Division of Large Numbers
Division of large numbers means splitting a number with four, five, or more digits into equal groups. In Class 5, students learn to divide 4-digit and 5-digit numbers by 2-digit and 3-digit divisors using the long division method.
Division answers two questions: How many groups? and How many in each group? For example, if 7,560 mangoes are packed equally into 36 boxes, division tells us each box gets 210 mangoes.
Long division is a step-by-step process of dividing, multiplying, subtracting, and bringing down — repeated until all digits are used.
What is Division of Large Numbers - Class 5 Maths (Operations)?
Division is the process of splitting a number into equal parts. The four parts of a division are:
| Dividend (number being divided) | ÷ | Divisor (number of groups) | = | Quotient (result) | R | Remainder (left over) |
Division algorithm (verification formula):
Dividend = (Divisor x Quotient) + Remainder
Key properties:
- Division is not commutative: 12 / 4 is not equal to 4 / 12
- Any number divided by 1 gives the number itself
- Any number divided by itself gives 1
- 0 divided by any number gives 0
- Division by 0 is not defined
Division of Large Numbers Formula
Dividend = (Divisor x Quotient) + Remainder
This formula is used to verify every division answer. The remainder must always be less than the divisor.
Types and Properties
Steps of Long Division (DMSB method):
- D — Divide: How many times does the divisor go into the first group of digits?
- M — Multiply: Multiply the divisor by the quotient digit.
- S — Subtract: Subtract the product from the current number.
- B — Bring down: Bring down the next digit from the dividend.
Repeat D-M-S-B until all digits are brought down.
Example: 8,456 / 23
| Step | Action | Working |
|---|---|---|
| 1 | 23 does not go into 8. Take 84. | 84 / 23 = 3 (since 23 x 3 = 69) |
| 2 | Subtract and bring down 5 | 84 - 69 = 15, bring down 5 → 155 |
| 3 | Divide 155 by 23 | 155 / 23 = 6 (since 23 x 6 = 138) |
| 4 | Subtract and bring down 6 | 155 - 138 = 17, bring down 6 → 176 |
| 5 | Divide 176 by 23 | 176 / 23 = 7 (since 23 x 7 = 161) |
| 6 | Subtract | 176 - 161 = 15 (remainder) |
Result: 8,456 / 23 = 367 remainder 15
Verification: 23 x 367 + 15 = 8,441 + 15 = 8,456
Solved Examples
Example 1: Example 1: 4-Digit by 1-Digit
Problem: Divide 9,576 by 8.
Solution:
Step 1: 9 / 8 = 1, remainder 1. Write 1 in quotient.
Step 2: Bring down 5 → 15. 15 / 8 = 1, remainder 7. Write 1.
Step 3: Bring down 7 → 77. 77 / 8 = 9, remainder 5. Write 9.
Step 4: Bring down 6 → 56. 56 / 8 = 7, remainder 0. Write 7.
Answer: 9,576 / 8 = 1,197
Verification: 8 x 1,197 = 9,576
Example 2: Example 2: 4-Digit by 2-Digit
Problem: Divide 7,854 by 26.
Solution:
Step 1: 26 does not go into 7. Take 78. 78 / 26 = 3 (26 x 3 = 78). Subtract: 0.
Step 2: Bring down 5 → 5. 5 / 26 = 0. Write 0.
Step 3: Bring down 4 → 54. 54 / 26 = 2 (26 x 2 = 52). Subtract: 2.
Answer: 7,854 / 26 = 302 remainder 2
Verification: 26 x 302 + 2 = 7,852 + 2 = 7,854
Example 3: Example 3: 5-Digit by 2-Digit
Problem: Divide 45,360 by 45.
Solution:
Step 1: 45 / 45 = 1, remainder 0.
Step 2: Bring down 3 → 3. 3 / 45 = 0.
Step 3: Bring down 6 → 36. 36 / 45 = 0.
Step 4: Bring down 0 → 360. 360 / 45 = 8 (45 x 8 = 360). Remainder 0.
Answer: 45,360 / 45 = 1,008
Example 4: Example 4: 5-Digit by 3-Digit
Problem: Divide 67,500 by 125.
Solution:
Step 1: 125 does not go into 67. Take 675. 675 / 125 = 5 (125 x 5 = 625). Subtract: 50.
Step 2: Bring down 0 → 500. 500 / 125 = 4 (125 x 4 = 500). Subtract: 0.
