Solved Exercises on Dividing 3-Digit Numbers in Grade 4

Master dividing 3-digit numbers: long division, mental math strategies, remainders, and word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic 3-Digit Division
Exercise 1
Use long division to solve: 248 ÷ 4
Definition:

Dividing 3-digit numbers: The process of dividing a number with three digits by a single-digit divisor using long division or mental math strategies.

3-Digit Division Method:
  1. Estimate: Think about how many times the divisor goes into the first digit
  2. Divide: How many times does the divisor go into the current number?
  3. Multiply: Multiply the divisor by the quotient digit
  4. Subtract: Subtract the product from the current number
  5. Bring Down: Bring down the next digit
  6. Repeat: Continue until all digits are processed
Setup
4 ) 248
Answer
?
Check
?
Step 1: Set up the long division

Write 4 ) 248 with the divisor (4) outside and dividend (248) inside

Step 2: Divide the first digit

How many times does 4 go into 2? Answer: 0 times
So consider the first two digits: 24

Step 3: Divide 24

How many times does 4 go into 24? Answer: 6 times
Write 6 above the 4

Step 4: Multiply

6 × 4 = 24
Write 24 below the 24

Step 5: Subtract

24 - 24 = 0
Write 0 below

Step 6: Bring down the next digit

Bring down the 8 to make 08

Step 7: Divide again

How many times does 4 go into 8? Answer: 2 times
Write 2 above the 8

Step 8: Multiply and subtract

2 × 4 = 8
8 - 8 = 0
No remainder

Step 9: Verify the answer

Check: 62 × 4 = 248 ✓

248 ÷ 4 = 62
Final answer:

248 ÷ 4 = 62

Applied rules:

• DMBS Rule: Divide, Multiply, Subtract, Bring Down

• Place value: Each digit's position affects its value in calculations

• Verification: Always check by multiplying quotient by divisor

2 3-Digit Division with Remainder
Exercise 2
Use long division to solve: 357 ÷ 6
Definition:

Division with remainder: When a 3-digit number is divided and there's a leftover amount that's smaller than the divisor.

Setup
6 ) 357
Quotient
?
Remainder
?
Step 1: Set up the long division

Write 6 ) 357

Step 2: Divide the first digit

How many times does 6 go into 3? Answer: 0 times
So consider the first two digits: 35

Step 3: Divide 35

How many times does 6 go into 35? Answer: 5 times (6 × 5 = 30)
Write 5 above the 5

Step 4: Multiply

5 × 6 = 30
Write 30 below the 35

Step 5: Subtract

35 - 30 = 5
Write 5 below

Step 6: Bring down the next digit

Bring down the 7 to make 57

Step 7: Divide again

How many times does 6 go into 57? Answer: 9 times (6 × 9 = 54)
Write 9 above the 7

Step 8: Multiply and subtract

9 × 6 = 54
57 - 54 = 3
Since 3 is less than 6, this is the remainder

Step 9: Verify the answer

Check: (6 × 59) + 3 = 354 + 3 = 357 ✓

357 ÷ 6 = 59 R3
Final answer:

357 ÷ 6 = 59 R3

Applied rules:

• Remainder rule: Remainder must be less than the divisor

• DMBS sequence: Follow the order of operations carefully

• Verification: Use Dividend = (Divisor × Quotient) + Remainder

3 Mental Math Strategies for 3-Digit Division
Exercise 3
Solve 480 ÷ 8 using mental math strategies. Explain your thought process.
Definition:

Mental math strategies: Techniques to solve division problems mentally by breaking them into easier parts or using known facts.

Problem
480 ÷ 8
Strategy
?
Process
?
Step 1: Break apart the dividend

Think of 480 as 48 tens (480 = 48 × 10)

Step 2: Use known facts

I know that 48 ÷ 8 = 6 (from multiplication facts)

Step 3: Apply the division

Since 480 = 48 tens, then 480 ÷ 8 = 6 tens = 60

Step 4: Alternative approach

Think of 480 as 400 + 80
400 ÷ 8 = 50
80 ÷ 8 = 10
50 + 10 = 60

Step 5: Verify the answer

Check: 60 × 8 = 480 ✓

480 ÷ 8 = 60
Final answer:

480 ÷ 8 = 60. I used the fact that 480 is 48 tens, and since 48 ÷ 8 = 6, then 480 ÷ 8 = 60. Alternatively, I broke 480 into 400 + 80, then divided each part by 8.

Applied rules:

• Distributive property: (a + b) ÷ c = a ÷ c + b ÷ c

• Mental math: Use known multiplication facts to solve division

• Place value: Recognize that 480 = 48 tens

3-Digit Division: Facts, Properties, and Methods
Dividend = (Divisor × Quotient) + Remainder
Division Formula
Division Example
248 ÷ 4 = 62
because 4 × 62 = 248
With Remainder
357 ÷ 6 = 59 R3
because (6 × 59) + 3 = 357
Mental Strategy
480 ÷ 8 = 60
because (400 ÷ 8) + (80 ÷ 8) = 50 + 10 = 60
Key definitions:

Dividend: The number being divided (the whole).

