Chapter focus
The distributive property connects multiplication with addition and subtraction. It explains mental calculation strategies and provides a bridge from arithmetic to algebra.
These original explanations and examples support practice; use your assigned Ganita Prakash edition for its full activity and exercise sequence.
1. Multiplication over addition
For numbers a, b and c, multiplication distributes over addition:
a(b + c) = ab + ac.
Both terms inside the brackets are multiplied by a. A rectangle model makes this visible: a rectangle of height a and total width b + c has area a(b + c); split it into widths b and c, and the areas add to ab + ac.
The same property works over subtraction: a(b − c) = ab − ac. Keep the negative sign with the term it affects.
2. Use distribution to calculate efficiently
Example: 19 × 7
Write 19 as 20 − 1: (20 − 1) × 7 = 20 × 7 − 1 × 7 = 140 − 7 = 133.
Example: 48 × 25
Write 48 as 50 − 2: 25 × (50 − 2) = 25 × 50 − 25 × 2 = 1,250 − 50 = 1,200.
Choose a nearby round number when it makes the arithmetic easier, then compensate for the amount added or subtracted.
3. Expand algebraic expressions
The same rule applies when letters represent numbers. Multiply every term in the bracket:
3(x + 5) = 3x + 15
−2(4y − 3) = −8y + 6
In the second example, (−2) × (−3) = +6. A sign chart or direct multiplication helps prevent sign errors.
Worked example: simplify 4(2x − 3) + 3(x + 2)
- Expand each bracket: 8x − 12 + 3x + 6.
- Collect like terms: (8x + 3x) + (−12 + 6).
- Simplify: 11x − 6.
Like terms have the same variable part; 11x and −6 cannot be combined further.
4. Factorisation reverses expansion
If each term has a common factor, take it outside the brackets. This reverses distribution:
6x + 18 = 6(x + 3), because 6(x + 3) expands to 6x + 18.
Worked example: factor 15a + 25
- Find the greatest common factor of 15 and 25: it is 5.
- Divide each term by 5: 15a ÷ 5 = 3a and 25 ÷ 5 = 5.
- Write 15a + 25 = 5(3a + 5).
- Check by expanding: 5 × 3a + 5 × 5 = 15a + 25.
Common errors to avoid
- 3(x + 4) is 3x + 12, not 3x + 4.
- Do not distribute multiplication across a product as though it were a sum: a(bc) = abc.
- Watch signs: −3(x − 2) = −3x + 6.
- When factoring, the factor outside must divide every term inside the expression.
Practice: expand, simplify or factor
1. Calculate 99 × 8 using distribution.
Answer: (100 − 1) × 8 = 800 − 8 = 792.
2. Expand 6(x + 4).
Answer: 6x + 24.
3. Expand and simplify 2(3m − 5) + 4(m + 1).
Answer: 6m − 10 + 4m + 4 = 10m − 6.
4. Factor 15a + 25 completely using the greatest common factor.
Answer: 5(3a + 5).
5. A learner writes 7(2p − 3) = 14p − 3. Correct the work.
Answer: Both terms must be multiplied by 7: 7(2p − 3) = 14p − 21.
Chapter summary
Distribution expands a product over a sum or difference, while factorisation takes a common factor back outside. Use the property to calculate mentally, simplify algebra, and verify work by reversing the operation.