags with remainder

Step 2: Determine how many times 7 goes into 47

\(7 × 1 = 7\), \(7 × 2 = 14\), \(7 × 3 = 21\), \(7 × 4 = 28\), \(7 × 5 = 35\), \(7 × 6 = 42\), \(7 × 7 = 49\)

Since 49 > 47, the largest multiple of 7 that fits is 42 (which is 7 × 6)

Step 3: Calculate the quotient and remainder

7 goes into 47 exactly 6 times, so Sarah can make 6 full bags

Leftover: \(47 - (7 × 6) = 47 - 42 = 5\) candies

Step 4: Interpret the answer in context

Sarah can make 6 full bags with 5 candies remaining

Sarah can make 6 full bags with 5 candies left over
Final answer:

Sarah can make 6 full bags with 5 candies left over

Applied rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder rule: Remainder must be less than the divisor

• Context interpretation: Quotient = complete groups, Remainder = leftovers

Division with Remainders: Rules and Methods
Dividend = (Divisor × Quotient) + Remainder
Division Algorithm
Rule 1
Remainder < Divisor
Remainder must be smaller
Rule 2
Quotient × Divisor ≤ Dividend
Product cannot exceed dividend
Rule 3
Remainder ≥ 0
Remainder is non-negative
Key definitions:

Dividend: The number being divided

Divisor: The number dividing the dividend

Quotient: The result of division (how many times divisor fits)

Remainder: Leftover amount that can't form another complete group

Complete methodology:
  1. Identify parts: Determine dividend, divisor, quotient, and remainder
  2. Find multiples: Determine largest multiple of divisor that fits in dividend
  3. Calculate quotient: How many times divisor fits completely
  4. Find remainder: Subtract (divisor × quotient) from dividend
  5. Verify: Check that Dividend = (Divisor × Quotient) + Remainder
Tip 1: Always check that your remainder is less than your divisor.
Tip 2: Use multiplication facts to quickly find how many times the divisor fits.
Tip 3: Draw pictures or use objects to visualize the division process.
Tip 4: Always verify your answer by multiplying and adding the remainder.
Common errors: Forgetting to check remainder is less than divisor, calculation mistakes in multiplication or subtraction.
Key to success: Practice multiplication tables, always verify answers, understand the relationship between multiplication and division.
Essential rules to remember:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder must always be less than the divisor

• Remainder must be greater than or equal to zero

• Always verify your answer by substituting back into the equation

Solution: Exercises 4 to 5
4 Multi-step division problem
Exercise 4
There are 89 students going on a field trip. Each bus can hold 12 students. How many buses are needed and how many seats will be empty?
Definition:

Multi-step problems: Require division with remainders followed by additional calculations.

Real-world context: May need to round up for practical applications.

Problem
\(89 ÷ 12\)
Quotient
\(7\)
Remainder
\(5\)
Step 1: Set up the division problem

Total students (89) ÷ Students per bus (12) = Number of full buses with remainder

Step 2: Determine how many times 12 goes into 89

\(12 × 1 = 12\), \(12 × 2 = 24\), \(12 × 3 = 36\), \(12 × 4 = 48\), \(12 × 5 = 60\), \(12 × 6 = 72\), \(12 × 7 = 84\), \(12 × 8 = 96\)

Since 96 > 89, the largest multiple of 12 that fits is 84 (which is 12 × 7)

Step 3: Calculate the quotient and remainder

12 goes into 89 exactly 7 times, so 7 full buses are filled

Leftover: \(89 - (12 × 7) = 89 - 84 = 5\) students

Step 4: Determine total buses needed

Since 5 students still need transportation, we need 1 more bus

Total buses needed: \(7 + 1 = 8\) buses

Step 5: Calculate empty seats

Total capacity: \(8 × 12 = 96\) seats

Empty seats: \(96 - 89 = 7\) seats

8 buses are needed with 7 empty seats
Final answer:

8 buses are needed with 7 empty seats

Applied rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder rule: Remainder must be less than the divisor

• Practical application: Round up when remainder exists for transportation needs

5 Pattern recognition with remainders
Exercise 5
What is the remainder when 123 is divided by 9? Use the digit sum method to verify.
Definition:

Digit sum method: For division by 9, the remainder equals the remainder when the sum of digits is divided by 9.

Pattern recognition: Identifying shortcuts in division problems.

