Solved Exercises on Dividing 2-Digit Numbers in Grade 4

Master dividing 2-digit numbers: long division, mental math strategies, remainders, and word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic 2-Digit Division
Exercise 1
Use long division to solve: 72 ÷ 6
Definition:

Dividing 2-digit numbers: The process of dividing a number with two digits by a single-digit divisor using long division or mental math strategies.

2-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
6 ) 72
Answer
?
Check
?
Step 1: Set up the long division

Write 6 ) 72 with the divisor (6) outside and dividend (72) inside

Step 2: Divide the first digit

How many times does 6 go into 7? Answer: 1 time
Write 1 above the 7

Step 3: Multiply

1 × 6 = 6
Write 6 below the 7

Step 4: Subtract

7 - 6 = 1
Write 1 below

Step 5: Bring down the next digit

Bring down the 2 to make 12

Step 6: Divide again

How many times does 6 go into 12? Answer: 2 times
Write 2 above the 2

Step 7: Multiply and subtract

2 × 6 = 12
12 - 12 = 0
No remainder

Step 8: Verify the answer

Check: 12 × 6 = 72 ✓

72 ÷ 6 = 12
Final answer:

72 ÷ 6 = 12

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 2-Digit Division with Remainder
Exercise 2
Use long division to solve: 85 ÷ 7
Definition:

Division with remainder: When a 2-digit number is divided and there's a leftover amount that's smaller than the divisor.

Setup
7 ) 85
Quotient
?
Remainder
?
Step 1: Set up the long division

Write 7 ) 85

Step 2: Divide the first digit

How many times does 7 go into 8? Answer: 1 time
Write 1 above the 8

Step 3: Multiply

1 × 7 = 7
Write 7 below the 8

Step 4: Subtract

8 - 7 = 1
Write 1 below

Step 5: Bring down the next digit

Bring down the 5 to make 15

Step 6: Divide again

How many times does 7 go into 15? Answer: 2 times (7 × 2 = 14)
Write 2 above the 5

Step 7: Multiply and subtract

2 × 7 = 14
15 - 14 = 1
Since 1 is less than 7, this is the remainder

Step 8: Verify the answer

Check: (7 × 12) + 1 = 84 + 1 = 85 ✓

85 ÷ 7 = 12 R1
Final answer:

85 ÷ 7 = 12 R1

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 2-Digit Division
Exercise 3
Solve 96 ÷ 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
96 ÷ 8
Strategy
?
Process
?
Step 1: Break apart the dividend

Think of 96 as 80 + 16

Step 2: Use known facts

I know that 80 ÷ 8 = 10 (from multiplication facts)

Step 3: Divide the remaining part

16 ÷ 8 = 2 (from multiplication facts)

Step 4: Add the partial quotients

10 + 2 = 12

Step 5: Verify the answer

Check: 12 × 8 = 96 ✓

Alternative strategy: Use multiples

Think: 8 × 10 = 80, so 96 - 80 = 16
Then 8 × 2 = 16
So 8 × 12 = 96

96 ÷ 8 = 12
Final answer:

96 ÷ 8 = 12. I used the break-apart strategy: 96 = 80 + 16, then 80 ÷ 8 = 10 and 16 ÷ 8 = 2, so 10 + 2 = 12.

Applied rules:

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

• Mental math: Use known multiplication facts to solve division

• Verification: Always check by multiplying quotient by divisor

2-Digit Division: Facts, Properties, and Methods
Dividend = (Divisor × Quotient) + Remainder
Division Formula
Division Example
72 ÷ 6 = 12
because 6 × 12 = 72
With Remainder
85 ÷ 7 = 12 R1
because (7 × 12) + 1 = 85
Mental Strategy
96 ÷ 8 = 12
because (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12
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.

2-digit number: A number with a tens place and a ones place (10-99).

2-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 × 12 = 96, then 96 ÷ 8 = 12.
Tip 2: Always check your work: (Divisor × Quotient) + Remainder should equal the Dividend.
Tip 3: When doing mental math, break the dividend into parts that are easily divisible by the divisor.
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 2-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 2-Digit Division
Strategy Effectiveness
Visualizing different mental math strategies for solving 2-digit division problems.

Analysis: The chart shows how different strategies work for various 2-digit division problems.

