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

Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 6 We Distribute, Yet Things Multiply Worksheet with Answers before your unit tests for better preparation.

We Distribute, Yet Things Multiply Worksheet Class 8

Class 8 Maths We Distribute, Yet Things Multiply Worksheet

We Distribute, Yet Things Multiply Class 8 Ganita Prakash Worksheet

Some Properties of Multiplication

The Puzzle of the Extra Pieces
Rohan and Meera were arranging a grid of flower pots in the garden with 17 rows and 19 columns. “That makes 17 x 19 = 323 pots,” Rokan calculated.

Meera: “If we add one more column, will it just increase by 1 pot?”

Rohan: No, it adds a full column of 17 pots. He explained it like this: 17 × (19 + 1) = (17 × 19) + (17 × 1). So, the total became 323 + 17 = 340.

Meera: What if we add both, a row and a column 9

Rohan: We can work it out as (17 + 1) × (19 + 1)
= (17 × 19) + (1 × 19)+ (17 × 1) + (1 × 1)
= 323 + 19 + 17 + 1
= 360 pots

Question 1.
Based on the above information, answer the following questions:
(a) If the product of 17 and 19 increases when the first number is change from 17 to 18, by how much does the product increase?
Answer:
19

(b) When the second number changes from 19 to 20 by, how much does the product increase?
Answer:
17

(c) What is the total increase in the product when both numbers increase by 1?
Answer:
37

(d) Expand (a + 1)(b + 1) using the distributive property.
Answer:
a + b + ab + 1

(e) Calculate (17 + 1)(19 – 1) and find the change from the original product of 1 7 and 19.
Answer:
Increase by 1

Question 2.
Use the identity (a + m) (b + n) = ab + an + mb + mn to solve the following questions:
(a) Find the increase in a x b from the above equation, when a = 10, b = 15, m = 2, and n = 3.
Answer:
66

(b) Verify the identity for a = 7, b = 9, m = -1, and n = 4.
Answer:
Do it yourself

(c) Find three pairs of numbers where increasing one by 2 and decreasing the other by 4 results in the same product. Explain how the identity justifies this.
Answer:
(1,6), (2, 8), (3, 10) (Answer may vary)

Question 3.
(a) Give an example where increasing one number by 2 and decreasing the other by 3 keeps the product unchanged.
(b) How does the product change when one number increases by 5 and the other decreases by 5? Show this algebraically.
Answer:
(a) (2,6)
(b) 5(b – a -5)

Question 4.
(a) If a = -4 and b = – 3, find (a + 2)(b – 1). What is the product, and how does it relate to ab?
(b) Expand (4 + u)(v – 4) completely using the distributive property.
Answer:
(a) 8,4 less than ab
(b) UV – 4K + 4V – 16

Question 5.
Expand (3a + 8) (b + 11 c – 5).
Answer:
3ab + 33ac – 15a + 8b + 88c – 40

Question 6.
For (a + b) (a² + 2ab + b²), expand and simplify the expression. Identify the pattern formed by the coefficients.
Answer:
a³ + 3a²b + 3ab² + b³

Fast Multiplications Using the Distributive Property

Neha was in charge of organizing gift packs for her school’s annual festive fundraiser.
Neha needed 11 chocolate bars per gift pack.
For × packs, she calculated:
11 × x = (10 + 1) × x = 10 × x + 1 × x = (10x + x) bars
Later, for y packs: 11 × y = (10 + 1) × y = (10y + y) bars

Thus, she observes that multiplying by 11 is the same as multiplying by 10 and adding one more group. Based on the above observation, answer the following questions:

Question 7.
Multiply 4826 by 11 using the distributive property and write the product as a numbers.
Answer:
53086, 48260 + 4826

Question 8.
Find the product of 7322 and 11 by expressLrLg 11 as 10 + 1.
Answer:
813142