Divisor: The number by which the dividend is divided (the size of each group).

Quotient: The result of division (the number of equal groups).

Remainder: What's left over after division that cannot form another complete group.

3-digit number: A number with a hundreds place, tens place, and ones place (100-999).

3-digit division methods:
  1. Standard long division: Divide, Multiply, Subtract, Bring Down, Repeat
  2. Mental math strategies: Break apart, use multiples, compensation
  3. Partial quotients: Estimate and subtract chunks
  4. Repeated subtraction: Keep subtracting the divisor
Tip 1: Use multiplication facts to help with division. If you know 8 × 60 = 480, then 480 ÷ 8 = 60.
Tip 2: Always check your work: (Divisor × Quotient) + Remainder should equal the Dividend.
Tip 3: When doing mental math with 3-digit numbers, break them into hundreds, tens, and ones.
Tip 4: The remainder is always smaller than the divisor. If you get a remainder larger than the divisor, you made a mistake!
Division properties: Division is not commutative (a÷b ≠ b÷a). Remainder is always 0 ≤ remainder < divisor.
Special cases: When remainder is 0, division is exact. Division by 0 is undefined. Any number divided by 1 equals itself.
Important 3-digit division rules:

• Remainder is always less than the divisor

• Dividend = (Divisor × Quotient) + Remainder

• When remainder is 0, division is exact

• Division by 0 is undefined

• Use known multiplication facts to solve division problems

Mental Math Strategies for 3-Digit Division
Strategy Effectiveness
Visualizing different mental math strategies for solving 3-digit division problems.

Analysis: The chart shows how different strategies work for various 3-digit division problems.

  • Breaking apart works well when dividends are multiples of the divisor
  • Using place value (hundreds, tens, ones) simplifies many problems
  • Compensation strategy is useful for numbers near round numbers
Solution: Exercises 4 to 5
4 Word Problem with 3-Digit Division
Exercise 4
A school has 432 students going on a field trip. Buses can hold 8 students each. How many buses are needed for all the students?
Definition:

Word problems with 3-digit division: Real-world situations that require dividing a 3-digit number by a single-digit number to solve.

Total Students
432
Students per Bus
8
Buses Needed
?
Step 1: Identify the problem

We need to divide 432 students by 8 students per bus: 432 ÷ 8

Step 2: Use long division

8 ) 432

Step 3: Divide the first two digits

How many times does 8 go into 43? Answer: 5 times (8 × 5 = 40)
Write 5 above the 3

Step 4: Multiply and subtract

5 × 8 = 40
43 - 40 = 3
Bring down the 2 to make 32

Step 5: Divide again

How many times does 8 go into 32? Answer: 4 times
Write 4 above the 2

Step 6: Multiply and subtract

4 × 8 = 32
32 - 32 = 0
No remainder

Step 7: State the answer

432 ÷ 8 = 54

Step 8: Verify the answer

Check: 54 × 8 = 432 ✓

Step 9: Interpret in context

54 buses are needed for all the students.

432 ÷ 8 = 54
Final answer:

54 buses are needed for all the students.

Applied rules:

• Problem-solving strategy: Identify total amount, group size, and what to find

• Connection to multiplication: Use known multiplication facts to solve division

• Verification: Always check your calculation with multiplication

5 Division Patterns with 3-Digit Numbers
Exercise 5
Find the pattern: 200÷4=50, 240÷4=60, 280÷4=70, 320÷4=?. Explain the pattern and predict 400÷4.
Definition:

Division patterns: Recognizing how quotients change when dividends increase by regular amounts while the divisor stays the same.

Pattern
200÷4→50, 240÷4→60, 280÷4→70, 320÷4→?
Prediction
400÷4→?
Rule
?
Step 1: Analyze the pattern

200 ÷ 4 = 50
240 ÷ 4 = 60
280 ÷ 4 = 70

Step 2: Notice the relationship

Each time the dividend increases by 40, the quotient increases by 10
200 → 240 (+40), 50 → 60 (+10)
240 → 280 (+40), 60 → 70 (+10)

Step 3: Apply the pattern

280 → 320 (+40), so 70 → 80 (+10)
Therefore, 320 ÷ 4 = 80

Step 4: Predict the next term

320 → 400 (+80), so 80 → 100 (+20)
Therefore, 400 ÷ 4 = 100

Step 5: Verify with actual division

320 ÷ 4: 4 goes into 32 eight times (8), remainder 0, bring down 0, so 80 ✓
400 ÷ 4: 4 goes into 4 once (1), remainder 0, bring down 00, so 100 ✓

Step 6: Generalize the pattern

When dividing by 4, increasing the dividend by 40 increases the quotient by 10

320 ÷ 4 = 80, 400 ÷ 4 = 100
Final answer:

The pattern shows that when dividing by 4, each time the dividend increases by 40, the quotient increases by 10. So 320 ÷ 4 = 80 and 400 ÷ 4 = 100.

