NCERT Solutions for Class 8th Maths Chapter 6 We Distribute, Yet Things Multiply
Updated on 2026-09-19
About this chapter
Distributivity, a (b + c) = ab + ac , is the one rule the whole chapter rests on. Read as an area picture, an a × ( b + c ) array splits into an a × b array and an a × c array. Applying it twice gives Identity 1: (a + m)(b + n) = ab + mb + an + mn — every term of the first bracket times every term of the second. Because the rules of integer multiplication handle the signs, the same identity also covers decreases: just take m or n negative. Three special cases are named in the chapter — 1A: (a + b)² = a² + 2ab + b², 1B: (a – b)² = a² + b² – 2ab, 1C: (a + b)(a – b) = a² – b². Each is proved twice: by expanding, and by cutting up a square. The same identities give fast mental arithmetic: 65² from 60² and 5², 397 × 403 from 400² – 3², and Sridharacharya's trick a² = (a + b)(a – b) + b². Sectio
- .1 Some Properties of Multiplication
- Fast Multiplications Using the Distributive Property
- .2 Special Cases of the Distributive Property
- Investigating Patterns
- .3 Mind the Mistake, Mend the Mistake
- .4 This Way or That Way, All Ways Lead to the Bay
- Coin Conjoin
Quick revision
| Name | Identity | Worked example | Where the chapter uses it |
|---|---|---|---|
| Distributivity | a (b + c) = ab + ac | 23 (27 + 1) = 23 × 27 + 23 | Increase in a product when one factor grows |
| Right form | (a + b) c = ac + bc | (20 + 3) × 7 = 140 + 21 = 161 | Brahmagupta's khaṇḍa-guṇanam |
| Identity 1 | (a + m)(b + n) = ab + mb + an + mn | (a + 1)(b + 1) = ab + a + b + 1 | Any increase or decrease in both factors |
| Identity 1 with n negative | (a + u)(b – v) = ab + ub – av – uv | (a + 1)(b – 1) = ab + b – a – 1 | One factor up, the other down |
| Identity 1A | (a + b)² = a² + 2ab + b² | 65² = 3600 + 600 + 25 = 4225 | Square of a sum; 104², 46² |
| Identity 1B | (a – b)² = a² + b² – 2ab | 55² = 3600 + 25 – 600 = 3025 | Square of a difference; 99², 91² |
| Identity 1C | (a + b)(a – b) = a² – b² | 397 × 403 = 400² – 3² = 159991 | Pattern 2; 45 × 55; 98 × 102 |
| Sum of two squares | 2(a² + b²) = (a + b)² + (a – b)² | 2(5² + 6²) = 11² + 1² | Pattern 1 |
| Sridharacharya | a² = (a + b)(a – b) + b² | 31² = 32 × 30 + 1 = 961 | Fast squaring |
| Multiply by 11 | N × 11 = N × 10 + N | 495 × 11 = 4|13|14|5 = 5445 | Add neighbouring digits |
| Multiply by 101 | N × 101 = N × 100 + N | 89 × 101 = 8989 | Add digits two places apart |
| Multiply by 99, 999 | N × 99 = N × 100 – N | 9734 × 99 = 973400 – 9734 = 963666 | Same rule, subtraction form |
Exercises
- .1 Some Properties of Multiplication — In-text Questions Page 1366
- .1 Some Properties of Multiplication — In-text Questions Page 1386
- .1 Some Properties of Multiplication — In-text Questions Page 1396
- .1 Some Properties of Multiplication — In-text Questions Page 1406
- .1 Some Properties of Multiplication — In-text Questions Page 1416
- .1 Some Properties of Multiplication — Figure it Out Page 1426
- Fast Multiplications Using the Distributive Property — In-text Questions Page 143
- Fast Multiplications Using the Distributive Property — In-text Questions Page 144
- .2 Special Cases of the Distributive Property — In-text Questions Page 1456
- .2 Special Cases of the Distributive Property — In-text Questions Page 1466
- .2 Special Cases of the Distributive Property — In-text Questions Page 1476
- Investigating Patterns — In-text Questions Page 148
- Investigating Patterns — In-text Questions Page 149
- Investigating Patterns — Figure it Out Page 149
- .3 Mind the Mistake, Mend the Mistake — Mind the Mistake, Mend the Mistake Page 1506
- .4 This Way or That Way, All Ways Lead to the Bay — In-text Questions Page 1506
- .4 This Way or That Way, All Ways Lead to the Bay — In-text Questions Page 1526
- .4 This Way or That Way, All Ways Lead to the Bay — In-text Questions Page 1536
- .4 This Way or That Way, All Ways Lead to the Bay — In-text Questions Page 1546
- .4 This Way or That Way, All Ways Lead to the Bay — Figure it Out Page 1546
- Coin Conjoin — Puzzle Time: Coin Conjoin Page 158