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CBSE · Class 8 · Mathematics Part 1

Part 1, Chapter 6: We Distribute, Yet Things Multiply

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Distributive property and increase in products

From daily life

A shop packs 18 boxes with 25 pens in each. If every box gets 1 more pen, the total rises by exactly 18 pens, one extra for each box.

Simple definition

Distributive property: a number multiplied by a sum can be multiplied with each part: a(b + c) = ab + ac. A statement like this, true for all numbers (even negative integers), is an identity.

abacbca

In the figure: The big rectangle a(b + c) is the sum of the two parts ab and ac.

Rule 1: Distribute — Split one number into easy parts (like 100 + 2 or 100 − 3) and multiply each part.

  • Easy: 6 × (10 + 4) = 60 + 24 = 84.
  • Medium: 47 × 102 = 4700 + 94 = 4794.
  • Exam-type: 97 × 15 = (100 − 3) × 15 = 1500 − 45 = 1455.

Rule 2: One number up by 1 — The product rises by the OTHER number: a(b + 1) = ab + a.

  • Easy: 7 × 30 = 210 and 7 × 31 = 217: rise 7.
  • Medium: 52 × 19 = 988 and 53 × 19 = 1007: rise 19.
  • Exam-type: 3200 = 50 × 64. If 50 becomes 51, the product is 3264: rise 64.

Rule 3: Both numbers up by 1 — The rise is a + b + 1: the two other numbers and one extra 1.

  • Easy: 5 × 6 = 30 and 6 × 7 = 42: rise 12 = 5 + 6 + 1.
  • Medium: 14 × 21 = 294 and 15 × 22 = 330: rise 36 = 14 + 21 + 1.
  • Exam-type: a = −3 and b = 8: (−3)(8) = −24 and (−2)(9) = −18: rise 6 = −3 + 8 + 1.

Rule 4: Up by m and by n — (a + m)(b + n) = ab + an + mb + mn, so the rise is an + mb + mn.

  • Easy: 10 × 10 → 11 × 12: 132 − 100 = 32 = 10 × 2 + 1 × 10 + 1 × 2.
  • Medium: 20 × 30 → 23 × 34: 782 − 600 = 182 = 80 + 90 + 12.
  • Exam-type: 8 × 5 → 12 × 8: 96 − 40 = 56 = 24 + 20 + 12.
Remember

An identity stays true for every integer, including negatives, because integers obey the distributive property too.

Common mistake

Saying that the rise is only a + b when both numbers go up by 1. The extra 1 (1 × 1) is also there.

Shortcut

Each number brings the OTHER along; together they add a bonus 1. Memory clue: "Each brings the other, both bring a bonus".

Spot the difference

PointOne number up by 1Both up by 1
Risethe other numbera + b + 1
Example 6 × 77 × 7 − 6 × 7 = 77 × 8 − 6 × 7 = 14
Extra 1 present?noyes

Practice questions

Easy
  1. 1Find 8 × (20 + 3).
  2. 2The product 9 × 50 changes to 9 × 51. Find the rise.
  3. 3Both numbers of 14 × 16 go up by 1. Find the rise.
Medium
  1. 4Use 60 + 1 and 100 + 1 to find 61 × 101.
  2. 512 × 18 becomes 14 × 19. Find the rise.
  3. 6Find 97 × 15 using 100 − 3.
Challenging
  1. 7For a = −4 and b = 7, both numbers go up by 1. Find the rise in the product.
  2. 8The product 10 × b rises by 25 when both numbers go up by 1. Find b.
Show answers (try them yourself first)
  1. 1184
  2. 29
  3. 331
  4. 46161
  5. 550
  6. 61455
  7. 74
  8. 814

Multiplying two brackets

From daily life

A garden of 12 m by 8 m grows to 15 m by 10 m. The new area 15 × 10 = 150 is made of four parts: 96 + 24 + 24 + 6, the old plot, two strips and a small corner.

