Divide 3-digit and 4-digit numbers by 2-digit numbers using long division
Understand the terms: dividend, divisor, quotient, and remainder
Use the long division algorithm step by step (divide, multiply, subtract, bring down)
Verify division answers using the formula: Dividend = Divisor × Quotient + Remainder
Estimate quotients by rounding the divisor
Apply division properties (dividing by 1, dividing by itself, zero divided by a number)
Divide by multiples of 10 using shortcuts
Solve word problems involving division in real-life Indian contexts
Introduction — Division in Daily Life
Division is one of the four basic operations in mathematics. It means sharing equally or finding out how many groups can be made. In Class 4, you learned to divide numbers by a single digit (1 to 9). Now in Class 5, we will divide larger numbers — 3-digit and 4-digit numbers by 2-digit numbers — using the long division method.
Think about these real-life situations:
Arjun has 756 laddoos to pack in boxes of 12 each. How many boxes does he need?
A school has 945 students. If each bus carries 15 students, how many buses are needed?
Meena's father earned ₹2,568 in 32 days. How much did he earn per day?
5,400 notebooks are to be distributed equally among 45 classrooms. How many per class?
All these problems need division by a 2-digit number. Let us learn how to solve them!
Division Vocabulary — Key Terms
Before we begin, let us understand the parts of a division problem:
Term
Meaning
Example (756 ÷ 12 = 63)
Dividend
The number being divided (the total)
756
Divisor
The number we divide by (group size)
12
Quotient
The answer (how many groups)
63
Remainder
What is left over after division
0 (exact division)
Image needed: A labelled diagram showing the parts of a division problem in the long division "house" format. Show 756 ÷ 12 = 63 with clear labels pointing to the dividend (inside the house), divisor (outside left), quotient (on top), and remainder (at bottom). Include the division equation: Dividend = Divisor × Quotient + Remainder. Style: Clean flat illustration, colour-coded labels (dividend=blue, divisor=green, quotient=red, remainder=orange), child-friendly Size: 700×400px
The Division Equation
Dividend = Divisor × Quotient + Remainder
This is also written as: D = d × Q + R
The remainder must always be less than the divisor (R < d)
Example: If 847 ÷ 23 = 36 remainder 19, then:
Check: 23 × 36 + 19 = 828 + 19 = 847 ✓
The remainder (19) is less than the divisor (23) ✓
Quick Revision — Division by a Single Digit
In Class 4, you learned to divide by single-digit numbers. Let us quickly revise with one example before moving to 2-digit divisors.
Example: Divide 456 ÷ 3
Step 1: 4 ÷ 3 = 1, remainder 1. Write 1 in quotient. Step 2: Bring down 5. Now we have 15. 15 ÷ 3 = 5. Write 5 in quotient. Step 3: Bring down 6. Now we have 06. 6 ÷ 3 = 2. Write 2 in quotient.
The same method applies when dividing by 2-digit numbers — but now we need to think about two digits at a time instead of one.
Long Division by 2-Digit Numbers — Step-by-Step Method
When dividing by a 2-digit number, we follow the same cycle: Divide → Multiply → Subtract → Bring Down. The key difference is that we now look at the first two or three digits of the dividend to begin.
Image needed: A circular diagram showing the 4-step long division cycle: DIVIDE → MULTIPLY → SUBTRACT → BRING DOWN, with arrows connecting each step in a loop. Each step shown in a different colour with a brief label explaining what to do. Style: Circular flowchart, colourful, child-friendly, bold text Size: 500×500px
Steps for Long Division
Step 1 — DIVIDE: Look at the first two (or three) digits of the dividend. Ask: "How many times does the divisor fit into this number?"
Step 2 — MULTIPLY: Multiply the divisor by the quotient digit you chose.
Step 3 — SUBTRACT: Subtract the product from the digits you were looking at.
Step 4 — BRING DOWN: Bring down the next digit of the dividend.
Repeat Steps 1–4 until all digits have been brought down.
Worked Example 1: 756 ÷ 12 (Exact Division)
Dividend: 756 Divisor: 12
Step 1: Look at the first two digits: 75. How many times does 12 go into 75?
