Division Problems 3 Digit by 2 Digit - Complete Guide for Grade 4 Students

Master division of 3-digit numbers by 2-digit numbers: long division method, quotients, remainders, and step-by-step solutions through comprehensive exercises and detailed explanations.

Division Problems 3 Digit by 2 Digit - 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 (if any)

3-Digit by 2-Digit Division: Division where the dividend is a 3-digit number (100-999) and the divisor is a 2-digit number (10-99)

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

Partial Quotient: The portion of the quotient determined at each step of long division

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

• Place Value Importance: Work from left to right (hundreds → tens → ones)

• Digit-by-Digit Process: Divide one digit at a time, bringing down next digit

• Estimation: Use rounding to estimate quotient before calculating

Step-by-Step Method
Long Division Process for 3 Digit by 2 Digit:
  1. Estimate: Round divisor and dividend to estimate the quotient
  2. Compare hundreds place: How many times does 2-digit divisor go into hundreds digit?
  3. If divisor is larger: Combine hundreds and tens digits
  4. Divide: How many times does divisor go into combined number?
  5. Multiply: Multiply quotient digit by divisor
  6. Subtract: Subtract result from current number
  7. Bring down: Bring down the next digit
  8. Repeat: Continue with remaining digits
  9. Check: Verify using: Divisor × Quotient + Remainder = Dividend
Examples and Tips
Simple Example: 120 ÷ 12 = 10 (no remainder)
With Remainder: 123 ÷ 12 = 10 remainder 3
Key Tip: Start with the largest place value and work left to right!
Check Method: Always verify: Quotient × Divisor + Remainder = Original Number
Solution: Exercises 1 to 3
1 Basic Division
Exercise 1
Find the quotient of:
\(144 \div 12\)
Understanding the Problem:

This is a basic 3-digit by 2-digit division problem where we need to find how many times 12 goes into 144

Solution Approach:
  1. Estimate: 144 ÷ 12 ≈ 140 ÷ 10 = 14
  2. Start with the hundreds and tens: 14 ÷ 12 = 1 remainder 2
  3. Continue with the ones place
Expression
\(144 \div 12\)
Division
\(12\)
Step 1: Set up the division

We need to divide 144 by 12

144 ÷ 12 = ?

Step 2: Estimate the quotient

Round: 144 ≈ 140, 12 ≈ 10

Estimate: 140 ÷ 10 = 14

So the answer should be around 12

Step 3: Compare hundreds and tens

How many times does 12 go into 14 (hundreds and tens)?

12 × 1 = 12, 12 × 2 = 24 (too big)

So 12 goes into 14 once (1)

Write 1 above the tens place

Step 4: Multiply and subtract

1 × 12 = 12

14 - 12 = 2

Step 5: Bring down the ones place

Bring down the 4

Now we have 24

Step 6: Divide the new number

How many times does 12 go into 24?

12 × 2 = 24

So 12 goes into 24 exactly two times (2)

Write 2 above the ones place

Step 7: Multiply and subtract

2 × 12 = 24

24 - 24 = 0

No remainder

\(144 \div 12 = 12\)
Final answer:

\(144 \div 12 = 12\) with no remainder

Applied rules:

• Estimation: Estimate before calculating to check reasonableness

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

• Multiplication check: 12 × 12 = 144 ✓

2 Division with Remainder
Exercise 2
Find the quotient and remainder of:
\(257 \div 13\)
Understanding the Problem:

This 3-digit by 2-digit division will have a remainder since 13 doesn't divide evenly into 257

Expression
\(257 \div 13\)
Division
\(19\) R\(10\)
Step 1: Set up the division

We need to divide 257 by 13

257 ÷ 13 = ?

Step 2: Estimate the quotient

Round: 257 ≈ 260, 13 ≈ 10

Estimate: 260 ÷ 10 = 26

So the answer should be around 20

Step 3: Compare hundreds and tens

How many times does 13 go into 25 (hundreds and tens)?

