Division Problems With Remainders and Answers - Complete Guide for Grade 4 Students

Master division problems with remainders and answers: understanding remainders, long division method, quotients, and step-by-step solutions through comprehensive exercises and detailed explanations with complete answers.

Division Problems With Remainders and Answers - Summary
Key Definitions
Essential Terms:

Division: The process of splitting a number into equal parts or groups

Dividend: The number being divided (the larger number)

Divisor: The number that divides the dividend (the smaller number)

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

Remainder: What's left over after division when the divisor doesn't divide evenly into the dividend

Remainder Division: Division where the divisor doesn't divide evenly into the dividend, leaving a remainder

Place Value: Position of digits (hundreds, tens, ones) that determines their value

Complete Answer: The quotient and remainder together (e.g., 3 remainder 2)

Core Rules and Laws
Fundamental Rules:

• Division Relationship: Dividend = Divisor × Quotient + Remainder

• Remainder Rule: Remainder is always less than the divisor

• Division by 1: Any number divided by 1 equals itself

• Zero Division: 0 divided by any non-zero number equals 0

• Division by Zero: Undefined (impossible)

• Inverse Operation: Division is the inverse of multiplication

• Remainder Constraint: Remainder must always be smaller than the divisor

• Complete Answer: Always provide both quotient and remainder when applicable

Step-by-Step Method
Long Division Process With Remainders and Answers:
  1. Divide: How many times does the divisor go into the current number?
  2. Multiply: Multiply the quotient digit by the divisor
  3. Subtract: Subtract the result from the current number
  4. Bring down: Bring down the next digit
  5. Repeat: Continue until all digits are processed
  6. Check remainder: If the final result is greater than 0, that's the remainder
  7. State complete answer: Include both quotient and remainder
  8. Verify: Use: Divisor × Quotient + Remainder = Dividend
Examples and Tips
Simple Example: 13 ÷ 4 = 3 remainder 1
With Remainder: 25 ÷ 6 = 4 remainder 1
Key Tip: The remainder is always less than the divisor!
Check Method: Always verify: Quotient × Divisor + Remainder = Original Number
Solution: Exercises 1 to 3
1 Basic Division With Remainder and Answer
Exercise 1
Find the quotient and remainder of:
\(19 \div 4\)
Understanding the Problem:

This is a basic division problem where 4 doesn't divide evenly into 19, so there will be a remainder. We need to find both the quotient and remainder.

Solution Approach:
  1. Find how many times 4 goes into 19
  2. 4 × 4 = 16, 4 × 5 = 20 (too big)
  3. So 4 goes into 19 four times with some left over
  4. 19 - 16 = 3, so remainder is 3
  5. Complete answer: 4 remainder 3
Expression
\(19 \div 4\)
Division
\(4\) R\(3\)
Step 1: Set up the division

We need to divide 19 by 4

19 ÷ 4 = ?

Step 2: Find how many times 4 goes into 19

Think: 4 × ? = how close to 19 without going over

4 × 4 = 16, 4 × 5 = 20 (too big)

So 4 goes into 19 four times

Step 3: Calculate the remainder

Multiply: 4 × 4 = 16

Subtract: 19 - 16 = 3

So the remainder is 3

Step 4: Check if remainder is valid

Is 3 less than 4? Yes, so 3 is our remainder

\(19 \div 4 = 4\) remainder \(3\)
Complete answer:

\(19 \div 4 = 4\) remainder \(3\)

Check: \(4 \times 4 + 3 = 16 + 3 = 19\) ✓

Applied rules:

• Remainder constraint: Remainder (3) is less than divisor (4)

• Division relationship: Dividend = Divisor × Quotient + Remainder

• Multiplication check: 4 × 4 + 3 = 19 ✓

2 Two-Digit Division With Remainder and Answer
Exercise 2
Find the quotient and remainder of:
\(53 \div 7\)
Understanding the Problem:

This 2-digit division will have a remainder since 7 doesn't divide evenly into 53. We need to find both the quotient and remainder.

Expression
\(53 \div 7\)
Division
\(7\) R\(4\)
Step 1: Set up the division

We need to divide 53 by 7

53 ÷ 7 = ?