Think: 12 × 6 = 72 (fits!) and 12 × 7 = 84 (too big!). So quotient digit = 6.
Step 2: Multiply: 12 × 6 = 72
Step 3: Subtract: 75 − 72 = 3
Step 4: Bring down the next digit (6). We now have 36.
Step 5: How many times does 12 go into 36? 12 × 3 = 36. Quotient digit = 3.
Step 6: Multiply: 12 × 3 = 36
Step 7: Subtract: 36 − 36 = 0. No more digits to bring down.
Answer: 756 ÷ 12 = 63, Remainder = 0
Check: 12 × 63 + 0 = 756 ✓
Worked Example 2: 945 ÷ 15 (Exact Division)
Dividend: 945 Divisor: 15
Step 1: First two digits: 94. How many times does 15 go into 94?
Think: 15 × 6 = 90 (fits!) and 15 × 7 = 105 (too big!). Quotient digit = 6.
Step 2: Multiply: 15 × 6 = 90
Step 3: Subtract: 94 − 90 = 4
Step 4: Bring down 5. We now have 45.
Step 5: How many times does 15 go into 45? 15 × 3 = 45. Quotient digit = 3.
Step 6: Multiply: 15 × 3 = 45
Step 7: Subtract: 45 − 45 = 0.
Answer: 945 ÷ 15 = 63, Remainder = 0
Check: 15 × 63 + 0 = 945 ✓
Worked Example 3: 847 ÷ 23 (With Remainder)
Dividend: 847 Divisor: 23
Step 1: First two digits: 84. How many times does 23 go into 84?
Think: 23 × 3 = 69, 23 × 4 = 92 (too big!). So quotient digit = 3.
Step 2: Multiply: 23 × 3 = 69
Step 3: Subtract: 84 − 69 = 15
Step 4: Bring down 7. We now have 157.
Step 5: How many times does 23 go into 157?
Think: 23 × 6 = 138, 23 × 7 = 161 (too big!). Quotient digit = 6.
Step 6: Multiply: 23 × 6 = 138
Step 7: Subtract: 157 − 138 = 19. No more digits to bring down.
Answer: 847 ÷ 23 = 36, Remainder = 19
Check: 23 × 36 + 19 = 828 + 19 = 847 ✓
Note: Remainder (19) is less than Divisor (23) ✓
Worked Example 4: 2,568 ÷ 32 (4-Digit ÷ 2-Digit)
Dividend: 2568 Divisor: 32
Step 1: First two digits: 25. Can 32 go into 25? No (32 > 25)!
So look at the first three digits: 256.
Step 2: How many times does 32 go into 256?
Think: 32 × 8 = 256. Perfect fit! Quotient digit = 8.
Step 3: Multiply: 32 × 8 = 256
Step 4: Subtract: 256 − 256 = 0
Step 5: Bring down 8. We now have 08 (which is just 8).
Step 6: How many times does 32 go into 8? Zero times! Quotient digit = 0.
Step 7: Multiply: 32 × 0 = 0. Subtract: 8 − 0 = 8. No more digits.
Answer: 2,568 ÷ 32 = 80, Remainder = 8
Check: 32 × 80 + 8 = 2,560 + 8 = 2,568 ✓
Worked Example 5: 5,400 ÷ 45 (With Zeros in Dividend)
Dividend: 5400 Divisor: 45
Step 1: First two digits: 54. How many times does 45 go into 54?
Think: 45 × 1 = 45 (fits!), 45 × 2 = 90 (too big!). Quotient digit = 1.
Step 2: Multiply: 45 × 1 = 45
Step 3: Subtract: 54 − 45 = 9
Step 4: Bring down 0. We now have 90.
Step 5: How many times does 45 go into 90? 45 × 2 = 90. Quotient digit = 2.
Step 7: Bring down 0. We now have 0. 45 goes into 0 zero times. Quotient digit = 0.