Question 9.
Evaluate the following by using the distributive property method:
(a) 86 × 11
(b) 672 × 11
(c) 4193 × 11
Answer:
(a) 946
(b) 7392
(c) 46123

Question 10.
How can you multiply a number by 101 using the distributive property? Explain with an example.
Answer:
37 × 101 = 37 × (100 + 1) = 37 × 100 + 37 = 3737

Question 11.
Evaluate the following by using the distributive property method:
(a) 73 × 101
(b) 842 × 101
(c) 5642 × 101
Answer:
(a) 7373
(b) 85042
(c) 569842

Question 12.
How do the multiplication shortcuts for 11, 101, and 1001 relate to the distributive property? Explain with examples.
Answer:
Do it yourself

Question 13.
Multiply the following without using the long multiplication:
(a) 2222 × 1001
(b) 38471 × 1001
Answer:
(a) 2224222
(b) 38509471

Question 14.
Calculate the products by applying the distributive property:
(a) 7685 × 99
(b) 14327 × 999
Answer:
(a) 760815
(b) 14312673

Special Cases of the Distributive Property

Square of the Sum/Difference of Two Numbers
One afternoon, two friends were learning how algebra connects to areas. The girl pointed to a large square and asked, “How do we find its total area?”

The boy explained that if each side measured (a + b), the area could be divided into smaller parts:
(a + b)² = a² + 2ab + b².

The next day, while studying another square, the girl remarked, “It feels like removing a smaller square from a bigger one.” The boy agreed and showed that this idea gives:
(a – b)² = a² – 2ab + b².

Later, the boy arranged rectangles into a square and said, “Here’s a shortcut—it saves time and skips long expansions.” Together they concluded: (a + b) (a – b ) = (a² – b²).

Question 15.
Using the identity (a + b)² = a² + 2ab + b², answer the following questions:
(a) Expand (x + 7)² using both the distributive property and the identity.
(b) A garden is of (20 + 5) m on each side. Use (a + b)² to find its area quickly.
(c) Using the identity find 108² without direct multiplication.
Answer:
(a) x²+ 14x + 49
(b) 625 m²
(c) 11664

Question 16.
Using tKe identity (a – b)² = a² – 2ab + b², answer the following questions:
(a) Expand (y-9)² using the identity.
(b) A rope of length (50-3) m is formed into a square. Find its area using (a-b)².
(c) Evaluate 68² by taking itas (70-2)².
Answer:
(a) y² – l8y + 81
(b) 138.06m²
(c) 4624

Question 17.
A square playground has a side length (a + b) units. A smaller square of side length (a – b) units is removed from it. Show that the remaining area equals 4ab.
Answer:
(a + b)² – (a – b)² = 4ab

Question 18.
Using the identity (a + b) (a – b) = (a² – b²), answer the following questions:
(a) Evaluate (p + 9)(p – 9).
Answer:
p² – 81

(b) Without multiplying directly, evaluate the following products:
(i) 99 × 101
(ii) 38 × 42
Answer:
(i) 9999
(ii) 1596

Question 19.
Find out without multiplying: which is larger, 49 × 51 or 50²?
Answer:
50² > 49 × 51

Question 20.
Use the identity (a + b) (a – b) = a² – b² to explain why 32 × 68 = 2176.
Answer:
Do it yourself

Investigating Patterns

Aryan and Tara sat under the banyan tree, the notebooks open.
Tara (looking seriously at her page), Why does 37 × 33 give 1221? It is not obvious.
Aryan (smiled): They are 2 away from 35. It is (35 – 2) (35 + 2) = 35² – 2².
Tara, So 83 × 87 is (85 – 2) (85 + 2) or 85² – 2² = 7221!

