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Division

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Chapter Notes

Overview
This chapter builds on the relationship between multiplication and division, teaching students how division is the inverse of multiplication. It covers division facts, patterns in division with place value, mental strategies for division (splitting numbers and repeated halving), the standard long division algorithm using partial quotients, and the concept of remainder. The one big idea is that every division problem can be understood as 'sharing equally' or 'grouping,' and that the dividend is always equal to divisor times quotient plus remainder (N = D × Q + R).
Key Concepts (13)
Division
Splitting a number (dividend) into a given number of equal groups (divisor) to find how many are in each group (quotient).
e.g. 35 ÷ 7 = 5 means 35 coconuts split into 7 groups has 5 coconuts in each group.
Dividend (N)
The number that is being divided or split into equal groups.
e.g. In 35 ÷ 7 = 5, 35 is the dividend.
Divisor (D)
The number by which the dividend is divided; it tells how many equal groups we are making.
e.g. In 35 ÷ 7 = 5, 7 is the divisor.
Quotient (Q)
The result of division; the number of items in each group.
e.g. In 35 ÷ 7 = 5, 5 is the quotient.
Remainder (R)
The amount left over when the dividend cannot be divided exactly by the divisor.
e.g. In 62 ÷ 5, each child gets 12 candies and 2 candies are left over; 2 is the remainder.
Division Fact
A division statement that is true, often derived from a multiplication fact.
e.g. From 5 × 7 = 35, we get two division facts: 35 ÷ 5 = 7 and 35 ÷ 7 = 5.
Multiplication Fact
A true multiplication statement that can be used to generate two division facts.
e.g. 5 × 7 = 35 is a multiplication fact.
Partial Quotients
A method of division where you subtract multiples of the divisor in steps, adding the partial quotients at the end.
e.g. 582 ÷ 6: subtract 120 (20 groups) five times, then 60 (10 groups), then 42 (7 groups); total quotient = 20+20+20+20+10+7 = 97.
Repeated Halving
A mental strategy for dividing by 4, 8, etc., by halving the number repeatedly.
e.g. 128 ÷ 4: half of 128 is 64, half of 64 is 32, so 128 ÷ 4 = 32.
Splitting the Number
A mental strategy where the dividend is split into convenient parts that are easy to divide by the divisor.
e.g. 1248 ÷ 6: split 1248 into 1200 + 48; 1200 ÷ 6 = 200, 48 ÷ 6 = 8, so total = 208.
Place Value Division
Dividing by considering the value of each digit (Hundreds, Tens, Ones) and regrouping when needed.
e.g. 324 ÷ 3: 3 Hundreds ÷ 3 = 1 Hundred, 2 Tens cannot be divided by 3, so regroup into 20 Ones + 4 Ones = 24 Ones ÷ 3 = 8 Ones; quotient = 108.
Regrouping
Converting a higher place value into a lower one to continue division when the digit is smaller than the divisor.
e.g. In 136 ÷ 6, 1 Hundred is regrouped into 10 Tens to make 13 Tens.
N = D × Q + R
The relationship between dividend, divisor, quotient, and remainder: Dividend = Divisor × Quotient + Remainder.
e.g. For 726 ÷ 4, quotient = 181, remainder = 2, so 726 = 4 × 181 + 2.
Formulas & Equations (2)
Dividend (N) = Divisor (D) × Quotient (Q)
Reads as: Dividend equals divisor times quotient
Variables: N = Dividend (the number being divided), D = Divisor (the number dividing), Q = Quotient (the result)
Use when: When checking if a division is exact (no remainder) or when finding a missing dividend.
⚠ Students forget that this formula only works when remainder is zero; they apply it even when there is a remainder.
N = D × Q + R
Reads as: Dividend equals divisor times quotient plus remainder
Variables: N = Dividend, D = Divisor, Q = Quotient, R = Remainder (must be less than divisor)
Use when: Always, for any division problem, especially to check the answer when there is a remainder.
⚠ Students write N = D × Q and forget to add the remainder, or they add the remainder but forget that R must be less than D.
How It Works
Writing Division Facts from a Multiplication Fact
  1. Step 1: Identify the multiplication fact (e.g., 5 × 7 = 35).
  2. Step 2: The product (35) becomes the dividend in both division facts.
  3. Step 3: The two factors (5 and 7) become the divisor and quotient in the two division facts.
  4. Step 4: Write 35 ÷ 5 = 7 and 35 ÷ 7 = 5.
💡 The product is always the dividend; the two factors swap roles as divisor and quotient.
Mental Strategy: Splitting the Number
  1. Step 1: Break the dividend into two or more parts that are easy to divide by the divisor (e.g., 1248 = 1200 + 48).
  2. Step 2: Divide each part separately (1200 ÷ 6 = 200, 48 ÷ 6 = 8).
  3. Step 3: Add the partial quotients (200 + 8 = 208).
💡 The parts must add up to the original dividend; choose parts that are multiples of the divisor.
Mental Strategy: Repeated Halving (for divisors 4, 8, 16, etc.)
  1. Step 1: If dividing by 4, halve the dividend once (e.g., 128 ÷ 4: half of 128 is 64).
  2. Step 2: Halve the result again (half of 64 is 32).
  3. Step 3: The final result is the quotient (32).
💡 This works only when the divisor is a power of 2 (2, 4, 8, 16...). For 8, halve three times.
Long Division Using Partial Quotients (Susie's Method)
  1. Step 1: Estimate how many groups of the divisor you can take away at once (e.g., 582 ÷ 6: take 20 groups = 120).
  2. Step 2: Subtract that amount from the dividend and write the partial quotient (20).
