Multiplication and Division

Overview

Multiplication and Division form the backbone of arithmetic computation at the primary level and are essential topics for PSTET Paper I. These operations build directly on addition and subtraction skills and serve as the foundation for fractions, ratios, and all higher mathematics. For the exam, you must demonstrate both computational fluency and the ability to apply these operations to real-life word problems.

PSTET questions typically test your understanding of multiplication tables (up to 10 or 12), properties of multiplication and division, long division procedures, and the ability to solve contextual problems involving equal groups, sharing, and repeated addition or subtraction. Expect questions that assess not just calculation but also conceptual understanding—why these operations work the way they do and how to teach them effectively to Classes I–V students.

Mastery here means quick recall of tables, error-free execution of multi-digit multiplication and long division, and confident translation of word problems into mathematical expressions.

Key Concepts

• **Multiplication as repeated addition**: 4 × 3 means adding 4 three times (4 + 4 + 4 = 12). This is the foundational idea for young learners.

• **Division as equal sharing or grouping**: 12 ÷ 3 can mean "sharing 12 items equally among 3 people" (each gets 4) or "how many groups of 3 in 12" (4 groups).

• **Inverse relationship**: Multiplication and division are inverse operations. If 6 × 7 = 42, then 42 ÷ 7 = 6 and 42 ÷ 6 = 7.

• **Commutative property of multiplication**: Order does not matter—5 × 8 = 8 × 5 = 40. Division is NOT commutative (12 ÷ 4 ≠ 4 ÷ 12).

• **Associative property of multiplication**: Grouping does not matter—(2 × 3) × 4 = 2 × (3 × 4) = 24.

• **Distributive property**: Multiplication distributes over addition—7 × 12 = 7 × (10 + 2) = 70 + 14 = 84. Useful for mental math.

• **Multiplicative identity**: Any number multiplied by 1 remains unchanged (n × 1 = n).

• **Zero property**: Any number multiplied by 0 equals 0; division by zero is undefined.

• **Remainder in division**: When a number does not divide evenly, the leftover is called the remainder. In 17 ÷ 5, quotient = 3, remainder = 2.

Formulas / Key Facts

| Fact | Explanation | |------|-------------| | Dividend = Divisor × Quotient + Remainder | The fundamental division equation—used to verify long division. | | n × 0 = 0 | Zero property of multiplication. | | n × 1 = n | Identity property of multiplication. | | n ÷ 1 = n | Any number divided by 1 is itself. | | n ÷ n = 1 (n ≠ 0) | Any non-zero number divided by itself equals 1. | | 0 ÷ n = 0 (n ≠ 0) | Zero divided by any non-zero number is 0. | | Division by zero is undefined | You cannot divide any number by 0. |

**Multiplication tables 1–10** must be memorised perfectly. Quick recall is tested directly and indirectly in almost every arithmetic problem.

Worked Examples

### Example 1: Multi-digit Multiplication

**Problem**: 47 × 36 = ?

**Step-by-step**: 1. Write 47 above 36, aligning place values. 2. Multiply 47 by 6 (ones digit of 36):

• 6 × 7 = 42 → write 2, carry 4
• 6 × 4 = 24, plus 4 = 28 → write 28
• First partial product = 282

3. Multiply 47 by 3 (tens digit of 36), shift one place left:

• 3 × 7 = 21 → write 1, carry 2
• 3 × 4 = 12, plus 2 = 14 → write 14
• Second partial product = 141, placed as 1410

4. Add partial products: 282 + 1410 = **1692**

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### Example 2: Long Division

**Problem**: 539 ÷ 7 = ?

**Step-by-step**: 1. 7 into 5? No. Take 53. 2. 7 × 7 = 49 (closest without exceeding 53). Write 7 above the 3. 3. 53 − 49 = 4. Bring down 9 → 49. 4. 7 × 7 = 49. Write 7 above the 9. 5. 49 − 49 = 0. 6. Quotient = **77**, Remainder = **0**.

**Verification**: 7 × 77 + 0 = 539 ✓

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### Example 3: Word Problem

**Problem**: A shopkeeper has 156 mangoes. He packs them equally into 12 boxes. How many mangoes are in each box?

**Solution**:

• This is an equal-sharing (division) problem.
• 156 ÷ 12 = ?
• 12 × 10 = 120; 156 − 120 = 36
• 12 × 3 = 36
• Quotient = 10 + 3 = **13 mangoes per box**.

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### Example 4: Word Problem with Remainder

**Problem**: 250 students are to be seated in rows of 8. How many complete rows can be formed, and how many students will be left?

**Solution**:

• 250 ÷ 8
• 8 × 31 = 248; Remainder = 250 − 248 = 2
• **31 complete rows**, **2 students left over**.

Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Assuming division is commutative (thinking 12 ÷ 4 = 4 ÷ 12). | Emphasise that dividend and divisor are not interchangeable. 12 ÷ 4 = 3, but 4 ÷ 12 is a fraction less than 1. | | Forgetting to add the carry in multiplication. | Always write the carry above the next column and add it immediately after the next product. | | Writing the remainder larger than the divisor. | The remainder must always be less than the divisor. If it is not, revise the quotient upward. | | Misplacing partial products in multi-digit multiplication. | Each successive partial product shifts one place to the left (or add a trailing zero). | | Ignoring the remainder in word problems. | Read the question carefully—sometimes you need to round up (e.g., number of buses needed) or report the remainder explicitly. |

Quick Reference

• **Multiplication = Repeated addition**; Division = Repeated subtraction or equal sharing.
• **Dividend = Divisor × Quotient + Remainder** — use this to check every division.
• Commutative and associative laws apply to multiplication, NOT to division.
• Remainder is always less than the divisor.
• Division by zero is undefined—never attempt it.
• Master tables 1–10 for speed; use distributive property for tables beyond 10 (e.g., 14 × 7 = 10 × 7 + 4 × 7).