N2: Four Operations
Add, subtract, multiply and divide integers, decimals and fractions; mixed numbers; place value; financial terms
📋 Key Concepts
The Four Operations: Addition (+), Subtraction (−), Multiplication (×), and Division (÷) are the foundation of arithmetic.
Place Value
| Place | Value |
|---|---|
| Thousands | 1000 |
| Hundreds | 100 |
| Tens | 10 |
| Ones | 1 |
| Tenths | 0.1 |
| Hundredths | 0.01 |
📝 Working with Integers
Adding Integers: Same signs → add and keep the sign. Different signs → subtract the smaller from the larger, keep the sign of the larger number.
Example 1
Calculate: (-7) + (-5)
Solution: -7 + (-5) = -12
Both negative, so add: 7 + 5 = 12, keep negative sign.
Example 2
Calculate: 15 + (-8)
Solution: 15 + (-8) = 7
Different signs: 15 - 8 = 7, positive is larger.
Subtracting Integers: Subtracting a negative is the same as adding a positive. a - (-b) = a + b
Example 3
Calculate: 12 - (-5)
Solution: 12 - (-5) = 12 + 5 = 17
📝 Decimal Operations
Addition/Subtraction: Line up decimal points vertically. Add zeros if needed to make the same number of decimal places.
Example 4
Calculate: 3.45 + 12.7
Solution:
3.45 +12.70 ------ 16.15
Multiplication: Multiply as whole numbers, then count decimal places in both numbers to place the decimal point.
Example 5
Calculate: 2.4 × 3.5
Solution: 24 × 35 = 840
Two decimal places total: 8.40
Division: Make the divisor a whole number by multiplying both numbers by the same power of 10.
Example 6
Calculate: 15.6 ÷ 0.4
Solution: Multiply both by 10: 156 ÷ 4 = 39
📝 Fraction Operations
Addition/Subtraction: Find a common denominator first, then add/subtract the numerators.
Example 7
Calculate: 2/3 + 1/4
Solution: Common denominator = 12
2/3 = 8/12, 1/4 = 3/12
8/12 + 3/12 = 11/12
Multiplication: Multiply numerators together, multiply denominators together. Simplify if possible.
Example 8
Calculate: 3/5 × 2/7
Solution: (3 × 2)/(5 × 7) = 6/35
Division: Multiply by the reciprocal (flip the second fraction).
Example 9
Calculate: 3/4 ÷ 2/5
Solution: 3/4 × 5/2 = 15/8 = 1 7/8
📝 Mixed Numbers
Method: Convert to improper fractions, calculate, then convert back to mixed numbers.
Example 10
Calculate: 2 1/3 × 1 1/2
Solution:
2 1/3 = 7/3, 1 1/2 = 3/2
7/3 × 3/2 = 21/6 = 7/2 = 3 1/2
📝 Financial Terms
| Term | Meaning | Formula |
|---|---|---|
| Profit | Money gained | Revenue - Cost |
| Loss | Money lost | Cost - Revenue |
| Simple Interest | Interest on principal only | P × R × T |
| Compound Interest | Interest on interest | P(1 + R)^T |
Example 11
A shop buys items for £15 and sells them for £23. Find the profit.
Solution: Profit = £23 - £15 = £8
🧮 Worked Examples — Exam Style
Four operations with mixed number types — integers, decimals, fractions. Each has one hazard: subtracting negatives, decimal alignment, common denominators, flip-and-multiply.
Calculate: −7 − (−12)
Subtracting a negative ADDS: −7 + 12 = 5
Number line: start at −7, move 12 right.
B1 for 5. 'Two minuses make a plus' — but only for subtraction of a negative and multiplication of two negatives.
Calculate: 2/3 × 3/4
Multiply straight across: (2 × 3)/(3 × 4) = 6/12
Simplify: ÷ 6 → 1/2
B1 for 1/2. Multiplication needs NO common denominator — that's only for + and −.
Exam wording to expect: "Work out …", "Give your answer as a fraction in its simplest form".
❓ Practice Questions
Q1: Calculate: (-8) + 15 - (-3)
Q2: Calculate: 4.56 + 2.8
Q3: Calculate: 0.6 × 2.4
Q4: Calculate: 3/8 + 1/6
Q5: Calculate: 5/6 ÷ 2/3
Q6: Calculate: 2¾ + 1½
Q7: Calculate: −3 × (−4) + 2 × (−5)
✅ Answers
- -8 + 15 + 3 = 10
- 4.56 + 2.80 = 7.36
- 6 × 24 = 144 → two decimal places → 1.44
- Common denom. 24: 9/24 + 4/24 = 13/24
- 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4
🧮 Worked Examples — Harder Exam Style
Work out 3½ × 2⅔, giving your answer as a mixed number.
