We Distribute, Yet Things Multiply Class 8 Maths Worksheet with Answers Chapter 6

Ganita Prakash Part-I, Grade 8, Textbook of Mathematics – Chapter 6 of the Class 8 NCERT book is “We Distribute, Yet Things Multiply.” This chapter covers some important concepts. You need to fully understand all the concepts, as they will be important in future classes. The topics that are covered in this chapter are –

• Distributive property basics
• Binomial multiplication
• Algebraic identities
• Square expansion shortcuts
• Fast mental multiplication

There are 4 parts in this worksheet – Basic, Standard, Advance, and HOTS. Answers are there at the bottom.

Class 8 Maths Worksheet – Chapter 6: We Distribute, Yet Things Multiply

Basic

1. Expand using distributive property:
   1. 7(3 + 5)
   2. 9(x + 2)
   3. −4(a − 3)
2. Expand:
   1. (a + 1)(b + 1)
   2. (a + 1)(b − 1)
3. Expand (a + b)(c + d).
4. Use a suitable identity to find 98 × 102.
5. Multiply using the “×11 trick”:
   1. 94 × 11
   2. 495 × 11
6. Expand and simplify:
   1. (x + 2)(x + 5)
   2. (p − 1)(p + 11)

Standard

1. Expand:
   1. (3 + u)(v − 3)
   2. (3 − x)(x − 6)
2. Expand:
   1. (−5a + b)(c + d)
   2. (5 + z)(y + 9)
3. Expand and simplify:
   1. (a + b)²
   2. (a − b)²
   3. (a + b)(a − b)
4. Find using identities (no long multiplication):
   1. 46²
   2. 91²
5. Use identity 2(a² + b²) = (a + b)² + (a − b)² to find:
   2(5² + 6²)
6. Like terms check: Simplify completely:
   a³ + 2a²b + a²b + 2ab² + ab² + b³

Advance

1. Expand using identity:
   1. (m + 3)²
   2. (6 + p)²
2. Expand and simplify:
   1. (−2a + 3)²
   2. (b − 6)²
3. Expand and simplify (keep as algebraic expression):
   (7y − (3/4)z)²
4. Mind the mistake: Correct the simplification and write the right answer:
   1. (5m + 6n)² = 25m² + 36n²
   2. (−q + 2)² = q² − 4q + 4
5. Pattern of circles: At Step k, the number of circles is k² + 2k.
   1. How many circles in Step 15?
   2. Find k if Step k has 224 circles.
6. Park tiling (self-contained):
   Two square lawns of side g are placed side-by-side (total lawn area = 2g²).
   A walking path of uniform width w surrounds the entire shape.
   The outer boundary becomes a rectangle of size (2g + 2w) by (g + 2w).
   Write an expression for the area that must be tiled.

HOTS

1. Calendar 2×2 block:
   The four numbers are a, (a + 1), (a + 7), (a + 8).
   Compare the diagonal products:
   1. a(a + 8)
   2. (a + 1)(a + 7)

   Which one is larger and by how much? (Show using algebra.)

2. Choose three consecutive integers: (n − 1), n, (n + 1).
   Show that:
   n² − (n − 1)(n + 1) = 1
3. Without fully multiplying, decide which is larger:
   1. 14 × 26 or 16 × 24
   2. 25 × 75 or 26 × 74

   (Use the idea (a + b)(a − b) = a² − b².)

4. Prove (in 2–3 lines):
   (a − b)² = (b − a)²
5. Fast multiplication:
   Use a distributive-identity method to compute:
   1. 3874 × 101
   2. 1111 × 1001
6. Coin Conjoin (logic puzzle):
   A triangle of 10 coins can be flipped upside down in 3 moves.
   Predict the minimum moves needed to flip a triangle of 15 coins.
   (Write your reasoning in 2–3 lines.)