Answer: 5,400 ÷ 45 = 120, Remainder = 0
Check: 45 × 120 + 0 = 5,400 ✓
Worked Example 6: 1,000 ÷ 25 (Exact Division)
Dividend: 1000 Divisor: 25
Step 1: First two digits: 10. Can 25 go into 10? No!
Look at first three digits: 100. How many times does 25 go into 100?
25 × 4 = 100. Quotient digit = 4.
Step 3: Bring down 0. We have 0. 25 goes into 0 zero times. Quotient digit = 0.
Answer: 1,000 ÷ 25 = 40, Remainder = 0
Check: 25 × 40 + 0 = 1,000 ✓
Think About It: When the first two digits of the dividend are smaller than the divisor, we look at the first three digits instead. This is the key difference when dividing by 2-digit numbers!
Estimating Quotients — How to Guess the Right Digit
The hardest part of dividing by a 2-digit number is finding the correct quotient digit. Here are some helpful tricks:
Estimation Tricks
Round the divisor: Round the divisor to the nearest 10, then divide mentally. This gives you a good starting guess.
Trial and error: Try a digit. If the product is too big, go one digit lower. If there is too much left over (more than the divisor), go one digit higher.
Use multiplication tables: Knowing tables up to 20 helps a lot!
Example: Estimate 847 ÷ 23
Method 1 — Round the divisor:
Round 23 to 20. Now think: 84 ÷ 20 ≈ 4. Try 4: 23 × 4 = 92. But 92 > 84, so try 3: 23 × 3 = 69 ✓
Method 2 — Think of nearby facts:
23 × 3 = 69, 23 × 4 = 92. Since 84 is between 69 and 92, the answer is 3.
Example: Estimate 5,400 ÷ 45
Round 45 to 50. Think: 5,400 ÷ 50 = 108. So the actual answer should be close to 108.
Let's check: 45 × 120 = 5,400. The actual answer is 120.
(Rounding gives us a rough idea — not the exact answer, but it helps us check if we're on the right track!)
Checking Your Division — Verification
After every division, you should check your answer using this formula:
Verification Formula
Dividend = Divisor × Quotient + Remainder
If this equation is true, your division is correct!
If Remainder = 0, then Dividend = Divisor × Quotient (exact division)
Image needed: A visual showing the checking formula with arrows. Show a completed division problem (847 ÷ 23 = 36 R 19) on the left side, and on the right side show the check: 23 × 36 = 828, then 828 + 19 = 847. Use arrows and colour coding to connect Divisor, Quotient, Remainder back to the Dividend. A green checkmark at the end. Style: Step-by-step visual with arrows, colour-coded, child-friendly Size: 750×350px
Example 1: Check if 945 ÷ 15 = 63 is correct.
Divisor × Quotient + Remainder = 15 × 63 + 0 = 945 ✓ (matches the dividend!)
Example 3: Ravi says 784 ÷ 24 = 33, remainder 2. Is he correct?
Check: 24 × 33 + 2 = 792 + 2 = 794. But the dividend is 784, not 794! Ravi is wrong. Let us find the correct answer:
784 ÷ 24: 78 ÷ 24 → 24 × 3 = 72. Subtract: 78 − 72 = 6. Bring down 4 → 64.
64 ÷ 24 → 24 × 2 = 48. Subtract: 64 − 48 = 16.
Correct answer: 784 ÷ 24 = 32, remainder 16
Check: 24 × 32 + 16 = 768 + 16 = 784 ✓
Properties of Division
Division has some important rules that always hold true:
Property
Rule
Example
Division by 1
Any number ÷ 1 = the number itself
756 ÷ 1 = 756
Division by itself
Any number ÷ itself = 1
45 ÷ 45 = 1
Zero divided
0 ÷ any number = 0
0 ÷ 23 = 0
Cannot divide by 0
Any number ÷ 0 = not defined (impossible!)
756 ÷ 0 = undefined
Dividend < Divisor
If dividend is smaller, quotient = 0 with remainder = dividend
5 ÷ 12 = 0 remainder 5
Dividend = Divisor
Quotient = 1, Remainder = 0
32 ÷ 32 = 1
Think About It: Why can we not divide by zero? Imagine you have 12 sweets and want to share them among 0 friends. How many does each friend get? The question makes no sense! There is no one to share with. That is why division by zero is undefined.
