The four operations on integers, decimals and fractions
The "four operations" are addition, subtraction, multiplication and division. You'll need them on every paper, often without a calculator. The methods below are the ones AQA examiners expect to see.
Integers — column methods
For multi-digit addition and subtraction, line up place values and work right-to-left, carrying or borrowing as needed.
Worked example: 4862 + 379. Line up units, then add column-by-column with carries: answer 5241.
For long multiplication, use either the grid method (foundation friendly) or column long multiplication (faster). Long division uses the "bus stop" or chunking method.
Negative numbers — sign rules
- Same sign × or ÷ same sign → positive.
- Different signs × or ÷ different signs → negative.
- For + and −: think of a number line. Two negatives next to each other = +. So 5 − (−3) = 5 + 3 = 8.
Worked example: (−4) × (−6) = +24. (−12) ÷ 3 = −4. 7 − (−2) = 9.
Decimals — keep place value tidy
Adding/subtracting decimals. Line up the decimal points, pad with zeros, then add or subtract as integers. The decimal point in the answer sits underneath.
Multiplying decimals. Ignore the decimal points and multiply as integers. Then count the total number of decimal places in the question and place that many in the answer. Example: 0.4 × 0.03. Multiply 4 × 3 = 12. Total d.p. = 1 + 2 = 3 → 0.012.
Dividing decimals. Multiply both numbers by 10, 100, … to make the divisor a whole number, then divide normally. Example: 6.4 ÷ 0.2. Multiply both by 10: 64 ÷ 2 = 32.
Fractions — the four operations
Same denominator + or −: add (or subtract) numerators, keep the denominator.
Different denominators: find a common denominator (LCM of the two denominators is most efficient), rewrite both fractions, then add or subtract. Example: ⅔ + ¼. LCM = 12. ⅔ = 8/12; ¼ = 3/12. Sum = 11/12.
Multiplication: multiply numerators, multiply denominators, simplify. Example: ⅔ × ⁹⁄₁₀ = ¹⁸⁄₃₀ = ⅗. (Cancel before multiplying for speed: cross-cancel the 3 and 9 → ⅔ × ⁹⁄₁₀ = ¹⁄₁ × ³⁄₅ = ⅗.)
Division: "keep, change, flip" — keep the first fraction, change ÷ to ×, flip the second. Example: ¾ ÷ ½ = ¾ × 2/1 = 6/4 = 1½.
Mixed numbers must be converted to improper fractions before any × or ÷.
⚠Common mistakes— Common mistakes (examiner traps)
- "Two negatives" rule applied to addition. Only × and ÷ flip signs for two negatives. With + and −, work along the number line.
- Adding fractions by adding numerators AND denominators. Wrong: ⅓ + ¼ ≠ 2/7. The correct answer is 7/12.
- Forgetting to count total d.p. in decimal multiplication.
- Forgetting to simplify the final fraction. AQA mark schemes nearly always want the answer in lowest terms.
- Mishandling negative subtraction. 6 − (−4) becomes 6 + 4 = 10, not 6 − 4 = 2.
➜Try this— Quick check
Calculate without a calculator: (a) 17.4 − 8.65 = ? (b) 0.06 × 0.5 = ? (c) ⅖ + ¾ = ?
Answers: (a) 8.75; (b) 0.030 (= 0.03); (c) 8/20 + 15/20 = 23/20 = 1³⁄₂₀.
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