The four operations

1. Overview

The four operations – addition, subtraction, multiplication, and division – are the foundation of numerical calculations in IGCSE Mathematics. This revision note covers performing these operations accurately with integers, fractions, and decimals, with a strong emphasis on the correct order of operations (BIDMAS/BODMAS) and the strategic use of brackets. Mastery of these skills is crucial for success in both Core and Extended level papers.

Key Definitions

• Sum: The result of adding two or more numbers together.
• Difference: The result of subtracting one number from another.
• Product: The result of multiplying two or more numbers together.
• Quotient: The result of dividing one number by another.
• Integer: A whole number that can be positive, negative, or zero (e.g., -3, 0, 7).
• Reciprocal: The result of dividing 1 by a number (e.g., the reciprocal of 5 is $\frac{1}{5}$).
• BIDMAS/BODMAS: An acronym used to remember the order of operations (Brackets, Indices/Orders, Division/Multiplication, Addition/Subtraction).

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Core Content

The Order of Operations (BIDMAS)

To get the correct answer, operations must be performed in this specific order:

1. Brackets (Work inside these first)
2. Indices (Powers and Square Roots)
3. Division and Multiplication (Left to right)
4. Addition and Subtraction (Left to right)

Worked Example 1 — Applying BIDMAS

Evaluate: $20 - (4 + 2) \times 3$

• Step 1 (Brackets): $4 + 2 = 6$
  • Reason: Evaluate the expression within the brackets.
• Step 2: $20 - 6 \times 3$
  • Reason: Substitute $(4 + 2)$ with $6$.
• Step 3 (Multiplication): $6 \times 3 = 18$
  • Reason: Perform the multiplication.
• Step 4: $20 - 18$
  • Reason: Substitute $6 \times 3$ with $18$.
• Step 5 (Subtraction): $20 - 18 = 2$
  • Reason: Perform the subtraction.

Final Answer: 2

Operations with Decimals

When adding or subtracting decimals, you must align the decimal points vertically. For multiplication, multiply as whole numbers then count the total decimal places.

Worked Example 2 — Subtraction with Decimals Calculate $23.75 - 8.92$

• Step 1: Align decimals:

    23.75
  - 08.92
  -------
  

  • Reason: Prepare for subtraction by aligning place values.
• Step 2: Subtract starting from the right (borrowing where necessary).

    23.75
  - 08.92
  -------
    14.83
  

  • Reason: Perform the subtraction column by column, borrowing when the top digit is smaller than the bottom digit.

Final Answer: 14.83

Operations with Fractions

• Addition/Subtraction: Find a Common Denominator first.
• Multiplication: Multiply numerators together and denominators together.
• Division: Multiply by the reciprocal of the second fraction ("Flip and Multiply").

Worked Example 3 — Fraction Division Calculate $\frac{2}{3} \div \frac{3}{4}$

• Step 1: Find the reciprocal of the second fraction: $\frac{4}{3}$
  • Reason: To divide by a fraction, we multiply by its reciprocal.
• Step 2: Rewrite the division as multiplication: $\frac{2}{3} \times \frac{4}{3}$
  • Reason: Applying the "flip and multiply" rule.
• Step 3: Multiply the numerators and the denominators: $\frac{2 \times 4}{3 \times 3} = \frac{8}{9}$
  • Reason: Multiplying fractions.

Final Answer: $\frac{8}{9}$

Worked Example 4 — Combining Operations with Fractions

Evaluate: $\frac{1}{2} + \frac{2}{5} \times \frac{5}{4}$

• Step 1 (Multiplication): $\frac{2}{5} \times \frac{5}{4} = \frac{2 \times 5}{5 \times 4} = \frac{10}{20}$
  • Reason: Perform multiplication before addition, according to BIDMAS.
• Step 2 (Simplify): $\frac{10}{20} = \frac{1}{2}$
  • Reason: Simplify the fraction by dividing numerator and denominator by their greatest common divisor (10).
• Step 3 (Addition): $\frac{1}{2} + \frac{1}{2} = \frac{1+1}{2} = \frac{2}{2}$
  • Reason: Add the fractions, since they now have a common denominator.
• Step 4 (Simplify): $\frac{2}{2} = 1$
  • Reason: Simplify the fraction.

Final Answer: 1

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Extended Content (Extended Only)

While the Core syllabus focuses on direct application of the four operations, the Extended syllabus requires you to apply these operations in more complex algebraic and problem-solving contexts. This includes manipulating expressions with multiple brackets, negative numbers, and fractional powers, often within the same problem. A deeper understanding of number properties and the ability to strategically apply BIDMAS are essential.