Convert to improper fractions: 3½ = 7/2 and 2⅔ = 8/3
Multiply: (7 × 8)/(2 × 3) = 56/6
Simplify: ÷ 2 → 28/3 = 9⅓
M1 for converting both, M1 for 56/6, A1 for 9⅓. Mixed-number multiplication REQUIRES improper conversion — multiplying the whole parts separately is the error.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. −8 + 15 − (−3) = −8 + 15 + 3 = 10
Q2. Line up decimal points: 4.56 + 2.80 = 7.36
Q3. 6 × 24 = 144, then place TWO decimal places: 1.44
Q4. Common denominator 24: 9/24 + 4/24 = 13/24 (already simplest)
Q5. Flip and multiply: 5/6 × 3/2 = 15/12 = 5/4 = 1¼
Q6. 2¾ + 1½ = 2¾ + 1¾ (½ = 2/4) = 3 + 5/4 = 4¼
Q7. (−3)(−4) = +12 and (2)(−5) = −10 → 12 − 10 = 2
📌 Key Facts to Memorise
- Minus × minus = plus; minus ÷ minus = plus; subtracting a negative adds
- Decimal + / −: line up the points; ×: count total decimal places; ÷: make the divisor a whole number
- Fractions + / −: common denominator first; ×: straight across; ÷: flip the second, then ×
- Mixed numbers: convert to improper fractions before multiplying/dividing
- Financial terms: profit, loss, deposit, balance, discount
🗺️ In the Exam
Non-calculator arithmetic is the backbone of Paper 1: nearly every question starts here. Decimal × decimal and fraction ÷ fraction are the two discriminating skills.
Links to: order of operations (N3), fractions of amounts (R3), money problems (N12).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always check your answer is sensible (e.g., multiplying by a number less than 1 gives a smaller answer)
- For negative numbers, draw a number line if stuck
- When dividing fractions, "keep it, flip it, multiply"
- Watch out for mixed numbers - convert them first
- In finance questions, profit = selling price - cost price
🧠 Problem-Solving Strategies
Problem-Solving
When tackling multi-step calculations with mixed formats, work step by step and convert everything to the same form first. For financial problems, identify whether you need profit/loss, then choose the correct operation. Always check if your answer is sensible — e.g., multiplying by a fraction less than 1 should give a smaller answer.
Multi-Step Problem
A shop buys 24 items at £3.50 each. It sells 18 of them at £5.25 each and the rest at £2 each in a clearance sale. Calculate the total profit or loss.
Solution: Total cost = 24 × £3.50 = £84. Revenue from full price = 18 × £5.25 = £94.50. Remaining items = 24 − 18 = 6. Clearance revenue = 6 × £2 = £12. Total revenue = £94.50 + £12 = £106.50. Profit = £106.50 − £84 = £22.50.
⚠️ Common Errors
Watch Out!
1. Wrong: 3 − (−5) = −2 Correct: 3 − (−5) = 3 + 5 = 8 (subtracting a negative is adding)
2. Wrong: 2/3 + 1/4 = 3/7 (adding numerators and denominators) Correct: Find common denominator: 8/12 + 3/12 = 11/12
3. Wrong: 0.6 × 0.4 = 2.4 (ignoring decimal places) Correct: 6 × 4 = 24, two decimal places total: 0.24
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A baker makes 3 1/2 batches of muffins. Each batch needs 2 1/4 cups of flour. He already has 4 cups. How many more cups of flour does he need? Show all working clearly.
Total flour needed = 3 1/2 × 2 1/4 = 7/2 × 9/4 = 63/8 = 7 7/8 cups.
Flour already has = 4 cups.
Extra needed = 7 7/8 − 4 = 3 7/8 cups.
Mark scheme: 1 mark for converting to improper fractions, 2 marks for correct multiplication, 1 mark for converting answer to mixed number, 2 marks for correct subtraction and final answer
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Two friends go out for a meal. The bill is £72. They split it in the ratio 5:4. The person paying the larger share also leaves a 10% tip on their portion.
(a) How much does each person pay before the tip?
(b) How much does the person paying the tip pay in total?
(c) Is it fair to tip only on one person's share? Give a mathematical reason for your answer.
Answers: (a) Total parts = 9. Larger share = 5/9 × £72 = £40. Smaller share = 4/9 × £72 = £32. (b) Tip = 10% of £40 = £4. Total = £44. (c) A tip is usually calculated on the whole bill. 10% of £72 = £7.20, so tipping only on one share means the tip is lower (£4 vs £7.20). Whether this is "fair" depends on context — they only ate their own portion.
📝 Exam Questions by Topic
🎬 Video Resources
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