Class 8 Mathematics

Chapter 6 Solutions — We Distribute, Yet Things Multiply

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Solutions

Overview

Step-by-step NCERT solutions for We Distribute, Yet Things Multiply (Chapter 6, NCERT Class 8 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the We Distribute, Yet Things Multiply textbook chapter.

Solved

What these solutions cover

All 20 questions in We Distribute, Yet Things Multiply are solved in the PDF. Here's what's inside, exercise by exercise:

Some Properties of Multiplication

Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the middle number of a 3 × 3 frame is given by the expression pq, as shown in the figure, write the expressions for the other numbers in the grid.
2.
Expand the following products.
- (i) (3 + u)(v − 3)
- (ii) (2/3)(15 + 6a)
- (iii) (10a + b)(10c + d)
- (iv) (3 − x)(x − 6)
- (v) (−5a + b)(c + d)
- (vi) (5 + z)(y + 9)
Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.
4.
Expand
- (i) (a + ab − 3b²)(4 + b), and
- (ii) (4y + 7)(y + 11z − 3).
5.
Expand
- (i) (a − b)(a + b),
- (ii) (a − b)(a² + ab + b²) and
- (iii) (a − b)(a³ + a²b + ab² + b³). Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Special Cases of the Distributive Property

Which is greater: (a − b)² or (b − a)²? Justify your answer.
Express 100 as the difference of two squares.
Find 406², 72², 145², 1097², and 124² using the identities you have learnt so far.
Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.

This Way or That Way, All Ways Lead to the Bay

1.
Compute these products using the suggested identity.
- (i) 46² using Identity 1A for (a + b)²
- (ii) 397 × 403 using Identity 1C for (a + b)(a − b)
- (iii) 91² using Identity 1B for (a − b)²
- (iv) 43 × 45 using Identity 1C for (a + b)(a − b)
2.
Use either a suitable identity or the distributive property to find each of the following products.
- (i) (p − 1)(p + 11)
- (ii) (3a − 9b)(3a + 9b)
- (iii) −(2y + 5)(3y + 4)
- (iv) (6x + 5y)²
- (v) (2x − 1/2)²
- (vi) (7p) × (3r) × (p + 2)
3.
For each statement identify the appropriate algebraic expression(s).
- (i) Two more than a square number. Options: 2 + s, (s + 2)², s² + 2, s² + 4, 2s², 2²s
- (ii) The sum of the squares of two consecutive numbers. Options: m² + n², (m + n)², m² + 1, m² + (m + 1)², m² + (m − 1)², (m + (m + 1))², (2m)² + (2m + 1)²
Consider any 2 by 2 square of numbers in a calendar. Find products of numbers lying along each diagonal — 4 × 12 = 48, 5 × 11 = 55. Do this for the other 2 by 2 squares. What do you observe about the diagonal products? Explain why this happens. Hint: Label the numbers in each 2 by 2 square as: a, (a+1) / (a+7), (a+8).
5.
Verify which of the following statements are true.
- (i) (k + 1)(k + 2) − (k + 3) is always 2.
- (ii) (2q + 1)(2q − 3) is a multiple of 4.
- (iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.
- (iv) (6n + 2)² − (4n + 3)² is 5 less than a square number.
A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?
Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same with other sets of numbers. What pattern do you notice? How do we write this as an algebraic equation? Expand both sides of the equation to check that it is a true identity.
What is the algebraic expression describing the following steps — add any two numbers. Multiply this by half of the sum of the two numbers? Prove that this result will be half of the square of the sum of the two numbers.
9.
Which is larger? Find out without fully computing the product.
- (i) 14 × 26 or 16 × 24
- (ii) 25 × 75 or 26 × 74
A tiny park is coming up in Dhauli. The plan is shown in the figure. The two square plots, each of area g² sq. ft., will have a green cover. All the remaining area is a walking path w ft. wide that needs to be tiled. Write an expression for the area that needs to be tiled.
11.
For each pattern shown below,
- (i) Draw the next figure in the sequence.
- (ii) How many basic units are there in Step 10?
- (iii) Write an expression to describe the number of basic units in Step y. [Pattern (a): Z/zigzag tile arrangement growing at each step. Pattern (b): rectangular grid of square tiles growing at each step.]

