NCERT Solutions For Class 8 Maths Chapter 6 – We Distribute, Yet Things Multiply

NCERT Solutions for Class 8 Maths Chapter 6 We Distribute, Yet Things Multiply helps students understand the distributive property of numbers and algebraic expressions. This chapter explains how multiplication is distributed over addition and subtraction. Students learn to simplify expressions, solve equations faster, and apply algebraic rules in calculations. These concepts are highly useful in algebra, factorisation, and higher mathematics. Regular practice improves speed, accuracy, and logical thinking.

Question & Answers

Q1. What is the distributive property of multiplication over addition?

Answer:
a × (b + c) = ab + ac

NCERT Solutions For Class 8 Maths Chapter 6 – We Distribute, Yet Things Multiply

We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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.
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
For each statement, identify the appropriate algebraic expression(s).
(i) Two more than a square number.
  • 2 + s
  • (s + 2)2
  • s2 + 2
  • s2 + 4
  • 2s2
  • 22s
(ii) The sum of the squares of two consecutive numbers
  • m2 + n2
  • (m + n)2
  • m2 + 1
  • m2 + (m + 1)2
  • m2 + (m – 1)2
  • (m + (m + 1))2
  • (2m)2 + (2m + 1)2
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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 he tiled. Write an expression for the area that needs to be tiled.

We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Which is larger? Find out without fully computing the product.
(i) 14 × 26 or 16 × 24
(ii) 25 × 75 or 26 × 74​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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.​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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.​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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?​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Verify which of the following statements are true.
(i) (k + 1) (k + 2) – (k + 3) is always a multiple of 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.​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Consider any 2 by 2 square of numbers in a calendar, as shown in the figure.
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
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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−12)2{\left( {2{\rm{x}} - \frac{1}{2}} \right)^2}​
(vi) (7p) × (3r) × (p + 2)​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Expand the following products.
(i) (3 + u) (v – 3)
(ii) 23\frac{2}{3}​(15 + 6a)
(iii) (10a + b) (10c + d)
(iv) (3 – x)(x – 6)
(v) (–5a + b) (c + d)
(vi) (5 + z) (y + 9)​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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)​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
Look at the following patterns.
Pattern 1:
2(2² + 1²) = 3² + 1²
2(3² + 1²) = 4² + 2²
2(6² + 5²) = 11² + 1²
2(5² + 3²) = 8² + 2²

Pattern 2:
9 × 9 - 1 × 1 = 10 × 8
8 × 8 - 6 × 6 = 14 × 2
7 × 7 - 2 × 2 = 9 × 5
10 × 10 - 4 × 4 = 14 × 6

​Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Find 4062, 722, 1452, 10972, and 1242 using the identities you have learned so far.
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Express 100 as the difference of two squares.
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​Which is greater: (a – b)2 or (b – a)2? Justify your answer.​
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
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?
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
Expand:
(i) (a + ab – 3b2\rm 3 b^2​​) (4 + b), and
(ii) (4y + 7) (y + 11z – 3).
We Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
​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 Distribute, Yet Things Multiply (New NCERT)
Medium
Q.
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.

NCERT Solutions For Class 8 Maths Chapter 6 – We Distribute, Yet Things Multiply

Q2. What is the distributive property of multiplication over subtraction?

Answer:
a × (b - c) = ab - ac

Q3. Simplify: 5 × (3 + 2)

Answer:
= 5 × 5
= 25

Q4. Simplify: 7 × (10 - 4)

Answer:
= 7 × 6
= 42

Q5. Expand: 3(x + 4)

Answer:
= 3x + 12

Q6. Expand: 6(y - 2)

Answer:
= 6y - 12

Q7. Simplify: 9(a + b)

Answer:
= 9a + 9b

Q8. Factorise: 4x + 8

Answer:
= 4(x + 2)

Q9. Factorise: 7y - 14

Answer:
= 7(y - 2)

Q10. Why is distributive property important?

Answer: It helps simplify calculations and solve algebraic expressions easily.

FAQs

1. What is Chapter 6 of Class 8 Maths?

Chapter 6 of Class 8 Maths is We Distribute, Yet Things Multiply, based on distributive property and algebraic expansion.

2. Why is this chapter important?

It builds the foundation for algebra, factorisation, and equations.

3. What is distributive property?

It means multiplying a number with each term inside brackets.

Example:
2(x + 3) = 2x + 6

4. Where is distributive property used?

It is used in simplification, algebra, equations, and mental maths calculations.