Division by Multiples of 10 — Shortcuts
Dividing by 10, 20, 30, 40, etc. can be done faster using a shortcut:
Shortcut for Dividing by Multiples of 10
To divide by 10: Remove one zero from the end of the dividend (or move the decimal point one place left).
To divide by 20: First divide by 10 (remove a zero), then divide by 2.
To divide by 30: First divide by 10, then divide by 3.
General: To divide by (N × 10): First divide by 10, then divide by N.
Another Way to Think: When both the dividend and divisor end in zeros, you can cancel equal zeros from both:
4,500 ÷ 20 → Cancel one zero each → 450 ÷ 2 = 225
6,300 ÷ 30 → Cancel one zero each → 630 ÷ 3 = 210
Word Problems — Division in Real Life
Division helps us solve many real-life problems. Here are some examples with Indian context:
Image needed: An illustration showing real-life division scenarios: (1) A person distributing laddoos equally into boxes, (2) Students boarding buses equally, (3) Rupee notes being split among friends, (4) Notebooks being distributed to classrooms. Four mini-scenes arranged in a 2×2 grid. Style: Colourful flat illustration, Indian context (Indian clothing, ₹ symbol), child-friendly Size: 750×500px
Problem 1: Arjun's mother made 756 laddoos for Diwali. She wants to pack them in boxes of 12 each. How many boxes will she fill completely?
Solution: 756 ÷ 12 = 63
She will fill 63 boxes completely with no laddoos left over.
Problem 2: A school has 945 students going on a picnic. Each bus can carry 15 students. How many buses are needed?
Solution: 945 ÷ 15 = 63
The school needs 63 buses.
Problem 3: Priya earned ₹2,568 by selling handmade greeting cards over 32 days. How much did she earn per day on average?
Solution: 2,568 ÷ 32 = 80, remainder 8
She earned approximately ₹80 per day (with ₹8 extra over the month).
Problem 4: A farmer harvested 5,400 kg of wheat. He wants to fill sacks of 45 kg each. How many sacks can he fill?
Solution: 5,400 ÷ 45 = 120
He can fill 120 sacks.
Problem 5: Ravi and his 24 classmates (25 students total) went on a trip. The total auto fare was ₹875. If they split it equally, how much does each student pay?
Solution: 875 ÷ 25 = 35
Each student pays ₹35.
Problem 6: A library has 1,848 books to arrange on shelves. Each shelf holds 22 books. How many shelves are needed?
The number we divide by (size of each group or number of groups)
Quotient
The answer to a division (how many groups or how much per group)
Remainder
The amount left over when division is not exact
Long Division
The step-by-step method for dividing large numbers
Exact Division
When the remainder is 0 (divides perfectly)
Estimation
Finding an approximate answer by rounding numbers
Verify / Check
Proving your answer is correct using Dividend = Divisor × Quotient + Remainder
Multiple
A number that can be divided evenly by another number (e.g., 45 is a multiple of 15)
Key Points to Remember
Division means sharing equally or finding how many groups.
Dividend = Divisor × Quotient + Remainder — use this to check every answer!
The remainder must always be less than the divisor. If it is equal or greater, your quotient is too small.
When dividing by a 2-digit number, start by looking at the first 2 digits of the dividend. If they are smaller than the divisor, look at the first 3 digits.
The cycle is: Divide → Multiply → Subtract → Bring Down. Repeat!
To estimate the quotient digit: round the divisor to the nearest 10 and divide mentally.
Any number ÷ 1 = itself | Any number ÷ itself = 1 | 0 ÷ any number = 0
Division by 0 is not possible (undefined).
When dividing by multiples of 10, cancel common zeros from dividend and divisor first.
In word problems, look for keywords: "share equally," "distribute," "split," "how many groups," "how much each," "per day/per person."
Finished learning? Now test what you know!
🎯 Quick Quiz — Test Yourself!
Answer these questions right here. You get instant feedback — no paper needed!
Score: 0 / 8
1. 144 ÷ 12 = ?