  • Breaking apart works well when dividends are close to multiples of the divisor
  • Compensation strategy is useful for numbers near round numbers
  • Using known facts is fastest when available
Solution: Exercises 4 to 5
4 Word Problem with 2-Digit Division
Exercise 4
A teacher has 56 books to distribute equally among 8 students. How many books will each student get?
Definition:

Word problems with 2-digit division: Real-world situations that require dividing a 2-digit number by a single-digit number to solve.

Total Books
56
Number of Students
8
Books per Student
?
Step 1: Identify the problem

We need to divide 56 books among 8 students: 56 ÷ 8

Step 2: Use mental math or long division

Using known facts: I know that 8 × 7 = 56

Step 3: State the answer

So 56 ÷ 8 = 7

Step 4: Verify the answer

Check: 7 × 8 = 56 ✓

Step 5: Interpret in context

Each student will get 7 books.

56 ÷ 8 = 7
Final answer:

Each student will get 7 books.

Applied rules:

• Problem-solving strategy: Identify total amount, number of groups, 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 2-Digit Numbers
Exercise 5
Find the pattern: 24÷4=6, 32÷4=8, 40÷4=10, 48÷4=?. Explain the pattern and predict 56÷4.
Definition:

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

Pattern
24÷4→6, 32÷4→8, 40÷4→10, 48÷4→?
Prediction
56÷4→?
Rule
?
Step 1: Analyze the pattern

24 ÷ 4 = 6
32 ÷ 4 = 8
40 ÷ 4 = 10

Step 2: Notice the relationship

Each time the dividend increases by 8, the quotient increases by 2
24 → 32 (+8), 6 → 8 (+2)
32 → 40 (+8), 8 → 10 (+2)

Step 3: Apply the pattern

40 → 48 (+8), so 10 → 12 (+2)
Therefore, 48 ÷ 4 = 12

Step 4: Predict the next term

48 → 56 (+8), so 12 → 14 (+2)
Therefore, 56 ÷ 4 = 14

Step 5: Verify with actual division

48 ÷ 4: 4 goes into 4 once (1), remainder 0, bring down 8, 4 goes into 8 twice (2), so 12 ✓
56 ÷ 4: 4 goes into 5 once (1), remainder 1, bring down 6 to make 16, 4 goes into 16 four times (4), so 14 ✓

Step 6: Generalize the pattern

When dividing by 4, increasing the dividend by 8 increases the quotient by 2

48 ÷ 4 = 12, 56 ÷ 4 = 14
Final answer:

The pattern shows that when dividing by 4, each time the dividend increases by 8, the quotient increases by 2. So 48 ÷ 4 = 12 and 56 ÷ 4 = 14.

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 2-Digit Numbers
Dividend = (Divisor × Quotient) + Remainder
Fundamental Division Formula
Key definitions:

Dividing 2-digit numbers: The process of dividing a number with two 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.

2-digit number: A number with a tens place and a ones place (10-99).

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 2-digit numbers into parts 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.
2-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 2-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 2-digit numbers:
f₁(x) = Time for mental math (for x ÷ 6, x = 12-96)
f₂(x) = Time for long division (for x ÷ 6, x = 12-96)
f₃(x) = Accuracy rate for each method (for x ÷ 6, x = 12-96)

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 2-digit numbers?

Answer: Use mental math when:

  • The division uses known multiplication facts (like 48 ÷ 6 = 8)
  • The number can be easily broken apart (like 72 ÷ 8 = 40÷8 + 32÷8 = 5 + 4 = 9)
  • You can use patterns or multiples you know well

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 division? I always make mistakes.

Answer: Practice these mental math strategies:

  • Memorize multiplication facts thoroughly - division relies on these
  • Break numbers apart: 63 ÷ 7 = (35 ÷ 7) + (28 ÷ 7) = 5 + 4 = 9
  • Use compatible numbers: 72 ÷ 8 = 72 ÷ 8 (think 8 × 9 = 72)
  • 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 2-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 17 cookies and 5 friends, each gets 3 cookies with 2 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 2-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 72 ÷ 6, think "about 70 ÷ 6 ≈ 11-12", so 12 is reasonable
  • Check that your answer makes sense in the problem context
  • Verify using multiplication: 12 × 6 = 72 ✓
  • Make sure remainder is less than divisor

Estimation helps catch major calculation errors before they become problems!