Simple definition

To multiply two brackets, multiply EVERY term of the first by EVERY term of the second, then add the like terms: (a + m)(b + n) = ab + an + mb + mn.

963616612382

In the figure: 15 × 10 = (12 + 3)(8 + 2): the four parts are 12 × 8, 3 × 8, 12 × 2 and 3 × 2.

Rule 1: (x + a)(x + b) — The middle term is the sum of the two numbers, the last term is their product.

  • Easy: (x + 2)(x + 3) = x² + 5x + 6.
  • Medium: (x − 4)(x + 9) = x² + 5x − 36.
  • Exam-type: (x − 5)(x − 6) = x² − 11x + 30.

Rule 2: Brackets with numbers in front — Multiply the front numbers as well: all four products are needed.

  • Easy: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3.
  • Medium: (3m − 2)(m + 5) = 3m² + 15m − 2m − 10 = 3m² + 13m − 10.
  • Exam-type: (4p − 3q)(p + 2q) = 4p² + 8pq − 3pq − 6q² = 4p² + 5pq − 6q².

Rule 3: Two different letters — Keep each product with its letters; like terms may not exist.

  • Easy: (a + 2)(b + 3) = ab + 3a + 2b + 6.
  • Medium: (a − 1)(b + 4) = ab + 4a − b − 4.
  • Exam-type: A rectangle (x + 5) by (y + 2) has area xy + 2x + 5y + 10.
Remember

Two terms times two terms always gives 4 products before you combine the like terms.

Common mistake

Multiplying only the first terms and the last terms: (x + 3)(x + 5) = x² + 15 is wrong. The correct answer is x² + 8x + 15.

Shortcut

Count 4 products every time. Memory clue: "Everybody shakes hands": every term meets every term of the other bracket.

Spot the difference

PointProduct (x + 3)(x + 5)Sum (x + 3) + (x + 5)
Resultx² + 8x + 152x + 8
Highest power of x21
Number of terms32

Practice questions

Easy
  1. 1Expand (x + 4)(x + 1).
  2. 2Expand (y + 2)(y + 6).
  3. 3How many products appear in (a + 2)(b + 3) before combining?
Medium
  1. 4Expand (2m + 3)(m − 1).
  2. 5Expand (x − 7)(x − 2).
  3. 6A rectangle has sides (x + 8) and (x + 3). Find its area.
Challenging
  1. 7Expand (3a − 2b)(a + 4b).
  2. 8(x + p)(x + q) = x² + 9x + 20. Find p and q.
Show answers (try them yourself first)
  1. 1x² + 5x + 4
  2. 2y² + 8y + 12
  3. 34
  4. 42m² + m − 3
  5. 5x² − 9x + 14
  6. 6x² + 11x + 24
  7. 73a² + 10ab − 8b²
  8. 8p = 4 and q = 5 (or the other way round)

Square of a sum and of a difference

From daily life

A square carpet has a side of 43 cm, which is 40 + 3. Its area is not 40² + 3². It is 1600 + 240 + 9 = 1849, because two rectangles of 40 by 3 also fit in.

Simple definition

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². Say it in words: first square, twice the product, last square.

Live figure

Square of a sum and of a difference

In the app you can move this figure with your finger.

Open in the app

In the figure: Move the sliders: the big square is made of a², b² and two rectangles ab.

Rule 1: Square of a sum — Use it to square numbers just above a round number.

  • Easy: 12² = (10 + 2)² = 100 + 40 + 4 = 144.
  • Medium: 53² = (50 + 3)² = 2500 + 300 + 9 = 2809.
  • Exam-type: (3x + 5)² = 9x² + 30x + 25.

Rule 2: Square of a difference — Use it for numbers just below a round number. The middle term is negative.

  • Easy: 19² = (20 − 1)² = 400 − 40 + 1 = 361.
  • Medium: 78² = (80 − 2)² = 6400 − 320 + 4 = 6084.
  • Exam-type: (4p − 3)² = 16p² − 24p + 9.