12 × 12 = 144, so 144 ÷ 12 = 12.
2. In 47 ÷ 5, what is the remainder?
5 × 9 = 45, and 47 − 45 = 2. So quotient 9, remainder 2.
3. Any number divided by 1 gives:
Dividing by 1 does not change the number.
4. Which division check is correct for 847 ÷ 23 = 36 R 19?
7. 300 chocolates shared equally among 12 children — how many each? (whole number)
300 ÷ 12 = 25 chocolates each (exactly, no remainder).
8. 56 students go on a trip in vans that hold 8 each. How many vans are needed?
56 ÷ 8 = 7 vans (each holds 8, and 7 × 8 = 56).
Practice Questions
A. Divide the following (show full working):
1. 936 ÷ 12
2. 784 ÷ 14
3. 675 ÷ 25
4. 891 ÷ 27
5. 2,496 ÷ 32
6. 3,780 ÷ 45
7. 4,928 ÷ 56
8. 7,200 ÷ 48
B. Divide and write the quotient and remainder:
1. 529 ÷ 16 Quotient = Remainder =
2. 853 ÷ 21 Quotient = Remainder =
3. 1,479 ÷ 34 Quotient = Remainder =
4. 2,345 ÷ 18 Quotient = Remainder =
5. 6,789 ÷ 53 Quotient = Remainder =
C. Check the following divisions using the formula (Dividend = Divisor × Quotient + Remainder):
1. 864 ÷ 24 = 36, R = 0. Is this correct?
2. 975 ÷ 13 = 75, R = 0. Is this correct?
3. 1,234 ÷ 28 = 43, R = 30. Is this correct?
4. 3,500 ÷ 42 = 83, R = 14. Is this correct?
D. Estimate the quotient (round the divisor to nearest 10):
1. 726 ÷ 18 ≈
2. 1,560 ÷ 31 ≈
3. 2,890 ÷ 49 ≈
4. 4,200 ÷ 58 ≈
E. Fill in the blanks:
1. In a division problem, if the dividend is 945, the divisor is 15, and the quotient is 63, then the remainder is .
2. Any number divided by 1 gives .
3. If the remainder is 0, the division is called division.
4. The remainder must always be than the divisor.
5. 0 ÷ 47 = .
F. True or False:
1. 450 ÷ 10 = 45
2. The remainder can be equal to the divisor.
3. If we divide any number by itself, the answer is always 1.
4. 0 ÷ 56 = 56
5. Division by 0 is possible and gives 0.
G. Choose the correct answer (MCQ):
1. What is 1,296 ÷ 36?
a) 34b) 36c) 38d) 32
2. If Dividend = 1,000, Divisor = 25, and Quotient = 40, what is the remainder?
a) 25b) 10c) 0d) 5
3. Which division has the largest remainder?
a) 100 ÷ 12b) 100 ÷ 15c) 100 ÷ 11d) 100 ÷ 13
4. 4,500 ÷ 50 = ?
a) 80b) 90c) 95d) 85
H. Word Problems:
1. Sita made 864 gulab jamuns for a wedding. She wants to put 24 in each tray. How many trays does she need?
2. A train covers 1,680 km in 14 hours. What is the speed of the train (distance per hour)?
3. Ravi has ₹3,250. He wants to buy notebooks that cost ₹25 each. How many notebooks can he buy?
4. A factory produces 4,536 biscuit packets in a day. They are packed in cartons of 36 each. How many cartons are filled?
5. 1,575 mangoes are distributed equally among 45 families in a village. How many mangoes does each family get?
6. An auto driver earns ₹2,730 in 21 days. How much does he earn per day? If he saves ₹50 per day, how much does he spend per day?
Activity — Fair Share Challenge!
Work with a partner or family member. Use the table below. For each row, you are given a total and the number of people. Find out how many each person gets and how much is left over.
Total Items
Number of People
Each Person Gets
Left Over
500 toffees
12 friends
₹1,000
15 students
365 days (1 year)
52 weeks
900 stickers
24 children
2,500 ml juice
16 glasses
Bonus Challenge: Verify each answer using the formula: Total = (Each person gets × Number of people) + Left over
Want to review the lesson again?