Sample

Worked examples

3 questions from We Distribute, Yet Things Multiply, solved step by step.

Some Properties of Multiplication

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

We need (a + 2)(b − 4) = ab.

Expanding the left side: (a + 2)(b − 4) = ab − 4a + 2b − 8.

Setting this equal to ab: −4a + 2b − 8 = 0, so 2b = 4a + 8, giving b = 2a + 4.

Choosing three values of a and computing b = 2a + 4:

a = 1: b = 2(1) + 4 = 6. Check: 1 × 6 = 6 and 3 × 2 = 6. ✓

a = 2: b = 2(2) + 4 = 8. Check: 2 × 8 = 16 and 4 × 4 = 16. ✓

a = 3: b = 2(3) + 4 = 10. Check: 3 × 10 = 30 and 5 × 6 = 30. ✓

Answer: Three examples: (1, 6), (2, 8), (3, 10) — in each case the product is unchanged when one number is increased by 2 and the other is decreased by 4.

Special Cases of the Distributive Property

Which is greater: (a − b)² or (b − a)²? Justify your answer.

Note that (b − a) = −(a − b).

When we square both sides: (b − a)² = (−(a − b))² = (a − b)².

Therefore (a − b)² = (b − a)² — they are equal.

Alternative: expand both.

(a − b)² = a² − 2ab + b².

(b − a)² = b² − 2ba + a² = a² − 2ab + b².

Both expressions are identical.

Answer: Neither is greater — (a − b)² = (b − a)² always.

This Way or That Way, All Ways Lead to the Bay

Compute these products using the suggested identity. (i) 46² using Identity 1A for (a + b)² (ii) 397 × 403 using Identity 1C for (a + b)(a − b) (iii) 91² using Identity 1B for (a − b)² (iv) 43 × 45 using Identity 1C for (a + b)(a − b)

(i) 46² using (a + b)² = a² + 2ab + b², with a = 40, b = 6:

46² = (40 + 6)² = 40² + 2 × 40 × 6 + 6² = 1600 + 480 + 36 = 2116.

(ii) 397 × 403 using (a + b)(a − b) = a² − b², with a = 400, b = 3:

397 × 403 = (400 − 3)(400 + 3) = 400² − 3² = 160000 − 9 = 159991.

(iii) 91² using (a − b)² = a² − 2ab + b², with a = 100, b = 9:

91² = (100 − 9)² = 100² − 2 × 100 × 9 + 9² = 10000 − 1800 + 81 = 8281.

(iv) 43 × 45 using (a + b)(a − b) = a² − b², with a = 44, b = 1:

43 × 45 = (44 − 1)(44 + 1) = 44² − 1² = 1936 − 1 = 1935.

Answer: (i) 46² = 2116; (ii) 397 × 403 = 159991; (iii) 91² = 8281; (iv) 43 × 45 = 1935.

All 20 questions in We Distribute, Yet Things Multiply are solved step by step in the Solutions PDF above.

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Chapter 6 Solutions — We Distribute, Yet Things Multiply

Overview

What these solutions cover

Some Properties of Multiplication

Special Cases of the Distributive Property

This Way or That Way, All Ways Lead to the Bay

Worked examples

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Which is greater: (a − b)² or (b − a)²? Justify your answer.

Compute these products using the suggested identity. (i) 46² using Identity 1A for (a + b)² (ii) 397 × 403 using Identity 1C for (a + b)(a − b) (iii) 91² using Identity 1B for (a − b)² (iv) 43 × 45 using Identity 1C for (a + b)(a − b)

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