Later, Tara asked about another trick: What about 2(a² + b²)?
Aryan wrote down the identity: 2 (a² + b²) = (a + b)² + (a – b)² Tara closed her book in excitement and said, “Algebraic identities are like magical shortcuts!” Based on the above, answer the following questions:

Question 21.
Look at these calculations:
46 × 44 = 2024
72 × 78 = 5616
105 × 105 = 11025

(a) What do you notice about the relationship between each pair of numbers being multiplied?
Answer:
The numbers being multiplied are always equally spaced around their arrange.

(b) Identify the specific algebraic identity that explains this pattern.
Answer:
(a – d)(a + d)= a¹ – d²

(c) Use the identity from part (b) to calculate 44 × 42 and 110 × 110 quickly.
Answer:
44 × 42 = 1848, 110 × 110 = 12100

Question 22.
Using the identity: 2(a² + b²) = (a + b)² + (a – b)² answer the following questions:
(a) Verify the above identity by choosing two numbers for a and b.
Answer:
Do it yourself

(b) Verify that (a + b)² + (a – b)², is identically equal to 2a² + 2b².
Answer:
Do it yourself

(d) 10032
Answer:
208

Question 23.
Fiad the following squares usLOEg suttabLe identities (without direct multiplication):
(a) 98²
(b) 203²
(c) 512²
(d) 1003²
Answer:
(a) 9604
(b) 41209
(c) 262144
(d) 1006009

Question 24.
Express 85 as the difference of two squares.
Answer:
11²-6² = 85 (The answer may very)

Mind the mistake, mend the mistake

One day in class, the teacher said,
“Children, I have some algebraic expressions that have been expanded and simplified. You have to check each one carefully. If you spot a mistake, think about what went wrong and then write the correct form. Let’s see who can find the errors fastest!”

Question 25.
Simplify the following expressions:

(a) -5x (2y – 3x)	(b) (4a – 3)2	(c) 7m2 + 3m + 5m2 – m	(d) (x + 4)(x – 5)	(e) \(\left(\frac{1}{2}\right)\)(8p – 6) + 10
Student’s Work: – 5x (2y – 3x) = (-5x × 2y) – (-5x × 3x) = -10xy – (-15x) = -10xy + 15x	Student’s Work: (4a – 3)2 = (4a)2 – (3)2 = 16a2 – 9	Student’s Work: 7m2 + 3 m + 5 m2 – m = (7 m2 + 5m2) + (3m – m) = 12m2 + 2m = 14m3	Student’s Work: (x + 4)(x – 5) = x(x – 5) + 4(x – 5) = x2 – 5x + 4x – 5 = x2 – X – 5	Student’s Work: \(\left(\frac{1}{2}\right)\)(8p – 6) + 10 = \(\frac{1}{2}\)(8p – 6) + 10 = 4p – 6 + 10 = 4p + 4
Find the mistake.	Which identity did the student misuse and how?	There are two distinct errors in the final two steps. Identify both.	Find the mistake in the muttiplication.	The distributive property was applied incorrectly. What was the error?
Explain the error.	Write the correct identity for (a-b)2.	Explain the rules for combining the like terms.	What is the correct result of (x + 4)(x – 5)?	Write the correct way to distribute \(\left(\frac{1}{2}\right)\)(8p – 6)
Provide the correct simplified expression.	Expand (4a 3)2 correctly.	simplify the expression correctly.	Provide the final, correct, expanded, and simplified form.	Solve the problem correctly.

Answer:
(a) -10xy + 15x²
(b) 160² – 24a +9
(c) 2m (1 + 6m)
(d) 112² – 6² = 85 (The answers may vary)

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

Question 26.
A staircase is built with blocks.
• Step 1: 1 block
• Step 2: 3 blocks (1 + 2)
• Step 3: 6 blocks (1 +2 + 3)
• Step 4: 10 blocks (1 +2 + 3 + 4)

(a) Write the number of blocks in step n.
(b) A famous formula that number of blocks in step n is \(\frac{n(n+1)}{2}\). Verify whether it is correct or not
by finding the number of blocks in step 10.
Answer:
(a) \(\frac{n(n+1)}{2}\)
(b) Number of blocks in step 10 = 55

