Division Problems 3 Digit by 1 - Complete Guide for Grade 4 Students

Master division of 3-digit numbers by 1-digit numbers: long division method, quotients, remainders, and step-by-step solutions through comprehensive exercises and detailed explanations.

Division Problems 3 Digit by 1 - 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 1-Digit Division: Division where the dividend is a 3-digit number (100-999) and the divisor is a 1-digit number (1-9)

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

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

Step-by-Step Method
Long Division Process for 3 Digit by 1:
  1. Compare hundreds place: How many times does 1-digit divisor go into hundreds digit?
  2. If divisor is larger: Combine hundreds and tens digits
  3. Divide: How many times does divisor go into combined number?
  4. Multiply: Multiply quotient digit by divisor
  5. Subtract: Subtract result from current number
  6. Bring down: Bring down the next digit
  7. Repeat: Continue with remaining digits
  8. Check: Verify using: Divisor × Quotient + Remainder = Dividend
Examples and Tips
Simple Example: 120 ÷ 4 = 30 (no remainder)
With Remainder: 123 ÷ 4 = 30 remainder 3
Key Tip: Start with the largest place value (hundreds) 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:
\(144 \div 6\)
Understanding the Problem:

This is a basic 3-digit by 1-digit division problem where we need to find how many times 6 goes into 144

Solution Approach:
  1. Start with the hundreds place: 1 ÷ 6 = 0 (since 6 > 1)
  2. Consider hundreds and tens together: 14 ÷ 6 = 2 remainder 2
  3. Continue with the ones place
Expression
\(144 \div 6\)
Division
\(24\)
Step 1: Set up the division

We need to divide 144 by 6

144 ÷ 6 = ?

Step 2: Compare hundreds place

How many times does 6 go into 1 (hundreds place)?

Since 6 > 1, 6 goes into 1 zero times

We consider the hundreds and tens together: 14

Step 3: Divide 14 (hundreds and tens)

How many times does 6 go into 14?

6 × 2 = 12, 6 × 3 = 18 (too big)

So 6 goes into 14 twice (2)

Write 2 above the tens place

Step 4: Multiply and subtract

2 × 6 = 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 6 go into 24?

6 × 4 = 24

So 6 goes into 24 exactly four times (4)

Write 4 above the ones place

Step 7: Multiply and subtract

4 × 6 = 24

24 - 24 = 0

No remainder

\(144 \div 6 = 24\)
Final answer:

\(144 \div 6 = 24\) with no remainder

Applied rules:

• Place value consideration: When divisor is larger than current digit, combine with next digit

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

• Multiplication check: 24 × 6 = 144 ✓

2 Division with Remainder
Exercise 2
Find the quotient and remainder of:
\(257 \div 4\)
Understanding the Problem:

This 3-digit by 1-digit division will have a remainder since 4 doesn't divide evenly into 257

Expression
\(257 \div 4\)
Division
\(64\) R\(1\)
Step 1: Set up the division

We need to divide 257 by 4

257 ÷ 4 = ?

Step 2: Compare hundreds place

How many times does 4 go into 2 (hundreds place)?

Since 4 > 2, 4 goes into 2 zero times

We consider the hundreds and tens together: 25

Step 3: Divide 25 (hundreds and tens)

How many times does 4 go into 25?

4 × 6 = 24, 4 × 7 = 28 (too big)

So 4 goes into 25 six times (6)

Write 6 above the tens place

Step 4: Multiply and subtract

6 × 4 = 24

25 - 24 = 1

Step 5: Bring down the ones place

Bring down the 7

Now we have 17

Step 6: Divide the new number

How many times does 4 go into 17?

4 × 4 = 16, 4 × 5 = 20 (too big)

So 4 goes into 17 four times (4)

Write 4 above the ones place

Step 7: Multiply and subtract

4 × 4 = 16

17 - 16 = 1

1 is less than 4, so 1 is our remainder

\(257 \div 4 = 64\) remainder \(1\)
Final answer:

\(257 \div 4 = 64\) remainder \(1\)

Check: \(64 \times 4 + 1 = 256 + 1 = 257\) ✓

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:
\(308 \div 7\)
Understanding the Problem:

This 3-digit by 1-digit division includes a zero in the dividend and will result in a 2-digit quotient

Expression
\(308 \div 7\)
Division
\(44\)
Step 1: Set up the division

We need to divide 308 by 7

308 ÷ 7 = ?

Step 2: Compare hundreds place

How many times does 7 go into 3 (hundreds place)?

Since 7 > 3, 7 goes into 3 zero times

We consider the hundreds and tens together: 30

Step 3: Divide 30 (hundreds and tens)

How many times does 7 go into 30?

7 × 4 = 28, 7 × 5 = 35 (too big)

So 7 goes into 30 four times (4)

Write 4 above the tens place

Step 4: Multiply and subtract

4 × 7 = 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 7 go into 28?

7 × 4 = 28

So 7 goes into 28 exactly four times (4)

Write 4 above the ones place

Step 7: Multiply and subtract

4 × 7 = 28

28 - 28 = 0

No remainder

\(308 \div 7 = 44\)
Final answer:

\(308 \div 7 = 44\) with no remainder

Check: \(44 \times 7 = 308\) ✓

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 books to distribute equally among 8 classrooms. How many books will each classroom receive?
Understanding the Problem:

This is a word problem requiring 3-digit by 1-digit division to solve an equal sharing situation

Problem
\(432 \div 8\)
Division
\(54\)
Step 1: Identify the operation

School has 432 books and 8 classrooms

To share equally: 432 ÷ 8 = ?