13 × 1 = 13, 13 × 2 = 26 (too big)

So 13 goes into 25 once (1)

Write 1 above the tens place

Step 4: Multiply and subtract

1 × 13 = 13

25 - 13 = 12

Step 5: Bring down the ones place

Bring down the 7

Now we have 127

Step 6: Divide the new number

How many times does 13 go into 127?

13 × 9 = 117, 13 × 10 = 130 (too big)

So 13 goes into 127 nine times (9)

Write 9 above the ones place

Step 7: Multiply and subtract

9 × 13 = 117

127 - 117 = 10

10 is less than 13, so 10 is our remainder

\(257 \div 13 = 19\) remainder \(10\)
Final answer:

\(257 \div 13 = 19\) remainder \(10\)

Check: \(19 \times 13 + 10 = 247 + 10 = 257\) ✓

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 Division with Zero in Quotient
Exercise 3
Find the quotient of:
\(308 \div 28\)
Understanding the Problem:

This 3-digit by 2-digit division includes a zero in the dividend and will result in a 2-digit quotient

Expression
\(308 \div 28\)
Division
\(11\)
Step 1: Set up the division

We need to divide 308 by 28

308 ÷ 28 = ?

Step 2: Estimate the quotient

Round: 308 ≈ 300, 28 ≈ 30

Estimate: 300 ÷ 30 = 10

So the answer should be around 10

Step 3: Compare hundreds and tens

How many times does 28 go into 30 (hundreds and tens)?

28 × 1 = 28, 28 × 2 = 56 (too big)

So 28 goes into 30 once (1)

Write 1 above the tens place

Step 4: Multiply and subtract

1 × 28 = 28

30 - 28 = 2

Step 5: Bring down the ones place

Bring down the 8

Now we have 28

Step 6: Divide the new number

How many times does 28 go into 28?

28 × 1 = 28

So 28 goes into 28 exactly one time (1)

Write 1 above the ones place

Step 7: Multiply and subtract

1 × 28 = 28

28 - 28 = 0

No remainder

\(308 \div 28 = 11\)
Final answer:

\(308 \div 28 = 11\) with no remainder

Check: \(11 \times 28 = 308\) ✓

Applied rules:

• Handling zeros: Zeros in dividend don't change the process

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

• Multiplication check: Verify result by multiplying quotient by divisor

Solution: Exercises 4 to 5
4 Word Problem Application
Exercise 4
A school has 432 students and needs to organize them into groups of 18. How many complete groups can be formed?
Understanding the Problem:

This is a word problem requiring 3-digit by 2-digit division to solve an equal grouping situation

Problem
\(432 \div 18\)
Division
\(24\)
Step 1: Identify the operation

School has 432 students and groups of 18

To find complete groups: 432 ÷ 18 = ?

Step 2: Estimate the quotient

Round: 432 ≈ 430, 18 ≈ 20

Estimate: 430 ÷ 20 = 21.5

So the answer should be around 24

Step 3: Set up the division

We need to divide 432 by 18

432 ÷ 18 = ?

Step 4: Compare hundreds and tens

How many times does 18 go into 43 (hundreds and tens)?

18 × 2 = 36, 18 × 3 = 54 (too big)

So 18 goes into 43 two times (2)

Write 2 above the tens place

Step 5: Multiply and subtract

2 × 18 = 36

43 - 36 = 7

Step 6: Bring down the ones place

Bring down the 2

Now we have 72

Step 7: Divide the new number

How many times does 18 go into 72?

18 × 4 = 72

So 18 goes into 72 exactly four times (4)

Write 4 above the ones place

Step 8: Multiply and subtract

4 × 18 = 72

72 - 72 = 0

No remainder

Step 9: Interpret the result

24 complete groups can be formed

All students are organized (no remainder)

24 complete groups can be formed
Final answer:

24 complete groups can be formed.

Check: \(24 \times 18 = 432\) ✓

Applied rules:

• Equal grouping: Division to organize items into equal groups

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

• Verification: Multiply quotient by divisor to check

5 Complex Division Problem
Exercise 5
Solve: \(563 \div 24\) 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 verify our answer.