Question 27.
Each figure is a square frame made of smaller squares.
• Figure 1: A 3 × 3 square with a 1 × 1 hole → 8 squares
• Figure 2: A 4 × 4 square with a 2 × 2 hole → 12 squares
• Figure 3: A 5 × 5 square with a 3 × 3 hole → 16 squares

Based on the above information, answer the following questions:
(a) Draw Figure 4.
Answer:

(b) Write an expression for the dumber of squares in figure n by subtracting the hole.
Answer:
(n + 2)² – n²

(c) Write another expression by breaidng the frame into 4 equal sida (Don’t forget the corners!)
Answer:
4(n + 2)

(d) Simplify both the expressions to show they are the same.
Answer:
(n + 2)² – n² = 4 (n + 2) -4

Question 28.
Read tHe given, information and then answer the questions that follow:
(a) A square garden has a side length of r metres. A uniform walkway of width m meters surrounds the garden on all four sides, forming a larger square.

(i) Express the total area covered by the walkway in terms of r and m.
Answer:
(i) 4rm + 4m²

(ii) If the side of the garden (r) is 14 metres and the walkway width (m) is 2 metres, calculate the area of the walkway.
Answer:
128m²

(iii) If the area of the walkway is exactly half the area of the garden, what is the relationship between r and m? Provide your answer algebraically and explain your reasoning.
Answer:
r = (r + 2√6)m

(b) A square of side length s has a square hole of side length t cut out from its centre.
(i) Find the area by subtracting the area of the inner square from the area of the outer square.

Answer:
s² – t²

(ii) Divide the square frame into 4 congruent rectangles and 4 small squares at the corners. Find the area of each part and add them together.
Answer:
Area of each rectangle = t \(\left(\frac{s-t}{2}\right)\)
Area of each square = \(\left(\frac{s-t}{2}\right)\)²
Total area = 2t(s — t) + (s — t)²

(iii) Show that both the methods lead to the same expression: s² – t².
Answer:
Do it yourself

(iv) If the square frame is made of a uniform metal strip, and s = 10 cm and t = 6 cm, what is the area of the uniform metal strip?
Answer:
64 m²

(c) A door is topped by a semicircular arch. The rectangular part of the door has height h and width
w. The semicircular arch has radius r = \(\frac{w}{2}\)
(i) Find the total area of the door.

Answer:
\(\frac{\pi w^2}{8}+wh\)

(ii) If h = 2m and w = 3 m, calculate the total area of the door.
Answer:
9.53 m²

Question 29.
Evaluate 47 x 53 using the identity (a + b) (a – b).
Answer:
2491

Question 30.
A square of side (x + y) has four rectangles of area xy removed from it, leaving a smaller square. Prove that the area of the remaining square is (x – y)².

Answer:
(x + y)² – 4xy = (x – y)²

Question 31.
A number leaves remainder of 4 when divided by 9, and another number leaves a remainder 7 when divided by 9. What is the remainder when their sum, and product are divided by 9? Write the numbers.
Answer:
Reminders: 2, 1, Numbers : (13, 16) (22. 25) (Answer may vary)

Question 32.
Which ts Larger? (Find out without fully computing the product).
(a) 18 × 32 or 20 × 30
(b) 33 × 67 or 34 × 66
Answer:
(a) 20 x 30 > 18 x 32
(b) 34 x 66 > 33 x 67

Worksheet On We Distribute, Yet Things Multiply Class 8

A. Choose the correct option.

1. If a and b are numbers, the increase in the product ab when both a and b are increased by 1 is
(a) a + b
(b) a + b + 1
(c) ab + 1
(d) 1
Answer:
(b) a + b + 1

2. What is the expanded form of (3 + u) ( v – 3)?
(a) 3v + uv – 9 – 3u
(b) 3v – 9 + uv + 3u
(c) 3v + uv – 9
(d) 3v + uv + 9 + 3u
Answer:
(a) 3v + uv – 9 – 3u

