The four operations
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Order of Operations and Brackets
When a calculation mixes several operations, the order they are carried out in changes the answer. A fixed set of rules - often remembered as BIDMAS - decides which operation happens first.
- BIDMAS gives the order to work through a calculation:
- Brackets, then Indices, then Division and Multiplication (left to right), then Addition and Subtraction (left to right)
- Brackets always take priority - whatever is inside them is worked out completely before anything outside is considered
Worked Example: Applying BIDMAS
- Question: Work out 3 + 4 × (7-2) ÷ 5.
- Step 1: Brackets first: 7 - 2 = 5
- Step 2: Multiplication and division, left to right: 4 × 5 = 20, then 20 ÷ 5 = 4
- Step 3: Addition last: 3 + 4 = 7
- Answer: 7
Worked Example: Inserting Brackets to Make a Calculation Correct
- Question: Put one pair of brackets into 5 × 6 - 2 + 4 = 40 to make it correct.
- Step 1: Without brackets, BIDMAS gives 5 × 6 - 2 + 4 = 30 - 2 + 4 = 32, not 40
- Step 2: To reach 40, try forcing the addition and subtraction to happen first: 5 × (6 - 2 + 4)
- Step 3: Check: 6 - 2 + 4 = 8, then 5 × 8 = 40 ✓
- Answer: 5 × (6 - 2 + 4) = 40
Common Mistakes
MistakeWorking strictly left to right, ignoring BIDMAS, e.g. evaluating 3 + 4 × 5 as (3 + 4) × 5 = 35
Fixmultiplication and division are always done before addition and subtraction, regardless of left-to-right position: 3 + 4 × 5 = 3 + 20 = 23
MistakeDoing addition before multiplication just because it appears first when reading left to right
Fixalways scan the whole calculation for brackets and indices first, then multiplication/division, and only then addition/subtraction
The Four Operations with Negative Numbers
Adding, subtracting, multiplying and dividing all follow consistent rules once negative numbers are involved. Practical contexts like temperature changes are a common place these rules are applied.
- Adding a negative number is the same as subtracting: a + (-b) = a - b
- Subtracting a negative number is the same as adding: a - (-b) = a + b
- Multiplying or dividing two numbers with the same sign gives a positive result
- Multiplying or dividing two numbers with different signs gives a negative result
Worked Example: Temperature Changes
- Question: The temperature at midnight was -6°C. By midday it had risen by 11°C, then by evening it had fallen by 9°C. What was the evening temperature?
- Step 1: Rising by 11°C from -6°C: -6 + 11 = 5
- Step 2: Falling by 9°C from 5°C: 5 - 9 = -4
- Answer: -4°C
Worked Example: Multiplying and Dividing Negative Numbers
- -8 × -3 = 24 (same signs, positive result)
- 6 × -7 = -42 (different signs, negative result)
- -20 ÷ 4 = -5 (different signs, negative result)
Common Mistakes
MistakeTreating "subtract a negative" as if it stays subtraction, e.g. working out -6 - (-4) as -6 - 4 = -10
Fixsubtracting a negative number is the same as adding: -6 - (-4) = -6 + 4 = -2
MistakeAssuming a negative multiplied by a negative stays negative, e.g. writing -8 × -3 = -24
Fixtwo numbers with the same sign always give a positive result when multiplied or divided: -8 × -3 = 24
The Four Operations with Fractions and Mixed Numbers
Each operation on fractions has its own method - addition and subtraction need a common denominator, while multiplication and division work directly with the numerators and denominators. Mixed numbers should always be converted to improper fractions first.
- To add or subtract fractions, first rewrite them with a common denominator, then add or subtract the numerators
- To multiply fractions, multiply the numerators together and multiply the denominators together - a common denominator is not needed
- To divide by a fraction, multiply by its reciprocal (flip the second fraction upside down)
- Before multiplying or dividing a mixed number, always convert it to an improper fraction first
Worked Example: Adding Mixed Numbers
- Question: Work out 2(1/3) + 1(3/4).
- Step 1: Convert to improper fractions: 2(1/3) = (7/3), 1(3/4) = (7/4)
- Step 2: Rewrite with a common denominator of 12: (28/12) + (21/12) = (49/12)
- Answer: (49/12) = 4(1/12)
Worked Example: Multiplying and Dividing Fractions
- Question: Work out (i) (3/5) × (2/9), (ii) (5/6) ÷ (2/3).
- Step 1 (i): Multiply numerators and denominators: (3 × 2/5 × 9) = (6/45), then simplify by dividing by 3: (2/15)
- Step 1 (ii): Multiply by the reciprocal of 2/3: (5/6) × (3/2) = (15/12), then simplify: (5/4) = 1(1/4)
- Answer: (i) 2/15, (ii) 1¼
Common Mistakes
MistakeAdding fractions by adding the numerators and denominators directly, e.g. writing 1/3 + 3/4 = 4/7
Fixfractions must share a common denominator before adding: convert to twelfths first, then add the numerators only
MistakeMultiplying mixed numbers directly without converting to improper fractions first
Fixalways convert a mixed number to an improper fraction before multiplying or dividing, then convert the answer back to a mixed number if needed
MistakeForgetting to flip the second fraction when dividing, e.g. working out 5/6 ÷ 2/3 as 5/6 × 2/3
Fixdividing by a fraction means multiplying by its reciprocal - only the second fraction gets flipped
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