Brackets first, then indices, then multiply and divide (left to right), then add and subtract (left to right). For fractions, change mixed numbers to improper fractions first; to divide, multiply by the reciprocal.
Fig. 1: from −7 °C to 5 °C is 7 degrees up to zero and 5 more, a rise of 5 − (−7) = 12 °C
Section 01
Negative numbers.
In the lab, set a starting number and a second number and step along the number line: adding a negative moves left, subtracting a negative moves right. Then try multiplying two negatives and check the sign rule below.
Subtracting a negative is adding: \(-3 - (-8) = -3 + 8 = 5\).
Adding a negative is subtracting: \(4 + (-9) = 4 - 9 = -5\).
Multiplying or dividing: same signs give a positive answer, different signs give a negative: \((-6) \times (-4) = 24\), \(-15 \div 3 = -5\).
Worked example Temperature changes
(a) The temperature rises from \(-7\) °C to 5 °C. By how much does it rise? (b) At 6 pm it is 3 °C; overnight it falls by 9 degrees. What is the new temperature?
\(-3^{2} = -9\) because the square is done before the minus sign, but \((-3)^{2} = 9\). On a calculator, type the brackets.
Section 02
Order of operations.
Brackets, Indices, Division and Multiplication, Addition and Subtraction. Division and multiplication have equal rank, so work left to right; the same for addition and subtraction.
\(20 - 12 + 2\) is 10, not 6. Addition does not come before subtraction: work left to right.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Work out \(5 + 3 \times 2\).
Quick check+10 XP
Work out \((-8) \div (-2)\).
Quick check+10 XP
Work out \(\dfrac{2}{5} + \dfrac{1}{3}\).
Quick check+10 XP
Work out \(0.2 \times 0.3\).
Section 03
Fractions: all four operations.
Change mixed numbers to improper fractions first. Add and subtract over a common denominator; multiply tops and bottoms; divide by multiplying by the reciprocal. Show every step: these are non-calculator questions.
Worked example Add and subtract mixed numbers
Without a calculator, work out (a) \(2\dfrac{1}{3} + 1\dfrac{3}{4}\) (b) \(3\dfrac{1}{5} - 1\dfrac{2}{3}\).
\(\dfrac{2}{5} + \dfrac{1}{3}\) is NOT \(\dfrac{3}{8}\). Never add the bottoms: use a common denominator, \(\dfrac{6}{15} + \dfrac{5}{15} = \dfrac{11}{15}\).
Section 04
Decimals without a calculator.
Line up the decimal points to add and subtract. To multiply, ignore the points, multiply, then put back as many decimal places as the question had in total. To divide, multiply both numbers by 10, 100, ... until you divide by a whole number.
Worked example Multiply and divide decimals
Without a calculator, work out (a) \(0.3 \times 0.04\) (b) \(4.8 \div 0.06\) (c) \(12.6 - 3.85\).
(a) \(3 \times 4 = 12\); there are \(1 + 2 = 3\) decimal places in the question, so \(0.012\).
(b) Multiply both by 100: \(480 \div 6 = 80\).
(c) Line up: \(12.60 - 3.85 = 8.75\).
Answer: (a) 0.012 (b) 80 (c) 8.75
Examiner trap
Dividing by a number less than 1 makes the answer BIGGER: \(4.8 \div 0.06 = 80\), not 0.8. Check the size of your answer makes sense.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
Without a calculator, work out \(2\dfrac{1}{2} \div 1\dfrac{1}{4}\).
Exam style+20 XP
At midnight the temperature is \(-4\) °C. By noon it has risen by 11 degrees, and by the next midnight it has fallen by 15 degrees from the noon value. What is the temperature at the next midnight?
Exam style+20 XP
Work out \(3 - 2 \times (5 - 8)^{2}\).
Exam style+20 XP
Without a calculator, work out \(1\dfrac{2}{3} - \dfrac{3}{4} \times \dfrac{2}{3}\).
Skill unlocked
The four operations, sorted.
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