3. The simplified form of -3p (-5p + 2q) is:
(a) 15p² + 6pq
(b) -15p² + 6pq
(c) 15p² + 6pq
(d) -8p – 2p
Answer:
(a) 15p² + 6pq

4. Which identity is equal to a² – b²?
(a) (a + b)(a – b)
(b) (a – b)²
(c) (a + b)²
(d) a² + b²
Answer:
(a) (a + b)(a – b)

5. Which of the following is NOT an identity?
(a) (p – q)² – p² – 2pq + q²
(b) (p + q)² – p² + 2pq + q²
(c) p² – q² = (p + q)(p + q)
(d) p(q + r) = pq + pr
Answer:
(d) p(q + r) = pq + pr

Directions. (6-7): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

6. Assertion (A): The product of two numbers always remains the same if one factor is increased by 2 and the other is decreased by 4.
Reason (R): If the numbers are a and b then the change in the product is given by the expression 2b – 4a – 8, which is not always zero.
Answer:
(d) Assertion (A) is false but Reason (R) is true.

7. Assertion (A): The expressions (m + n)² – 4mn and (n – m)² are equivalent.
Reason (R): Both expressions are simplified to the same polynomial, n² – 2mn + m².
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

B. Fill in the blanks:

1. The expansion of (2x + 3y)² is _____.
Answer:
4x² + 9y² + 12xy

2. By using the identity (a + b) (a – b) = a² – b², the value of 406² – 394² is _____.
Answer:
9600

3. If Q number a Is increased by 4 and another number b is decreased by 2, the increase in their product is 4b – 2a – 8. The original product was _____.
Answer:
ab

4. The next figure in the sequence (Step 1, Step 2, Step 3..) has a number of units given by the expression y² + 2y. The number of units in Step 5 will be _____.

Answer:
35

5. The expression for the area of a square with side length (5p – 2q) is _____.
Answer:
25p² + 4q² – 20pq

C. State whether the following statements are True (T) or False (F).

1. The expression (x – 4)² is equal to x² – 16.
Answer:
False

2. The distributive property can be applied to expressions with more than two terms inside the brackets.
Answer:
True

3. The product of two numbers remains the same if one is increased by k and the other is decreased by k.
Answer:
False

4. The identity (a – b)² = a² – 2ab + b² also holds when a and b are negative numbers.
Answer:
True

5. The expression (a +1) (b -1) equals ab + a + b – 1 always.
Answer:
False

D. Solve the following.

Question 1.
Find the product of (10a + b) and (10c + d) using the distributive property.
Answer:
100ac + 10ad + 10bc + bd

Question 2.
Find 3 examples where the product of two numbers remains unchanged when one ts increased by 2 arid the other is decreased by 4.
Answer:
Do it yourself

Question 3.
Simplify: (a – b)(a³ + a²b + ab² + b³). What is the pattern? PredLct the next identity.
Answer:
a⁴ – b⁴, pattern:aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b + ….+ ab^n-2 + b^n-1)

Question 4.
Simplify: (5m + 6n)² – (5m – 6n)²
Answer:
120 mn

Question 5.
The figure shows a pattern of squares. Write an expression for the number of squares in Step n and justify your answer.
(Step 1: 2 squares, Step 2: 4 squares, Step 3: q squares, …)

Answer:
n² where n ≥ 2

Question 6.
The product of two rLurnbers is 120. If one number ts increased by 5 and the other is decreased by 2, the new product is still 120. Find the possible pairs of original numbers.
Answer:
(15, 8)

Question 7.
Simplify the expression: 2(x — 1) + 3(x + 4) — (5x — 2)
Answer:
12

Question 8.
Identify and correct the error: (5m + 6n)² = 25m² + 36n²
Answer:
(5m + 6m)² = 25m² + 36n² + 60mn