Step 2: Find how many times 7 goes into 53

Think: 7 × ? = how close to 53 without going over

7 × 7 = 49, 7 × 8 = 56 (too big)

So 7 goes into 53 seven times

Step 3: Calculate the remainder

Multiply: 7 × 7 = 49

Subtract: 53 - 49 = 4

So the remainder is 4

Step 4: Check if remainder is valid

Is 4 less than 7? Yes, so 4 is our remainder

Step 5: Verify the answer

Check: 7 × 7 + 4 = 49 + 4 = 53 ✓

\(53 \div 7 = 7\) remainder \(4\)
Complete answer:

\(53 \div 7 = 7\) remainder \(4\)

Check: \(7 \times 7 + 4 = 49 + 4 = 53\) ✓

Applied rules:

• Division algorithm: Dividend = Divisor × Quotient + Remainder

• Remainder constraint: Remainder must be less than divisor

• Estimation: Use multiplication facts to determine how many times divisor fits

3 Three-Digit Division With Remainder and Answer
Exercise 3
Find the quotient and remainder of:
\(147 \div 9\)
Understanding the Problem:

This 3-digit division will have a remainder. We need to find both the quotient and remainder.

Expression
\(147 \div 9\)
Division
\(16\) R\(3\)
Step 1: Set up the division

We need to divide 147 by 9

147 ÷ 9 = ?

Step 2: Compare hundreds and tens

How many times does 9 go into 14 (hundreds and tens)?

9 × 1 = 9, 9 × 2 = 18 (too big)

So 9 goes into 14 once (1)

Write 1 above the tens place

Step 3: Multiply and subtract

1 × 9 = 9

14 - 9 = 5

Step 4: Bring down the ones place

Bring down the 7

Now we have 57

Step 5: Divide the new number

How many times does 9 go into 57?

9 × 6 = 54, 9 × 7 = 63 (too big)

So 9 goes into 57 six times (6)

Write 6 above the ones place

Step 6: Multiply and subtract

6 × 9 = 54

57 - 54 = 3

3 is less than 9, so 3 is our remainder

\(147 \div 9 = 16\) remainder \(3\)
Complete answer:

\(147 \div 9 = 16\) remainder \(3\)

Check: \(16 \times 9 + 3 = 144 + 3 = 147\) ✓

Applied rules:

• Long division: Divide, multiply, subtract, bring down, repeat

• Remainder constraint: Remainder (3) is less than divisor (9)

• Multiplication check: Verify result by multiplying quotient by divisor

Solution: Exercises 4 to 5
4 Word Problem With Remainder and Answer
Exercise 4
Tom has 67 apples and wants to pack them in boxes that hold 8 apples each. How many full boxes can he fill and how many apples will be left over?
Understanding the Problem:

This word problem requires division with remainder to find both complete groups and leftovers. We need to find the quotient (full boxes) and remainder (leftover apples).

Problem
\(67 \div 8\)
Division
\(8\) R\(3\)
Step 1: Identify the operation

Tom has 67 apples and packs 8 per box

To find full boxes: 67 ÷ 8 = ?

Step 2: Find how many times 8 goes into 67

Think: 8 × ? = how close to 67 without going over

8 × 8 = 64, 8 × 9 = 72 (too big)

So 8 goes into 67 eight times

Step 3: Calculate the remainder

Multiply: 8 × 8 = 64

Subtract: 67 - 64 = 3

So the remainder is 3

Step 4: Interpret the results

Quotient (8): Number of full boxes

Remainder (3): Apples left over

Step 5: Verify the answer

Check: 8 × 8 + 3 = 64 + 3 = 67 ✓

8 full boxes, 3 apples left over
Complete answer:

Tom can fill 8 full boxes and will have 3 apples left over.

Check: \(8 \times 8 + 3 = 64 + 3 = 67\) ✓

Applied rules:

• Equal grouping: Division to organize items into equal groups

• Real-world context: Interpret the quotient and remainder in the problem's context

• Verification: Multiply quotient by divisor plus remainder to check

5 Complex Division With Remainder and Answer
Exercise 5
Solve: \(318 \div 15\) and verify your answer
Understanding the Problem:

This 3-digit by 2-digit division will have a remainder. We need to find the quotient and remainder, then verify our answer.