Step 2: Set up the division

We need to divide 432 by 8

432 ÷ 8 = ?

Step 3: Compare hundreds place

How many times does 8 go into 4 (hundreds place)?

Since 8 > 4, 8 goes into 4 zero times

We consider the hundreds and tens together: 43

Step 4: Divide 43 (hundreds and tens)

How many times does 8 go into 43?

8 × 5 = 40, 8 × 6 = 48 (too big)

So 8 goes into 43 five times (5)

Write 5 above the tens place

Step 5: Multiply and subtract

5 × 8 = 40

43 - 40 = 3

Step 6: Bring down the ones place

Bring down the 2

Now we have 32

Step 7: Divide the new number

How many times does 8 go into 32?

8 × 4 = 32

So 8 goes into 32 exactly four times (4)

Write 4 above the ones place

Step 8: Multiply and subtract

4 × 8 = 32

32 - 32 = 0

No remainder

Step 9: Interpret the result

Each classroom gets 54 books

All books are distributed (no remainder)

Each classroom gets 54 books
Final answer:

Each classroom gets 54 books.

Check: \(54 \times 8 = 432\) ✓

Applied rules:

• Equal sharing: Division to distribute items equally

• 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: \(563 \div 9\) and verify your answer
Understanding the Problem:

This 3-digit by 1-digit division will have a remainder. We need to find the quotient and verify our answer.

Expression
\(563 \div 9\)
Division
\(62\) R\(5\)
Step 1: Set up the division

We need to divide 563 by 9

563 ÷ 9 = ?

Step 2: Compare hundreds place

How many times does 9 go into 5 (hundreds place)?

Since 9 > 5, 9 goes into 5 zero times

We consider the hundreds and tens together: 56

Step 3: Divide 56 (hundreds and tens)

How many times does 9 go into 56?

9 × 6 = 54, 9 × 7 = 63 (too big)

So 9 goes into 56 six times (6)

Write 6 above the tens place

Step 4: Multiply and subtract

6 × 9 = 54

56 - 54 = 2

Step 5: Bring down the ones place

Bring down the 3

Now we have 23

Step 6: Divide the new number

How many times does 9 go into 23?

9 × 2 = 18, 9 × 3 = 27 (too big)

So 9 goes into 23 two times (2)

Write 2 above the ones place

Step 7: Multiply and subtract

2 × 9 = 18

23 - 18 = 5

5 is less than 9, so 5 is our remainder

Step 8: Verify the answer

Check: 62 × 9 + 5 = 558 + 5 = 563 ✓

Alternative check: 563 ÷ 9 = 62 remainder 5 ✓

\(563 \div 9 = 62\) remainder \(5\)
Final answer:

\(563 \div 9 = 62\) remainder \(5\)

Check: \(62 \times 9 + 5 = 558 + 5 = 563\) ✓

Applied rules:

• Division verification: Always check your answer

• Multiplication facts: Knowing 9 × 62 = 558 helps solve this

• Place value: Work from hundreds to tens to ones

Questions & Answers

Question: When I divide 3-digit numbers by 1-digit numbers, sometimes I get confused about when to combine digits. Can you explain this?

Answer: Great question! You combine digits when the divisor is larger than the current digit you're working with. Here's how:

Rule: When the divisor is larger than the current digit, you combine that digit with the next digit to form a larger number.

Example with 257 ÷ 4:

  • 4 cannot go into 2 (hundreds place), so we combine 2 and 5 to get 25
  • 4 goes into 25 six times (24), so we write 6 above the tens place
  • Then continue with the ones place

Why we do this: We need enough "value" to divide by the divisor. If we only have 2, we can't divide by 4, but if we have 25, we can divide by 4.

Another Example: In 144 ÷ 6, 6 cannot go into 1, so we look at 14 (hundreds and tens together).

Question: I sometimes get the digits mixed up in my quotient. How do I make sure I put the numbers in the right places?

Answer: This is a common challenge! Here are strategies to keep your digits in the right places:

Strategy 1: Use Place Value Lines

  • Draw lines above your dividend to show where each digit of the quotient goes
  • For 3-digit dividends, draw three lines: _ _ _ (hundreds, tens, ones)
  • When you divide the hundreds and tens together, write the result above the tens place
  • When you divide the final number, write the result above the ones place

Strategy 2: Think About Size

  • Look at your dividend. If it's 432 ÷ 8, your quotient should be around 50 (since 400 ÷ 8 = 50)
  • Estimate: 432 ÷ 8 should be close to 400 ÷ 8 = 50

Strategy 3: Check Your Work

  • Always verify: Quotient × Divisor + Remainder = Dividend
  • If this doesn't work, check your digit placement

Question: Why do I sometimes get 2-digit quotients when dividing 3-digit numbers by 1-digit numbers?

Answer: This happens when the divisor is relatively large compared to the dividend. The number of digits in your quotient depends on how many times the divisor fits into your dividend:

Examples:

  • 144 ÷ 6 = 24 (2-digit quotient) - because 6 fits into 144 many times
  • 432 ÷ 8 = 54 (2-digit quotient) - because 8 fits into 432 many times
  • 120 ÷ 2 = 60 (2-digit quotient) - because 2 fits into 120 many times

Contrast with:

  • 432 ÷ 2 = 216 (3-digit quotient) - because 2 fits into 432 fewer times per place value

Rule of Thumb: The larger the divisor, the smaller the quotient. If the divisor is relatively large compared to the dividend, you'll often get a quotient with fewer digits than the dividend.