Applied rules:

• Pattern recognition: Observe how changes in dividend affect quotient

• Proportional reasoning: Understand the relationship between dividend and quotient

• Verification: Check predictions with actual calculations

Complete Summary: Dividing 3-Digit Numbers
Dividend = (Divisor × Quotient) + Remainder
Fundamental Division Formula
Key definitions:

Dividing 3-digit numbers: The process of dividing a number with three digits by a single-digit divisor using various strategies including long division and mental math.

Dividend: The number being divided (the whole).

Divisor: The number by which the dividend is divided (the size of each group).

Quotient: The result of division (the number of equal groups).

Remainder: The amount left over after division that cannot form another complete group.

3-digit number: A number with a hundreds place, tens place, and ones place (100-999).

Complete methodology:
  1. Identify the problem: Determine what needs to be divided and by what.
  2. Choose a strategy: Mental math (if possible) or long division.
  3. Execute the strategy: Apply the chosen method step-by-step.
  4. Handle remainders: If any remainders exist, note them appropriately.
  5. Verify the answer: Use multiplication to check your work.
  6. Interpret the result: Apply the answer to the original problem context.
Tip 1: Strengthen your multiplication facts, as they directly help with division.
Tip 2: For mental math, break 3-digit numbers into hundreds, tens, and ones that are easily divisible.
Tip 3: When using long division, keep numbers aligned in columns to avoid errors.
Tip 4: Always verify your answer by multiplying quotient by divisor and adding remainder.
Common errors: Forgetting to bring down digits in long division, making subtraction errors, not checking that remainder is less than divisor.
Exam preparation: Practice both mental math and long division methods, master multiplication facts, understand remainders, always verify answers.
3-digit division rules:

• Remainder is always less than the divisor

• Dividend = (Divisor × Quotient) + Remainder

• Use known multiplication facts to solve division efficiently

• Mental math strategies can simplify many 3-digit division problems

• Always verify your answer using multiplication

Exercise with Visualization: Division Strategy Comparison
Exercise 6: Efficiency Analysis
Comparing the efficiency of different strategies for dividing 3-digit numbers:
f₁(x) = Time for mental math (for x ÷ 8, x = 100-500)
f₂(x) = Time for long division (for x ÷ 8, x = 100-500)
f₃(x) = Accuracy rate for each method (for x ÷ 8, x = 100-500)

Analysis: The chart compares the effectiveness of different division strategies.

  • Mental math is faster for familiar facts
  • Long division is more reliable for complex problems
  • Combining strategies often yields the best results

Questions & Answers

Question: When should I use mental math versus long division for 3-digit numbers?

Answer: Use mental math when:

  • The division uses known multiplication facts (like 480 ÷ 8 = 60)
  • The number can be easily broken apart (like 480 = 400 + 80)
  • You can use place value patterns (like 480 = 48 tens)

Use long division when the numbers don't match familiar facts or when you need a systematic approach.

Question: How can I get better at mental math for 3-digit division? I always make mistakes.

Answer: Practice these mental math strategies:

  • Memorize multiplication facts thoroughly - division relies on these
  • Break numbers apart by place value: 480 ÷ 8 = (400 ÷ 8) + (80 ÷ 8) = 50 + 10 = 60
  • Use compatible numbers: 480 ÷ 8 = 48 tens ÷ 8 = 6 tens = 60
  • Practice with simple problems first, then gradually increase difficulty
  • Always verify your mental answer with multiplication

Regular practice will improve your mental math skills!

Question: My child is having trouble with remainders in 3-digit division. Any suggestions?

Answer: Help your child understand remainders with concrete examples:

  • Use physical objects like counters or blocks to show "leftover" items
  • Emphasize that remainders are always smaller than the divisor
  • Practice the verification step: (Divisor × Quotient) + Remainder = Dividend
  • Create stories: "If you have 357 cookies and 6 bags, each bag holds 59 cookies with 3 left over"
  • Use visual models like arrays to show complete groups and leftovers

Concrete experiences help build conceptual understanding of remainders.

Question: Why do I need to learn long division when I can just use multiplication facts?

Answer: Long division is important because:

  • Not all division problems use familiar multiplication facts
  • It provides a systematic method that works for any numbers
  • It builds foundational skills for more advanced math topics
  • It helps develop problem-solving and logical thinking skills
  • It's necessary for dividing larger numbers that aren't in your memory

Both mental math and long division are valuable tools in your math toolkit!

Question: How do I know if my 3-digit division answer is reasonable? Sometimes I get answers that seem too big or too small.

Answer: Use estimation to check reasonableness:

  • Round the dividend to a nearby number that's easy to divide
  • For 432 ÷ 8, think "about 400 ÷ 8 ≈ 50", so 54 is reasonable
  • Check that your answer makes sense in the problem context
  • Verify using multiplication: 54 × 8 = 432 ✓
  • Make sure remainder is less than divisor

Estimation helps catch major calculation errors before they become problems!