Rule 3: Use it backwards — Spot a² + 2ab + b² to write it as one bracket squared, or use (a + b)² − 2ab = a² + b².

  • Easy: x² + 14x + 49 = (x + 7)².
  • Medium: 9a² − 12a + 4 = (3a − 2)².
  • Exam-type: a + b = 8 and ab = 15: a² + b² = 64 − 30 = 34.
Remember

(a + b)² − (a − b)² = 4ab. Check with a = 6 and b = 2: 64 − 16 = 48 = 4 × 12.

Common mistake

Writing (a + b)² = a² + b². Test: (3 + 4)² = 49 but 3² + 4² = 25. The middle term 2ab = 24 is missing.

Shortcut

For the middle term, double the product and keep the sign of the bracket. Memory clue: "Square, twice, square".

Spot the difference

Point(a + b)²a² + b²
For a = 3, b = 44925
Middle term 2abpresent (24)absent
Equal?nono

Practice questions

Easy
  1. 1Find 31² using 30 + 1.
  2. 2Expand (m + 5)².
  3. 3Find 29² using 30 − 1.
Medium
  1. 4Expand (2x − 7)².
  2. 5Find 102².
  3. 6Expand (5a + 2b)².
Challenging
  1. 7a + b = 9 and ab = 18. Find a² + b².
  2. 8Write x² − 16x + 64 as a square.
Show answers (try them yourself first)
  1. 1961
  2. 2m² + 10m + 25
  3. 3841
  4. 44x² − 28x + 49
  5. 510404
  6. 625a² + 20ab + 4b²
  7. 745
  8. 8(x − 8)²

Difference of squares and quick products

From daily life

From an 18 cm square sheet a 3 cm square is cut. The rest has area 324 − 9 = 315 cm². You can also get it as 21 × 15, because (18 + 3)(18 − 3) = 315.

Simple definition

(a + b)(a − b) = a² − b². The two middle terms +ab and −ab cancel, so only the difference of two squares is left.

(a−b)²b(a−b)b(a−b)b²a − bba − bb

In the figure: Remove b² from the big a² square. The L-shape is a² − b² and can be re-cut into one rectangle (a + b) by (a − b).

Rule 1: Expand and factorise — Same signs inside two brackets with + in one and − in the other give a difference of squares.

  • Easy: (x + 5)(x − 5) = x² − 25.
  • Medium: (3p + 4)(3p − 4) = 9p² − 16.
  • Exam-type: (7a − 2b)(7a + 2b) = 49a² − 4b².

Rule 2: Quick products — If two numbers are equally far from an easy number, square the easy number and subtract the square of the gap.

  • Easy: 47 × 53 = 50² − 3² = 2500 − 9 = 2491.
  • Medium: 96 × 104 = 100² − 4² = 9984.
  • Exam-type: 7.2 × 6.8 = 7² − 0.2² = 49 − 0.04 = 48.96.

Rule 3: Difference of two squares — Write a² − b² as (a + b)(a − b) and compute the two easy numbers.

  • Easy: 15² − 5² = 20 × 10 = 200.
  • Medium: 61² − 39² = 100 × 22 = 2200.
  • Exam-type: 41² − 40² = 81 (this is 41 + 40); and x² − 81 = (x + 9)(x − 9).
Remember

The difference of the squares of two consecutive numbers is their sum: (n + 1)² − n² = 2n + 1.

Common mistake

Mixing up (a + b)(a − b) with (a − b)². Test a = 8 and b = 3: (a + b)(a − b) = 55 but (a − b)² = 25.

Shortcut

Opposite signs kill the middle term. Memory clue: "Plus-minus pair, middle goes away".