Answer Key

Basic – Answers

1. 1. 7(3 + 5) = 21 + 35 = 56
   2. 9(x + 2) = 9x + 18
   3. −4(a − 3) = −4a + 12
2. 1. (a + 1)(b + 1) = ab + a + b + 1
   2. (a + 1)(b − 1) = ab − a + b − 1
3. (a + b)(c + d) = ac + ad + bc + bd
4. 98 × 102 = (100 − 2)(100 + 2) = 100² − 2² = 10000 − 4 = 9996
5. 1. 94 × 11 = 1034
   2. 495 × 11 = 5445
6. 1. (x + 2)(x + 5) = x² + 7x + 10
   2. (p − 1)(p + 11) = p² + 10p − 11

Standard – Answers

1. 1. (3 + u)(v − 3) = 3v − 9 + uv − 3u
   2. (3 − x)(x − 6) = 3x − 18 − x² + 6x = −x² + 9x − 18
2. 1. (−5a + b)(c + d) = −5ac − 5ad + bc + bd
   2. (5 + z)(y + 9) = 5y + 45 + zy + 9z
3. 1. (a + b)² = a² + 2ab + b²
   2. (a − b)² = a² − 2ab + b²
   3. (a + b)(a − b) = a² − b²
4. 1. 46² = (40 + 6)² = 40² + 2·40·6 + 6² = 1600 + 480 + 36 = 2116
   2. 91² = (100 − 9)² = 100² − 2·100·9 + 9² = 10000 − 1800 + 81 = 8281
5. 2(5² + 6²) = (5 + 6)² + (6 − 5)² = 11² + 1² = 121 + 1 = 122
6. a³ + 2a²b + a²b + 2ab² + ab² + b³
   = a³ + 3a²b + 3ab² + b³

Advance – Answers

1. 1. (m + 3)² = m² + 6m + 9
   2. (6 + p)² = p² + 12p + 36
2. 1. (−2a + 3)² = (−2a)² + 2(−2a)(3) + 3² = 4a² − 12a + 9
   2. (b − 6)² = b² − 12b + 36
3. (7y − (3/4)z)²
   = (7y)² − 2·7y·(3/4)z + ((3/4)z)²
   = 49y² − (21/2)yz + (9/16)z²
4. 1. (5m + 6n)² = 25m² + 60mn + 36n²
   2. (−q + 2)² = (2 − q)² = q² − 4q + 4 (this one is actually correct)
5. 1. Step 15: 15² + 2·15 = 225 + 30 = 255
   2. k² + 2k = 224 → k² + 2k − 224 = 0 → (k + 16)(k − 14) = 0 → k = 14
6. Tiled area = (2g + 2w)(g + 2w) − 2g²
   = 2(g + w)(g + 2w) − 2g²
   = 6gw + 4w²

HOTS – Answers

1. (a + 1)(a + 7) = a² + 8a + 7
   a(a + 8) = a² + 8a
   So (a + 1)(a + 7) is larger by 7.
2. (n − 1)(n + 1) = n² − 1
   So n² − (n² − 1) = 1
3. 1. 14×26 = (20 − 6)(20 + 6) = 20² − 6² = 400 − 36 = 364
      16×24 = (20 − 4)(20 + 4) = 20² − 4² = 400 − 16 = 384
      So 16×24 is larger.
   2. 25×75 = (50 − 25)(50 + 25) = 50² − 25² = 2500 − 625 = 1875
      26×74 = (50 − 24)(50 + 24) = 50² − 24² = 2500 − 576 = 1924
      So 26×74 is larger.
4. (a − b)² = (−(b − a))² = (b − a)²
5. 1. 3874×101 = 3874×(100 + 1) = 387400 + 3874 = 391274
   2. 1111×1001 = 1111×(1000 + 1) = 1111000 + 1111 = 1112111
6. One valid reasoning:
   10-coin needs 3 moves, next bigger triangle (15 coins) increases by one “row”.
   The pattern suggests minimum moves increase by 1, so answer: 4 moves.

Worksheet From Other Chapters

Ganita Prakash Part-I

• A Square and a Cube Class 8 Maths Worksheet with Answers
• Power Play Class 8 Maths Worksheet with Answers
• A Story of Numbers Class 8 Maths Worksheet with Answers
• Number Play Class 8 Maths Worksheet with Answers
• Proportional Reasoning-1 Class 8 Maths Worksheet with Answers

Ganita Prakash Part-II

• Fractions in Disguise Class 8 Maths Worksheet with Answers
• Proportional Reasoning–2 Class 8 Maths Worksheet with Answers
• Algebra Play Class 8 Maths Worksheet with Answers