You use addition, subtraction, multiplication, and division every day, from splitting a bill to scaling a recipe. This guide will help you master these four essential operations with whole numbers, decimals, and fractions, giving you a solid toolkit for any maths challenge!
Operations by Number Type
The key to success is understanding the specific rules for each type of number. Let’s break them down.
🔢 Integers (Whole Numbers)
For large numbers, use formal written methods like column addition/subtraction. For negative numbers, remember the sign rules:
- Subtracting a negative is the same as adding a positive: $2 – (-5) = 2 + 5 = 7$.
- Multiplication/Division Sign Rules:
(+) × (+) = (+)
(−) × (−) = (+)
(+) × (−) = (−)
🔵 Decimals
The crucial rule is to keep track of the decimal point.
- Addition/Subtraction: Always line up the decimal points.
- Multiplication: Multiply as whole numbers, then count the total decimal places in the question and apply that to the answer.
- Division: Make the divisor (the number you’re dividing by) a whole number by multiplying both numbers by 10, 100, etc.
Mastering Fractions
Adding & Subtracting Fractions
You can only add or subtract fractions when they have a common denominator.
- Find a common denominator (the LCM).
- Convert the fractions to their equivalents.
- Add or subtract the numerators only.
- Simplify the result.
Example: $\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$
Multiplying & Dividing Fractions
Always convert mixed numbers to improper fractions first!
- Multiplication: Multiply the numerators together, then multiply the denominators together. Simplify.
- Division (Keep, Change, Flip): Keep the first fraction, change the ÷ to a ×, and flip the second fraction upside down. Then multiply.
Example: $\frac{4}{9} \div \frac{2}{3} = \frac{4}{9} \times \frac{3}{2} = \frac{12}{18} = \frac{2}{3}$
Worked Examples
Integer Addition
$348 + 175$
3 ¹4 ¹8
+ 1 7 5
---------
5 2 3
Start from the right, adding each column and carrying over any tens.
Decimal Subtraction
$9.2 – 3.75$
9 . 2 0
- 3 . 7 5
---------
5 . 4 5
Line up the decimal points, add a trailing zero to 9.2, then borrow where needed working right to left.
Decimal Multiplication
$2.4 \times 1.5$
- Multiply as whole numbers: $24 \times 15 = 360$.
- Count decimal places in the question: $2.4$ (1 place) and $1.5$ (1 place) = 2 places total.
- Apply to the answer: $360 \rightarrow 3.60$.
Answer: 3.6
Context Problem (VAT)
A headset costs £75 + VAT at 20%. What is the total cost?
- Calculate VAT: 20% of £75. (10% is £7.50, so 20% is £15.)
- Add VAT to original cost: £75 + £15.
Answer: £90
Tutor Insights
🤔 Common Misunderstandings
- Sign Rules: Forgetting that subtracting a negative is the same as adding a positive ($a – (-b) = a + b$).
- Decimal Points: Forgetting to line them up for addition/subtraction, or miscounting the places for multiplication.
- Fractions: Trying to add fractions without a common denominator, or forgetting to convert mixed numbers before multiplying/dividing.
📝 Common Exam Mistakes
- Simple calculation errors: Mistakes in times tables or basic arithmetic under pressure.
- Not showing working: You can get method marks for logical steps even if your final answer is wrong.
- Not simplifying fractions: Leaving an answer as $\frac{10}{20}$ instead of $\frac{1}{2}$ will lose marks.
Practice Questions
- Calculate: (a) $275 + 189$, (b) $-8 \times 6$, (c) $56 \div (-7)$
- Calculate: (a) $4.75 + 2.8$, (b) $12.3 – 5.67$, (c) $9.6 \div 0.4$
- Calculate: (a) $\frac{1}{5} + \frac{3}{10}$, (b) $\frac{3}{4} \times \frac{2}{9}$, (c) $\frac{5}{8} \div \frac{3}{4}$
- A shop buys 25 chocolate bars for £10.50. They sell each bar for 65p. What is their total profit if they sell all 25 bars?
Show Answers
- (a) 464, (b) −48, (c) −8
- (a) 7.55, (b) 6.63, (c) 24
- (a) $\frac{1}{2}$, (b) $\frac{1}{6}$, (c) $\frac{5}{6}$
- Total sales = $0.65 \times 25 = £16.25$. Profit = £16.25 − £10.50 = £5.75.
FAQs
Q: Why do I need written methods when I have a calculator?
A: Exams always include a non-calculator paper where you must show your formal written methods. Understanding these methods also helps you spot errors and builds your number sense, which is useful even when using a calculator.
Q: What’s an easy way to remember the sign rules?
A: For multiplication and division: if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
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