Expression
\(318 \div 15\)
Division
\(21\) R\(3\)
Step 1: Set up the division

We need to divide 318 by 15

318 ÷ 15 = ?

Step 2: Compare hundreds and tens

How many times does 15 go into 31 (hundreds and tens)?

15 × 2 = 30, 15 × 3 = 45 (too big)

So 15 goes into 31 two times (2)

Write 2 above the tens place

Step 3: Multiply and subtract

2 × 15 = 30

31 - 30 = 1

Step 4: Bring down the ones place

Bring down the 8

Now we have 18

Step 5: Divide the new number

How many times does 15 go into 18?

15 × 1 = 15, 15 × 2 = 30 (too big)

So 15 goes into 18 one time (1)

Write 1 above the ones place

Step 6: Multiply and subtract

1 × 15 = 15

18 - 15 = 3

3 is less than 15, so 3 is our remainder

Step 7: Verify the answer

Check: 21 × 15 + 3 = 315 + 3 = 318 ✓

Alternative check: 318 ÷ 15 = 21 remainder 3 ✓

\(318 \div 15 = 21\) remainder \(3\)
Complete answer:

\(318 \div 15 = 21\) remainder \(3\)

Check: \(21 \times 15 + 3 = 315 + 3 = 318\) ✓

Applied rules:

• Division verification: Always check your answer

• Multiplication facts: Knowing 15 × 21 = 315 helps solve this

• Remainder constraint: Remainder (3) is less than divisor (15)

Questions & Answers

Question: I understand remainders, but I'm confused about how to write the complete answer. Should I write "remainder" or "R" or what?

Answer: Great question! There are several correct ways to write division answers with remainders:

Method 1: Using "remainder"

  • 19 ÷ 4 = 4 remainder 3
  • This is the most formal way and clearly explains what the numbers mean

Method 2: Using "R"

  • 19 ÷ 4 = 4 R 3
  • This is a shorter way that's commonly used

Method 3: Using fraction notation

  • 19 ÷ 4 = 4 3/4
  • This shows the remainder as a fraction of the divisor

Most teachers prefer: 4 remainder 3 or 4 R 3. Always ask your teacher which format they prefer!

Question: How do I know if I got the right answer when I have a remainder? Sometimes I'm not sure if my remainder is correct.

Answer: There are several ways to check if your answer with remainder is correct:

Rule 1: Remainder Must Be Smaller Than Divisor

  • If you get a remainder that's equal to or larger than your divisor, you made a mistake
  • Example: In 19 ÷ 4, if you got remainder 5, that's wrong because 5 > 4
  • The correct answer is 4 remainder 3 (because 3 < 4)

Rule 2: Verify Using the Division Formula

  • Check: Divisor × Quotient + Remainder = Original Dividend
  • Example: For 19 ÷ 4 = 4 remainder 3, check: 4 × 4 + 3 = 16 + 3 = 19 ✓

Rule 3: Reasonableness Check

  • Make sure your answer makes sense in the context of the problem
  • The remainder should represent what's left after equal distribution

Always verify your answers! This helps you catch mistakes and builds confidence.

Question: When I solve word problems, how do I know what the quotient and remainder mean in the story?

Answer: Understanding what the quotient and remainder represent in word problems is crucial! Here's how to identify them:

Look for Keywords:

  • Quotient usually means: "complete groups," "full sets," "number of boxes," "equal shares"
  • Remainder usually means: "left over," "remaining," "extra," "not enough for another group"

Example Analysis:

  • "67 apples packed in boxes of 8": Quotient (8) = full boxes, Remainder (3) = leftover apples
  • "25 students in teams of 6": Quotient (4) = complete teams, Remainder (1) = students not on a team
  • "$34 shared equally among 5 people": Quotient (6) = dollars each person gets, Remainder (4) = cents left over

Key Strategy: Read the problem carefully and ask yourself: "What am I dividing into equal groups?" The quotient tells you how many complete groups, and the remainder tells you what's left over.