  3. Step 3: Repeat with the remainder until the remainder is less than the divisor.
  4. Step 4: Add all partial quotients to get the final quotient.
💡 You can take away larger multiples (like 100 groups) to reduce steps; the sum of partial quotients must equal the final quotient.
Long Division Using Place Value (Standard Algorithm)
  1. Step 1: Write the dividend in a place value chart (Th H T O).
  2. Step 2: Start from the highest place value. Divide the digit by the divisor. If it is smaller, regroup to the next lower place.
  3. Step 3: Write the quotient digit above the corresponding place.
  4. Step 4: Multiply the quotient digit by the divisor, subtract from the current number, and bring down the next digit.
  5. Step 5: Repeat until all digits are used. The final number left is the remainder.
💡 If a digit cannot be divided (e.g., 2 Tens ÷ 3), write 0 in the quotient for that place and regroup the digit into the next lower place.
Checking Division with Remainder
  1. Step 1: Multiply the divisor by the quotient.
  2. Step 2: Add the remainder to the product.
  3. Step 3: The result must equal the dividend. If it does, the division is correct.
💡 The remainder must always be less than the divisor. If remainder ≥ divisor, the quotient is too small.
Exam Traps — Don't Make These Mistakes
⚠ Forgetting the Remainder in the Check Formula
✗ N = D × Q (e.g., for 726 ÷ 4, they write 726 = 4 × 181)
→ This is only true when remainder is 0. Here, 4 × 181 = 724, not 726. The remainder 2 is missing.
✓ N = D × Q + R, so 726 = 4 × 181 + 2.
⚠ Writing the Wrong Division Fact from Multiplication
✗ From 5 × 7 = 35, they write 35 ÷ 35 = 1 or 35 ÷ 1 = 35
→ The division facts must use the two factors (5 and 7) as divisor and quotient, not the product itself.
✓ 35 ÷ 5 = 7 and 35 ÷ 7 = 5.
⚠ Placing Zero Incorrectly in the Quotient (Place Value Division)
✗ For 324 ÷ 3, they write quotient as 18 instead of 108
→ They forget that 2 Tens ÷ 3 gives 0 Tens, so a zero must be placed in the Tens place of the quotient.
✓ Quotient is 108: 1 Hundred, 0 Tens, 8 Ones.
⚠ Ignoring the Remainder in Word Problems
✗ For 535 ÷ 25, they write quotient = 21, so they say 21 bags are needed
→ The remainder 10 means 10 coconuts are left, so an extra bag is needed. The answer is 22 bags, not 21.
✓ 21 full bags + 1 more bag for the remaining 10 coconuts = 22 bags.
⚠ Confusing Dividend and Divisor in Word Problems
✗ For '582 coconuts sold equally to 6 customers,' they write 6 ÷ 582
→ The dividend is the total (582), and the divisor is the number of groups (6). They reverse the order.
✓ 582 ÷ 6 = 97 coconuts per customer.
Compare & Contrast
Meaning of the divisor
Division as Sharing (Partitive Division)
Division as Grouping (Quotative Division)
In sharing, the divisor tells the number of groups (e.g., 35 split into 7 groups). In grouping, the divisor tells the size of each group (e.g., 35 split into groups of 5). Both give the same quotient but the interpretation differs.
Size of each subtraction step
Susie's Method (Partial Quotients)
Sunitha's Method (More Efficient Partial Quotients)
Susie subtracts small multiples (like 20 groups at a time), requiring many steps. Sunitha subtracts larger multiples (like 90 or 500 groups at a time), reducing the number of steps. Both are correct, but Sunitha's method is faster and closer to the standard algorithm.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Fill in the blanks (1 mark), 1 Short Answer (2 marks) on division facts or patterns, 1 Long Answer (4-5 marks) on word problem involving multi-step division or remainder.
Most asked: Division facts from multiplication facts · Long division with remainder (N = D × Q + R) · Word problems involving division (sharing equally, grouping, remainder handling) · Place value division (especially when quotient has a zero)
Don't skip: Word problems where remainder requires an extra unit (e.g., number of buses, number of bags); Checking division using N = D × Q + R; Finding missing dividend, divisor, or quotient in a division statement
One-Line Revision (read aloud before the exam)
  1. Division is the inverse of multiplication: every multiplication fact gives two division facts.
  2. Dividend (N) ÷ Divisor (D) = Quotient (Q); if not exact, remainder (R) is left.
  3. The relationship is N = D × Q + R, and R must always be less than D.
  4. To divide using place value, start from the highest place and regroup when a digit is smaller than the divisor.
  5. If a digit cannot be divided, write 0 in the quotient for that place and regroup.
  6. Mental strategy: split the dividend into convenient parts, divide each, and add the results.
  7. Repeated halving works for dividing by 4, 8, 16, etc.: halve the number as many times as the power of 2.
  8. In partial quotients, subtract larger multiples of the divisor to reduce steps.
  9. In word problems, if there is a remainder, decide whether to round up (extra bag/bus) or ignore it (money left over).
  10. From 5 × 7 = 35, the two division facts are 35 ÷ 5 = 7 and 35 ÷ 7 = 5.
  11. To check division, multiply divisor by quotient and add remainder; the result must equal the dividend.
  12. When dividing by 10, 100, etc., the quotient is the dividend with the same number of zeros removed.
  13. A remainder of 0 means the divisor divides the dividend exactly.
  14. In place value division, always write the quotient digits above the correct place (Th, H, T, O).

