We Distribute Yet Things Multiply
This Class 8 Ganita Prakash chapter uncovers why sharing into groups and multiplying are linked through the distributive property: a × (b + c) = a × b + a × c. Using area models and number examples, it builds up to multiplying algebraic expressions and expanding brackets — a skill at the heart of algebra.
Learning objectives
- State and use the distributive property of multiplication over addition.
- Use the area model to picture multiplication.
- Multiply a number or term by a sum.
- Multiply two binomials by expanding brackets.
Key concepts
The distributive property
Multiplication distributes over addition: a × (b + c) = a×b + a×c. For example 6 × 23 = 6 × (20 + 3) = 120 + 18 = 138. The same idea works with subtraction: a × (b − c) = a×b − a×c. This lets us break hard products into easy parts.
The area model
A rectangle of length (b + c) and width a has area a(b + c). Splitting the length into b and c gives two rectangles of areas a×b and a×c, and their sum equals the whole. This picture explains the distributive property and makes expanding brackets feel natural.
Multiplying a term by a sum
To multiply a single term by an expression in brackets, multiply it by each term inside: 3(x + 4) = 3x + 12, and 2a(a + 5) = 2a² + 10a. Each part is handled separately and then added, exactly as the distributive property says.
Multiplying two binomials
To multiply two sums, each term of the first multiplies every term of the second: (x + 2)(x + 3) = x×x + x×3 + 2×x + 2×3 = x² + 5x + 6. This is just the distributive property applied twice.
Important formulas
Distributive over addition
a × (b + c) = a×b + a×c
Distributive over subtraction
a × (b − c) = a×b − a×c
Product of two binomials
(x + a)(x + b) = x² + (a+b)x + ab
Key definitions
- Distributive property
- Multiplying a sum by a number equals multiplying each addend and adding the results.
- Term
- A single number or product of numbers and variables in an expression.
- Binomial
- An algebraic expression with two terms, such as x + 3.
- Expand
- To remove brackets by multiplying out using the distributive property.
Solved examples
Q1. Use the distributive property to find 7 × 104.
Solution: 7 × 104 = 7 × (100 + 4) = 700 + 28 = 728.
Q2. Expand 5(2x + 3).
Solution: 5 × 2x + 5 × 3 = 10x + 15.
Q3. Multiply (x + 4)(x + 2).
Solution: x×x + x×2 + 4×x + 4×2 = x² + 2x + 4x + 8 = x² + 6x + 8.
Common mistakes to avoid
- Multiplying only the first term inside the bracket, e.g. 3(x + 4) = 3x + 4.
- Dropping a sign when distributing over subtraction.
- Forgetting to multiply every pair of terms when expanding two binomials.
- Adding exponents wrongly: x × x = x², not 2x.
We Distribute Yet Things Multiply — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The distributive property says a(b + c) =
Practice questions
Short answer
State the distributive property.
a × (b + c) = a×b + a×c; multiplication distributes over addition.
Expand 6(x + 2).
6x + 12.
Why does the area model explain distribution?
A rectangle of width a and length (b+c) splits into areas a×b and a×c that add to a(b+c).
Long answer
Show how the distributive property makes 23 × 7 easy, and explain the area model behind it.
Write 23 as 20 + 3, so 23 × 7 = (20 + 3) × 7 = 20×7 + 3×7 = 140 + 21 = 161. The area model pictures a rectangle 7 units wide and 23 units long; splitting the length into 20 and 3 gives two rectangles of areas 140 and 21, whose sum is the whole area 161 — exactly what distribution computes.
Multiply (x + 5)(x + 2) step by step and state the rule used.
Using the distributive property twice, each term of the first bracket multiplies each term of the second: x×x = x², x×2 = 2x, 5×x = 5x, 5×2 = 10. Adding gives x² + 2x + 5x + 10 = x² + 7x + 10. The rule is (x + a)(x + b) = x² + (a+b)x + ab.
HOTS (Higher Order Thinking)
Explain why 99 × 47 can be found as 47 × 100 − 47.
Since 99 = 100 − 1, distribution gives 99 × 47 = (100 − 1) × 47 = 4700 − 47 = 4653, turning a hard product into an easy subtraction.
A square of side (x + 3) has what area, and how does distribution show it?
Its area is (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9; distribution multiplies each term of one factor by each term of the other.
Quick revision
Revision notes
- a(b + c) = ab + ac; a(b − c) = ab − ac.
- Area model: a rectangle split lengthwise shows distribution.
- Multiply a term by a sum: multiply each inside term.
- Two binomials: every term times every term, then add.
Key takeaways
- Distribution turns hard products into easy parts.
- Expanding brackets is repeated distribution.
- x × x = x², a common slip to avoid.
Frequently asked questions
What is the distributive property?
Multiplying a sum by a number equals multiplying each part and adding: a(b+c) = ab + ac.
How do I expand two brackets?
Multiply each term in the first bracket by each term in the second, then add the products.
Does x + x equal x²?
No — x + x = 2x, while x × x = x².