Spot the difference

Point(a + b)(a − b)(a − b)²
For a = 8, b = 35525
Resulta² − b²a² − 2ab + b²
Middle termnone−2ab

Practice questions

Easy
  1. 1Expand (m + 6)(m − 6).
  2. 2Find 52 × 48 using 50 ± 2.
  3. 3Find 25² − 15².
Medium
  1. 4Find 103 × 97 using 100 ± 3.
  2. 5Expand (4x + 9)(4x − 9).
  3. 6Find 74² − 26².
Challenging
  1. 7Find 9.9 × 10.1.
  2. 8Write 25a² − 49 as a product of two brackets.
Show answers (try them yourself first)
  1. 1m² − 36
  2. 22496
  3. 3400
  4. 49991
  5. 516x² − 81
  6. 64800
  7. 799.99
  8. 8(5a + 7)(5a − 7)

Mending mistakes, patterns and proofs

From daily life

A teacher marks homework and finds 3(x + 4) written as 3x + 4. A quick test with x = 1 shows 3 × 5 = 15 but 3 + 4 = 7, so the step is wrong. Letters let us catch such slips and also prove claims for all numbers.

Simple definition

To check a step, put in a small number on both sides. Many patterns can be counted in more than one way, and different ways give equal expressions. To prove something for all numbers, write even numbers as 2k and odd numbers as 2k + 1.

In the figure: Count the ring as 5 × 5 − 3 × 3 = 16 or as 4 sides of 4 dots = 16. Two ways, one answer.

Rule 1: Mend the mistake — Test with a number, find the faulty step, rewrite it correctly.

  • Easy: 3(x + 4) = 3x + 12, not 3x + 4.
  • Medium: (a + 3)² = a² + 6a + 9, not a² + 9.
  • Exam-type: −2(m − 5) = −2m + 10, not −2m − 10.

Rule 2: Count a pattern in two ways — A ring of dots around a k by k square has (k + 2)² − k² = 4k + 4 dots.

  • Easy: k = 3: 25 − 9 = 16 and 4 × 3 + 4 = 16.
  • Medium: k = 10: 144 − 100 = 44 and 4 × 10 + 4 = 44.
  • Exam-type: n(n + 2) = (n + 1)² − 1; for n = 6: 48 = 49 − 1.

Rule 3: Prove with letters — One example never proves a claim for all numbers. Letters do.

  • Easy: Two odd numbers: (2a + 1) + (2b + 1) = 2(a + b + 1), which is even.
  • Medium: Three consecutive numbers n, n + 1, n + 2 add to 3n + 3 = 3(n + 1): 7 + 8 + 9 = 24 = 3 × 8.
  • Exam-type: (2k + 1)² = 4k(k + 1) + 1 is 1 more than a multiple of 8: 9² = 81 = 80 + 1.
Remember

Only like terms (same letters, same powers) can be added: 5w² + 6w cannot become 11w².

Common mistake

Multiplying only one term of a bracket, or adding unlike terms. Also, believing a claim after two or three examples.

Shortcut

Plug in 1 or 2 to test every step. Memory clue: "Plug a number, catch a blunder".

Spot the difference

PointChecking with numbersProving with letters
Can show a claim is false?yes (one failure)yes
Can show it is true for all?noyes
Examplex = 1: 3 × 5 ≠ 3 + 42k + 1

Practice questions

Easy
  1. 1Is 4(x + 3) = 4x + 3 correct? Give the right answer.
  2. 2Dots in a ring around a 5 by 5 square: find (k + 2)² − k² for k = 5.
  3. 3Simplify 2(y + 5) + 3y.
Medium
  1. 4Write (m − 4)² correctly.
  2. 5Step n of a pattern has (n + 1)² − 1 dots. How many in step 9?
  3. 6Write the sum of three consecutive numbers starting at n.
Challenging
  1. 7Simplify (x + 3)(x + 2) − (x + 4).
  2. 8Write 15² as a multiple of 8 plus 1.
Show answers (try them yourself first)
  1. 14x + 12
  2. 224
  3. 35y + 10
  4. 4m² − 8m + 16
  5. 599
  6. 63n + 3
  7. 7x² + 4x + 2
  8. 88 × 28 + 1