What this chapter is about
This chapter explores the distributive property of multiplication over addition and subtraction. You have already learnt how to multiply and add numbers, but now you will see how these two operations work together in a beautiful way. When you multiply a number with a sum (or difference), you can "distribute" the multiplication to each part separately and then combine the results.
Understanding the distributive property is essential for simplifying algebraic expressions and solving equations efficiently. It helps you break difficult calculations into easier parts. For instance, multiplying 7 × 102 becomes simpler when you think of it as 7 × (100 + 2) = 700 + 14 = 714.
After studying this chapter, you should be able to apply the distributive property to simplify numerical calculations, expand algebraic expressions like 3(x + 5), and recognise when this property makes problem-solving faster. These skills form the foundation for working with polynomials in higher classes.
Key ideas
- The distributive property states that a × (b + c) = a × b + a × c. Multiplication "spreads" over each term inside the bracket.
- The property also works with subtraction: a × (b − c) = a × b − a × c.
- You can use this property in both directions: expand brackets (distribute) or factor out a common term (the reverse process).
- For algebraic expressions, 2(x + 4) means 2 × x + 2 × 4 = 2x + 8.
- The distributive property helps in mental maths by breaking numbers into convenient parts.
- When multiplying two binomials like (a + b)(c + d), you distribute each term of the first bracket to each term of the second.
- The property holds for all real numbers, including negative numbers and fractions.
Formulas and facts to remember
- Distributive law (addition): a × (b + c) = a × b + a × c — multiply the outside number with each term inside, then add.
- Distributive law (subtraction): a × (b − c) = a × b − a × c — multiply the outside number with each term inside, then subtract.
- Expanding two brackets: (a + b)(c + d) = ac + ad + bc + bd — each term in the first bracket multiplies each term in the second.
- Factoring out: ab + ac = a(b + c) — the reverse of distribution; take out the common factor.
Worked examples
Example 1: Simplify 8 × 53 using the distributive property.
Break 53 into 50 + 3.
8 × 53 = 8 × (50 + 3)
= 8 × 50 + 8 × 3
= 400 + 24
= 424
Example 2: Expand 5(2y − 7).
Multiply 5 with each term inside the bracket.
5 × 2y = 10y
5 × 7 = 35 (but there is a minus sign, so it becomes −35)
Therefore, 5(2y − 7) = 10y − 35.
Example 3: Meera buys 4 notebooks at ₹45 each and 4 pens at ₹12 each. Find the total cost using distribution.
Total cost = 4 × 45 + 4 × 12
Factor out 4: = 4 × (45 + 12)
= 4 × 57
= 228
So Meera spends ₹228 in total.
Common mistakes
- Forgetting to multiply every term inside the bracket → Always distribute the multiplier to each term, not just the first one.
- Ignoring the minus sign when distributing → Carry the sign of each term carefully; 3(x − 4) gives 3x − 12, not 3x + 12.
- Writing 2(x + 5) = 2x + 5 instead of 2x + 10 → The multiplier acts on every term, including the constant.
- Thinking the distributive property only works with addition → It works equally with subtraction.
- Mixing up distribution with simple addition → 2 + (3 × 4) is not the same as (2 + 3) × (2 + 4); distribution applies when multiplication is outside the bracket.
Quick revision
- a × (b + c) = a × b + a × c distributes multiplication over addition.
- The same rule works for subtraction: a × (b − c) = a × b − a × c.
- Use distribution to make mental calculations easier by breaking numbers apart.
- When expanding brackets in algebra, multiply the outside term with every term inside.
- Factoring is the reverse: pull out a common factor to write a sum as a product.