Problem
\(123 ÷ 9\)
Quotient
\(13\)
Remainder
\(6\)
Step 1: Standard division method

\(9 × 13 = 117\), and \(123 - 117 = 6\)

So \(123 ÷ 9 = 13\) remainder \(6\)

Step 2: Verify using digit sum method

Sum of digits in 123: \(1 + 2 + 3 = 6\)

Now divide sum by 9: \(6 ÷ 9 = 0\) remainder \(6\)

Step 3: Compare results

Both methods give remainder 6, confirming our answer

Step 4: Verify with division algorithm

Check: \(123 = (9 × 13) + 6 = 117 + 6 = 123\) ✓

\(123 ÷ 9 = 13\) remainder \(6\)
Final answer:

The remainder when 123 is divided by 9 is 6

Applied rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Digit sum property: For division by 9, remainder equals remainder of digit sum

• Verification: Multiple methods confirm accuracy

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

Division: The operation of splitting a number into equal parts

Remainder: The amount left over after division that doesn't form a complete group

Exact division: When remainder is 0, the division is exact

Complete methodology:
  1. Analyze the problem: Identify dividend, divisor, and what you're solving for
  2. Estimate: Think about how many times the divisor might fit into the dividend
  3. Calculate: Use multiplication facts to find the largest multiple that fits
  4. Find remainder: Subtract the product from the dividend
  5. Verify: Check using the division algorithm
Tip 1: Practice multiplication tables regularly to speed up division.
Tip 2: Use visual models like arrays or grouping to understand remainders.
Tip 3: Always check that remainder is less than divisor - if not, adjust quotient.
Tip 4: In word problems, consider if remainder requires rounding up for practical purposes.
Common errors: Forgetting to check remainder is less than divisor, miscalculating multiplication facts, misinterpreting word problems.
Exam preparation: Master basic division facts, practice word problems, understand when to round up remainders.
Essential formulas and rules:

• Division algorithm: Dividend = (Divisor × Quotient) + Remainder

• Remainder rule: 0 ≤ Remainder < Divisor

• Verification: Always substitute back to check your work

• Digit sum property: For division by 9, remainder equals digit sum remainder

Division Patterns: Remainders Analysis
Exercise 6: Remainder Patterns
Analyze division patterns with remainders:
Numbers 1-20 divided by 3, 4, and 5
Observe the repeating patterns in remainders.

Analysis: The chart shows how remainders cycle in predictable patterns.

  • Division by 3: Remainders cycle 0, 1, 2, 0, 1, 2, ...
  • Division by 4: Remainders cycle 0, 1, 2, 3, 0, 1, 2, 3, ...
  • Division by 5: Remainders cycle 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, ...

Questions & Answers

Question: I don't understand why the remainder has to be smaller than the divisor. Can you explain this?

Answer: Great question! The remainder must be smaller than the divisor because if it were equal to or larger than the divisor, we could make another complete group.

Think of it this way: If you're dividing 17 by 5 and got a remainder of 5, that means you could have made one more complete group of 5. So 17 ÷ 5 = 3 remainder 2, not 2 remainder 5.

Here's a visual example: With 17 objects and groups of 5:

  • Group 1: 5 objects
  • Group 2: 5 objects
  • Group 3: 5 objects
  • Remaining: 2 objects (this is the remainder)
Since 2 < 5, we can't make another complete group, so 2 is the correct remainder.

Question: How do I know if I need to round up when I have a remainder in a word problem?

Answer: Look for keywords in the problem that indicate you need complete units:

  • Rounding up needed: "how many buses," "how many containers," "how many boxes," "at least," "minimum required"
  • Rounding up NOT needed: "how many complete groups," "how many full sets," "what remains"

For example: "If 25 students need to ride in cars that hold 4 people each, how many cars are needed?" You'd need 7 cars (not 6), because the 1 remaining student still needs transportation.

Always read the question carefully to determine what the answer represents in the context of the problem!

Question: I sometimes get confused about which number is the dividend, divisor, and quotient. Is there a way to remember?

Answer: Here are memory aids for each part:

  • Dividend: "The one being DIVIDED" - it's the first number in the division sentence
  • Divisor: "Does the dividing" - it's the number doing the action
  • Quotient: "What you QUOTE as the answer" - it's the result of division

In the expression 24 ÷ 6 = 4:

  • 24 is the dividend (being divided)
  • 6 is the divisor (doing the dividing)
  • 4 is the quotient (the answer)
Remember: Dividend ÷ Divisor = Quotient