We Distribute, Yet Things Multiply

Overview

This chapter develops the distributive law and uses it to multiply expressions and numbers efficiently. The idea that a × (b + c) = a × b + a × c underlies mental maths and all of algebraic multiplication.

Key concepts

• Multiplication distributes over addition and subtraction.
• Splitting a number (e.g. 103 = 100 + 3) makes multiplication easier.
• The area model of a rectangle shows distribution visually.
• The same law expands products of binomials in algebra.

Important formulae

• a × (b + c) = a × b + a × c
• a × (b − c) = a × b − a × c
• (a + b)(c + d) = ac + ad + bc + bd

Solved example

1. Compute 25 × 104 using distribution.
2. Write 104 as 100 + 4.
3. 25 × (100 + 4) = 25 × 100 + 25 × 4 = 2500 + 100 = 2600.

Important questions

1. Use the distributive law to evaluate 99 × 47.
2. Expand (x + 5)(x + 3).
3. Show that 12 × (8 + 2) = 12 × 8 + 12 × 2.
4. Find 17 × 6 + 17 × 4 by taking out the common factor.

Quick revision

The distributive law lets you break a multiplication into easier parts and rebuild it. Use the area model to picture it. The same rule expands binomial products, the bridge to algebraic identities.