MCQs — 10 Questions with Answers

Q1. In the division problem 35 ÷ 7 = 5, which number is the Divisor?

  • A. 35
  • B. 7 ✓
  • C. 5
  • D. 0

Answer: B — The divisor is the number we divide by (the number of groups), which is 7.

Q2. If 6 × 8 = 48, which of these is NOT a correct division fact?

  • A. 48 ÷ 6 = 8
  • B. 48 ÷ 8 = 6
  • C. 48 ÷ 6 = 6 ✓
  • D. 48 ÷ 4 = 12

Answer: C — 48 ÷ 6 = 8, not 6, because 6 × 8 = 48 gives us two correct division facts: 48 ÷ 6 = 8 and 48 ÷ 8 = 6.

Q3. What is 1200 ÷ 6?

  • A. 200 ✓
  • B. 180
  • C. 120
  • D. 2000

Answer: A — Since 6 × 200 = 1200, we know that 1200 ÷ 6 = 200.

Q4. Sabina cycles 160 km in 20 days at the same distance each day. How many kilometres does she cycle each day?

  • A. 6 km
  • B. 8 km ✓
  • C. 10 km
  • D. 12 km

Answer: B — 160 ÷ 20 = 8 km per day, because she cycles the same distance each day.

Q5. What pattern do you see when dividing by 10? 100 ÷ 10 = ____, 1000 ÷ 10 = ____, 1200 ÷ 10 = ____

  • A. 10, 100, 120 (remove one zero) ✓
  • B. 10, 100, 120 (add one zero)
  • C. 100, 1000, 12000 (multiply by 10)
  • D. 1, 10, 12 (remove all zeros)

Answer: A — When dividing by 10, one zero is removed from the end of the number: 100 ÷ 10 = 10, 1000 ÷ 10 = 100, 1200 ÷ 10 = 120.

Q6. To divide 128 by 4 using the halving strategy, what is the first step?

  • A. Subtract 4 from 128
  • B. Half of 128 = 64 ✓
  • C. Multiply 128 by 2
  • D. Add 128 + 4

Answer: B — To divide by 4, you halve twice: half of 128 is 64, then half of 64 is 32, so 128 ÷ 4 = 32.

Q7. How many ₹100 notes does Seema need to carry if she wants to buy coconuts worth ₹4200?

  • A. 42 notes ✓
  • B. 40 notes
  • C. 420 notes
  • D. 4 notes

Answer: A — 4200 ÷ 100 = 42 notes, because 100 × 42 = 4200.

Q8. In the division 1248 ÷ 6 = 208, what is the Quotient?

  • A. 1248
  • B. 6
  • C. 208 ✓
  • D. 1254

Answer: C — The quotient is the answer to the division problem, which is 208.

Q9. Fill in the blank: 630 ÷ 63 = ____

  • A. 7
  • B. 10 ✓
  • C. 63
  • D. 630

Answer: B — 630 ÷ 63 = 10, following the pattern where you can cancel common factors: 630 ÷ 63 = (63 × 10) ÷ 63 = 10.

Q10. When 97 coconuts are packed into bags that hold 25 coconuts each, how many complete bags can be filled and what is left over?