Step 3: Bring down 0 → 0. 0 / 125 = 0.
Answer: 67,500 / 125 = 540
Example 5: Example 5: Word Problem — Equal Distribution
Problem: A school collected ₹84,500 as donations. The amount is to be distributed equally among 25 orphanages. How much does each orphanage get?
Solution:
Step 1: Amount per orphanage = 84,500 / 25
Step 2: 84 / 25 = 3 (25 x 3 = 75). Subtract: 9.
Step 3: Bring down 5 → 95. 95 / 25 = 3 (25 x 3 = 75). Subtract: 20.
Step 4: Bring down 0 → 200. 200 / 25 = 8 (25 x 8 = 200). Subtract: 0.
Step 5: Bring down 0 → 0. 0 / 25 = 0.
Answer: Each orphanage gets ₹3,380.
Example 6: Example 6: Word Problem — Packing
Problem: A factory produced 15,624 biscuits. They are packed in boxes of 36 each. How many full boxes can be made? How many biscuits are left over?
Solution:
Step 1: 15,624 / 36
Step 2: 156 / 36 = 4 (36 x 4 = 144). Subtract: 12.
Step 3: Bring down 2 → 122. 122 / 36 = 3 (36 x 3 = 108). Subtract: 14.
Step 4: Bring down 4 → 144. 144 / 36 = 4 (36 x 4 = 144). Subtract: 0.
Answer: 434 full boxes with 0 biscuits left over.
Example 7: Example 7: Word Problem — Travel
Problem: Dev drove 2,340 km in 18 days. If he drove the same distance each day, how many km did he drive per day?
Solution:
Step 1: Distance per day = 2,340 / 18
Step 2: 23 / 18 = 1 (18 x 1 = 18). Subtract: 5.
Step 3: Bring down 4 → 54. 54 / 18 = 3 (18 x 3 = 54). Subtract: 0.
Step 4: Bring down 0 → 0. 0 / 18 = 0.
Answer: Dev drove 130 km per day.
Example 8: Example 8: Division with Zero in Quotient
Problem: Divide 52,164 by 52.
Solution:
Step 1: 52 / 52 = 1. Subtract: 0.
Step 2: Bring down 1 → 1. 1 / 52 = 0. Write 0 in quotient.
Step 3: Bring down 6 → 16. 16 / 52 = 0. Write 0.
Step 4: Bring down 4 → 164. 164 / 52 = 3 (52 x 3 = 156). Subtract: 8.
Answer: 52,164 / 52 = 1,003 remainder 8
Example 9: Example 9: Word Problem — Sharing Equally
Problem: Neha's class collected 4,725 stickers. They want to share them equally among 45 students. How many stickers does each student get?
Solution:
Step 1: 4,725 / 45
Step 2: 47 / 45 = 1 (45 x 1 = 45). Subtract: 2.
Step 3: Bring down 2 → 22. 22 / 45 = 0.
Step 4: Bring down 5 → 225. 225 / 45 = 5 (45 x 5 = 225). Subtract: 0.
Answer: Each student gets 105 stickers.
Example 10: Example 10: Estimation Before Division
Problem: Estimate 8,743 / 42, then find the exact answer.
Solution:
Estimate: 8,743 ≈ 8,800 and 42 ≈ 40. So 8,800 / 40 = 220.
Exact:
Step 1: 87 / 42 = 2 (42 x 2 = 84). Subtract: 3.
Step 2: Bring down 4 → 34. 34 / 42 = 0.
Step 3: Bring down 3 → 343. 343 / 42 = 8 (42 x 8 = 336). Subtract: 7.
Answer: 8,743 / 42 = 208 remainder 7 (close to estimate of 220)
Real-World Applications
Real-life uses of division of large numbers:
- Equal sharing: Distributing sweets, money, or books equally among students or groups. If 2,400 sweets are shared among 48 students, each gets 2,400 / 48 = 50 sweets.
- Packing: Finding how many boxes are needed. If a factory has 15,000 items and each box holds 36 items, they need 15,000 / 36 = 416 boxes (with 24 items left over).
- Average: Total marks / number of subjects = average. If a student scores 425 across 5 subjects, the average is 425 / 5 = 85.
- Speed and distance: Speed = distance / time. A train covers 1,200 km in 15 hours at 1,200 / 15 = 80 km/h.
- Budgeting: Annual income / 12 = monthly income. ₹6,00,000 per year means ₹50,000 per month.