Expression
\(563 \div 24\)
Division
\(23\) R\(11\)
Step 1: Set up the division

We need to divide 563 by 24

563 ÷ 24 = ?

Step 2: Estimate the quotient

Round: 563 ≈ 560, 24 ≈ 25

Estimate: 560 ÷ 25 = 22.4

So the answer should be around 23

Step 3: Compare hundreds and tens

How many times does 24 go into 56 (hundreds and tens)?

24 × 2 = 48, 24 × 3 = 72 (too big)

So 24 goes into 56 two times (2)

Write 2 above the tens place

Step 4: Multiply and subtract

2 × 24 = 48

56 - 48 = 8

Step 5: Bring down the ones place

Bring down the 3

Now we have 83

Step 6: Divide the new number

How many times does 24 go into 83?

24 × 3 = 72, 24 × 4 = 96 (too big)

So 24 goes into 83 three times (3)

Write 3 above the ones place

Step 7: Multiply and subtract

3 × 24 = 72

83 - 72 = 11

11 is less than 24, so 11 is our remainder

Step 8: Verify the answer

Check: 23 × 24 + 11 = 552 + 11 = 563 ✓

Alternative check: 563 ÷ 24 = 23 remainder 11 ✓

\(563 \div 24 = 23\) remainder \(11\)
Final answer:

\(563 \div 24 = 23\) remainder \(11\)

Check: \(23 \times 24 + 11 = 552 + 11 = 563\) ✓

Applied rules:

• Division verification: Always check your answer

• Multiplication facts: Knowing 24 × 23 = 552 helps solve this

• Place value: Work from hundreds to tens to ones

Questions & Answers

Question: When I divide 3-digit numbers by 2-digit numbers, it seems harder. How can I make it easier?

Answer: Dividing by 2-digit numbers can seem harder, but there are strategies to make it easier:

Strategy 1: Estimation First

  • Round the divisor and dividend to simpler numbers
  • Calculate the approximate answer first
  • Example: For 257 ÷ 13, round to 260 ÷ 10 = 26

Strategy 2: Use Multiplication Facts

  • Think about what multiples of the divisor are close to your current number
  • For 13 × ?: Think 13 × 1 = 13, 13 × 2 = 26, etc.

Strategy 3: Work Digit by Digit

  • Focus on one part of the division at a time
  • Don't try to do everything at once

Remember: The process is the same as dividing by 1-digit numbers, just with a slightly larger divisor!

Question: I get confused about when to use the hundreds and tens together versus just the hundreds. How do I know?

Answer: The key is to compare the divisor with the current digit or digits you're working with:

Rule: When the divisor is larger than the current digit you're working with, you need to include more digits.

Examples:

  • In 144 ÷ 12: 12 is larger than 1 (hundreds digit), so we use 14 (hundreds and tens)
  • In 257 ÷ 13: 13 is larger than 2 (hundreds digit), so we use 25 (hundreds and tens)
  • In 308 ÷ 28: 28 is larger than 3 (hundreds digit), so we use 30 (hundreds and tens)

Why this works: We need enough "value" to divide by the divisor. If we only have 1, we can't divide by 12, but if we have 14, we can divide by 12.

Quick check: If your divisor is larger than the current digit, combine it with the next digit.

Question: How do I know if my answer is reasonable when dividing 3-digit by 2-digit numbers?

Answer: There are several ways to check if your answer is reasonable:

Method 1: Estimation Check

  • Round the divisor and dividend to simpler numbers
  • Calculate the approximate answer
  • Compare with your actual answer
  • Example: For 563 ÷ 24, round to 560 ÷ 25 ≈ 22, so 23 is reasonable

Method 2: Multiplication Verification

  • Multiply your quotient by the divisor
  • Add the remainder
  • You should get back to your original dividend
  • Example: 23 × 24 + 11 = 552 + 11 = 563 ✓

Method 3: Reasonableness Check

  • Your quotient should be smaller than your dividend
  • For 3-digit ÷ 2-digit, expect a 1-digit or 2-digit answer
  • The remainder should be smaller than the divisor

Always use the verification method: Divisor × Quotient + Remainder = Dividend