Mathematics
Division — 3-Digit by 2-Digit (Long Division)
Class 5 | CBSE / NCERT / ICSE
Name: Class: Date:
Learning Objectives
Divide 3-digit and 4-digit numbers by 2-digit numbers using long division
Understand dividend, divisor, quotient, and remainder
Division means sharing equally or finding how many groups. In Class 4, you divided by single digits. Now in Class 5, we divide 3-digit and 4-digit numbers by 2-digit numbers using long division.
Real-life examples: packing 756 laddoos in boxes of 12, splitting ₹2,568 equally over 32 days, distributing 5,400 notebooks among 45 classrooms.
Division Vocabulary
Term
Meaning
Example (756 ÷ 12 = 63)
Dividend
Number being divided
756
Divisor
Number we divide by
12
Quotient
The answer
63
Remainder
What is left over
0
[Image: Labelled diagram — long division house format showing dividend, divisor, quotient, remainder with 756 ÷ 12 = 63]
Division Equation: Dividend = Divisor × Quotient + Remainder (R must be less than Divisor)
Shortcut: Cancel equal zeros from both dividend and divisor, then divide.
4,500÷20 → 450÷2 = 225 | 6,300÷30 → 630÷3 = 210 | 8,000÷50 → 800÷5 = 160
Word Problems — Division in Real Life
[Image: Real-life division — distributing laddoos in boxes, students in buses, splitting ₹ among friends, notebooks to classrooms]
1. 756 laddoos in boxes of 12: 756÷12 = 63 boxes 2. 945 students, 15 per bus: 945÷15 = 63 buses 3. ₹2,568 over 32 days: 2568÷32 = ₹80/day (R=8) 4. 5,400 kg wheat in 45 kg sacks: 5400÷45 = 120 sacks 5. ₹875 split among 25 students: 875÷25 = ₹35 each 6. 1,848 books on shelves of 22: 1848÷22 = 84 shelves
Key Words
Term
Meaning
Dividend
The number being divided
Divisor
The number we divide by
Quotient
The answer to division
Remainder
Amount left over (must be < divisor)
Long Division
Step-by-step method for dividing large numbers
Exact Division
When remainder is 0
Estimation
Approximate answer by rounding
Verify
Check using D = d×Q + R
Key Points:
• Dividend = Divisor × Quotient + Remainder — always check!
• Remainder must be less than divisor.
• When first 2 digits < divisor, use first 3 digits.
• Cycle: Divide → Multiply → Subtract → Bring Down.
• a÷1=a | a÷a=1 | 0÷a=0 | a÷0=undefined.
• For multiples of 10: cancel common zeros first.
• Word problem keywords: share, distribute, split, per day, each, equally.
Practice Questions
A. Divide the following (show full working):
1. 936 ÷ 12
2. 784 ÷ 14
3. 675 ÷ 25
4. 891 ÷ 27
5. 2,496 ÷ 32
6. 3,780 ÷ 45
7. 4,928 ÷ 56
8. 7,200 ÷ 48
B. Divide and write quotient and remainder:
1. 529 ÷ 16 → Q = R =
2. 853 ÷ 21 → Q = R =
3. 1,479 ÷ 34 → Q = R =
4. 2,345 ÷ 18 → Q = R =
5. 6,789 ÷ 53 → Q = R =
C. Check using formula (Dividend = Divisor × Quotient + Remainder):
1. 864 ÷ 24 = 36, R=0. Correct?
2. 975 ÷ 13 = 75, R=0. Correct?
3. 1,234 ÷ 28 = 43, R=30. Correct?
4. 3,500 ÷ 42 = 83, R=14. Correct?
D. Estimate the quotient (round divisor to nearest 10):
1. 726 ÷ 18 ≈
2. 1,560 ÷ 31 ≈
3. 2,890 ÷ 49 ≈
4. 4,200 ÷ 58 ≈
E. Fill in the blanks:
1. If dividend=945, divisor=15, quotient=63, remainder =