Worked Example 5 — Algebraic Simplification with Multiple Brackets Simplify: $3[2(x + 1) - 4(2 - x)]$

• Step 1 (Inner Brackets - First Term): $2(x + 1) = 2x + 2$
  • Reason: Expand the first inner bracket using the distributive property.
• Step 2 (Inner Brackets - Second Term): $-4(2 - x) = -8 + 4x$
  • Reason: Expand the second inner bracket using the distributive property.
• Step 3 (Substitute): $3[2x + 2 - 8 + 4x]$
  • Reason: Substitute the expanded terms back into the original expression.
• Step 4 (Combine Like Terms): $3[6x - 6]$
  • Reason: Combine the 'x' terms ($2x + 4x = 6x$) and the constant terms ($2 - 8 = -6$) inside the outer brackets.
• Step 5 (Outer Bracket): $3 \times 6x - 3 \times 6 = 18x - 18$
  • Reason: Expand the outer bracket using the distributive property.

Final Answer: $18x - 18$

Worked Example 6 — Combining Fractions and Algebra Simplify: $\frac{3x - 1}{2} + \frac{x + 3}{5}$

• Step 1 (Common Denominator): The lowest common denominator of 2 and 5 is 10.
  • Reason: To add fractions, they must have a common denominator.
• Step 2 (Rewrite Fractions): $\frac{5(3x - 1)}{10} + \frac{2(x + 3)}{10}$
  • Reason: Multiply the first fraction by $\frac{5}{5}$ and the second by $\frac{2}{2}$ to obtain the common denominator.
• Step 3 (Expand Numerators): $\frac{15x - 5}{10} + \frac{2x + 6}{10}$
  • Reason: Expand the numerators using the distributive property.
• Step 4 (Combine Fractions): $\frac{(15x - 5) + (2x + 6)}{10}$
  • Reason: Combine the fractions over the common denominator.
• Step 5 (Simplify Numerator): $\frac{15x - 5 + 2x + 6}{10}$
  • Reason: Remove the brackets in the numerator.
• Step 6 (Combine Like Terms): $\frac{17x + 1}{10}$
  • Reason: Combine the 'x' terms ($15x + 2x = 17x$) and the constant terms ($-5 + 6 = 1$) in the numerator.

Final Answer: $\frac{17x + 1}{10}$

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Key Equations

While there are no formulas on the IGCSE formula sheet for basic operations, the following rules must be memorised:

Adding Fractions: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Multiplying Fractions: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Dividing Fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

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Common Mistakes to Avoid

• ❌ Incorrect Order of Operations: Evaluating $5 + 2 \times 3$ as $(5 + 2) \times 3 = 21$ is wrong.
  • ✓ Right: $5 + 2 \times 3 = 5 + 6 = 11$ (Multiplication before addition).
• ❌ Forgetting the Negative Sign: When expanding brackets, especially with subtraction, forgetting to distribute the negative sign can lead to errors. E.g., $4 - (x - 2)$ becomes $4 - x - 2$ (incorrect) instead of $4 - x + 2$.
  • ✓ Right: $4 - (x - 2) = 4 - x + 2 = 6 - x$.
• ❌ Decimal Alignment Errors: When adding or subtracting decimals, not aligning the decimal points vertically will result in an incorrect answer.
  • ✓ Right: Align the decimal points:

      12.34
    + 0.567
    -------
      12.907
    

• ❌ Incorrect Reciprocal: When dividing fractions, inverting the wrong fraction or failing to invert at all.
  • ✓ Right: $\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$.

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Exam Tips

• Show Your Working: Even if you use a calculator, write down the intermediate steps. If you make one small typo but your method is correct, you can still get "Method Marks."
• Command Words:
  • Evaluate/Calculate: Find the final numerical value.
  • Simplify: Reduce the expression to its smallest form.
• Calculator vs. Non-Calculator: In non-calculator sections, use "long multiplication" and "bus stop division." In calculator sections, use the fraction button ($\frac{\square}{\square}$) to ensure the order of operations is handled correctly by the device.
• Real-world Context: These questions often appear as "money" problems. Always give money answers to 2 decimal places (e.g., $$5.20$, not $$5.2$).
• Typical Values: Be familiar with common squares (up to $15^2$) and cubes (up to $5^3$) to speed up non-calculator calculations.
• Double Check: After completing a calculation, especially in non-calculator papers, quickly estimate the answer to see if your result is reasonable. This can help you catch arithmetic errors.