- Cost per unit: Total cost / number of items = price per item. ₹7,500 for 25 books means each book costs ₹300.
When the remainder matters:
| Situation | How to Handle Remainder |
|---|---|
| Packing items into boxes | Round UP — you need one extra box for leftover items |
| Buying whole items (books, chairs) | Round DOWN — you cannot buy a partial item |
| Sharing money equally | State the remainder — it is the amount left over |
| Finding averages | Express as a decimal if needed |
Key Points to Remember
- Follow the DMSB cycle: Divide → Multiply → Subtract → Bring down.
- The remainder must always be less than the divisor.
- If the divisor does not go into the current group of digits, write 0 in the quotient and bring down the next digit.
- Verify using: Dividend = Divisor x Quotient + Remainder.
- Division is not commutative: 20 / 4 is not the same as 4 / 20.
- Division by 0 is undefined — it has no answer.
- Estimate first to check if your answer is reasonable.
- When dividing by multiples of 10, simplify by cancelling trailing zeros in both dividend and divisor.
Practice Problems
- Divide 8,436 by 12.
- Find the quotient and remainder: 56,789 / 34
- Divide 72,000 by 125.
- Arjun has ₹24,500. He wants to buy books costing ₹175 each. How many books can he buy?
- A train travelled 4,860 km in 36 hours. What was its speed in km per hour?
- 48,620 oranges are packed in crates of 65 each. How many full crates are made?
- Divide 1,00,000 by 250.
- A school has 2,160 students sitting in rows of 48. How many rows are needed?
Frequently Asked Questions
Q1. What is long division?
Long division is a step-by-step method to divide large numbers. It involves four repeated steps: Divide, Multiply, Subtract, and Bring down the next digit. This cycle continues until all digits of the dividend have been used.
Q2. What do dividend, divisor, quotient, and remainder mean?
The dividend is the number being divided. The divisor is the number you divide by. The quotient is the result (answer). The remainder is what is left over after division. For example, in 47 / 5 = 9 R 2, the dividend is 47, divisor is 5, quotient is 9, and remainder is 2.
Q3. How do you check if a division answer is correct?
Use the formula: Dividend = (Divisor x Quotient) + Remainder. Multiply the divisor by the quotient, then add the remainder. If the result equals the dividend, your answer is correct.
Q4. What happens when the divisor does not go into the current digits?
Write 0 in the quotient for that position and bring down the next digit. For example, in 3,024 / 15, after dividing 30 by 15 to get 2, the next digit gives 02. Since 15 does not go into 2, write 0 and bring down 4 to get 24.
Q5. Can the remainder be larger than the divisor?
No, the remainder must always be less than the divisor. If the remainder is equal to or greater than the divisor, the quotient digit is too small and you must increase it.
Q6. Why can we not divide by zero?
Division by zero is undefined because no number multiplied by 0 gives a non-zero result. For example, there is no number that satisfies 5 / 0 = ? because ? x 0 can never equal 5.
Q7. How is division related to multiplication?
Division is the inverse (reverse) of multiplication. If 15 x 8 = 120, then 120 / 15 = 8 and 120 / 8 = 15. Knowing multiplication tables makes division faster.
Q8. What is the shortcut for dividing by 10, 100, or 1000?
Remove the corresponding number of trailing zeros. 5,600 / 10 = 560 (remove 1 zero). 5,600 / 100 = 56 (remove 2 zeros). If the dividend does not end in zeros, the result will have a decimal or remainder.
Q9. How do you divide a 5-digit number by a 3-digit number?
Use the same long division method. Start by checking if the divisor goes into the first 3 digits of the dividend. If not, take 4 digits. Divide, multiply, subtract, and bring down. Continue until done.
Q10. When should you use estimation in division?
Estimate before the exact calculation to get a rough idea of the answer. Round both numbers to simpler values. For 8,743 / 42, round to 8,800 / 40 = 220. After finding the exact answer (208 R 7), compare with the estimate to check reasonableness.
Related Topics
- Multiplication of Large Numbers
- Long Division
- Addition of Large Numbers
- Subtraction of Large Numbers
- Order of Operations (BODMAS)
- Word Problems on Four Operations
- Mental Math (Grade 5)
- Multiplication of 4-Digit Numbers
- Division of 4-Digit by 2-Digit Numbers
- Simplification Using BODMAS
- Properties of Operations (Grade 5)
- Unitary Method (Grade 5)