  • A. 3 bags with 22 coconuts left ✓
  • B. 4 bags with 0 left
  • C. 2 bags with 47 left
  • D. 3 bags with 25 left

Answer: A — 97 ÷ 25 = 3 remainder 22, because 25 × 3 = 75 and 97 - 75 = 22 leftover.

Flashcards

What is the Dividend in a division problem?

The dividend is the total number being divided or split into groups.

What is the Divisor in a division problem?

The divisor is the number that tells us how many groups we are making or dividing by.

What is the Quotient in a division problem?

The quotient is the answer we get after dividing—how many are in each group.

If 5 × 7 = 35, write two division facts from this.

35 ÷ 5 = 7 and 35 ÷ 7 = 5 are the two division facts from 5 × 7 = 35.

What is the formula that shows dividend, divisor, quotient, and remainder?

N = D × Q + R, where N is dividend, D is divisor, Q is quotient, and R is remainder.

How do you divide 1248 by 6 using splitting?

Split 1248 into 1200 + 48, then divide: 1200 ÷ 6 = 200 and 48 ÷ 6 = 8, so 1248 ÷ 6 = 208.

What strategy can you use to divide 128 by 4?

Halve twice: half of 128 is 64, half of 64 is 32, so 128 ÷ 4 = 32.

If 900 ÷ 10 = 90, what pattern do you see?

When dividing by 10, the quotient is the same number but with one zero removed from the end.

Sabina cycles 160 km in 20 days equally. How far does she cycle each day?

160 ÷ 20 = 8 km per day.

If 582 ÷ 6 is done using repeated subtraction, what fact helps you?

You can subtract groups of 60 (or 120) repeatedly from 582 to find how many groups of 6 fit in.

Important Board Questions

Write the two division facts from the multiplication fact 9 × 8 = 72. [1 mark]

If 9 × 8 = 72, then 72 ÷ 9 = 8 AND 72 ÷ 8 = 9. Every multiplication fact creates exactly two division facts.

Solve 1248 ÷ 6 by splitting the number into convenient parts. Show your working. [2 marks]

Split 1248 into 1200 and 48. Divide each part by 6 separately, then add the quotients. 1200 ÷ 6 = 200, and 48 ÷ 6 = 8, so answer = 200 + 8 = 208.

The owner of a store wants to distribute ₹5500 equally among 5 employees as a gift. (a) How much will each employee get? (b) If he distributes the same amount among 10 employees instead, will each get more or less? Explain. [3 marks]

Part (a): Divide 5500 ÷ 5 = 1100. Part (b): When you divide the same amount among more people, each person gets less. Use: 5500 ÷ 10 = 550, which is less than 1100.

Susie harvested 1117 coconuts in April. She sold 582 to 6 customers equally. (a) How many coconuts did each customer get? (b) How many coconuts were left? (c) She needs to pack the remaining coconuts in bags holding 25 each. How many complete bags can she fill and how many are left over? Show all working using the division formula N = D × Q + R. [5 marks]

Part (a): 582 ÷ 6 = 97. Part (b): 1117 - 582 = 535 left. Part (c): 535 ÷ 25 gives quotient 21 and remainder 10. Use formula: 535 = 25 × 21 + 10. She needs 21 complete bags and has 10 coconuts left.

True or False? When dividing 1000 by 100, the quotient is 10. Give a reason for your answer. [2 marks]

True. Because 100 × 10 = 1000, following the place value pattern that dividing by 100 removes two zeros from the number.

Assertion-Reason Questions for Division

Q1. Assertion (A):

Every multiplication fact can be written as two division facts.

Reason (R):

If A × B = C, then C ÷ A = B and C ÷ B = A.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the assertion is true. Multiplication and division are inverse operations.

Q2. Assertion (A):

When we divide 1200 by 6, we get 200.

Reason (R):

Because 1200 is twelve hundreds, and twelve divided by 6 is 2, so we get 2 hundreds which is 200.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true, and the reason provides the correct explanation using place value understanding.

Q3. Assertion (A):

The remainder must always be smaller than the divisor.

Reason (R):

This is shown in the division formula N = D × Q + R where R must be less than D.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the rule through the division formula.

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Case-Based Questions (CBQ) for Division

Case Study:

Ravi's mother is preparing food for a school event. She has 240 coconut sweets to pack equally into 8 boxes. After packing, she has 15 coconuts left to make oil. How many sweets will each box contain?

Q1. How many sweets should be packed in each box?

  • A. 30 ✓
  • B. 20
  • C. 25
  • D. 35

Answer: A

Q2. How many total coconut sweets did Ravi's mother start with before packing?

  • A. 225
  • B. 240
  • C. 255 ✓
  • D. 265

Answer: C

Q3. If she wanted to pack the remaining 15 coconuts into packets of 3 each, how many packets would she need?

  • A. 5 ✓
  • B. 6
  • C. 7
  • D. 8

Answer: A

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