Four operations with integers, fractions and decimals

What you'll learn

This topic forms the foundation of numerical computation in CIE IGCSE Mathematics, tested directly in Paper 1 (non-calculator) and underpinning all quantitative questions across Papers 2, 3 and 4. You'll master the four operations—addition, subtraction, multiplication and division—applied to integers (positive and negative whole numbers), fractions and decimals, both with and without a calculator.

Key terms and definitions

Integer — a whole number that can be positive, negative or zero (e.g., -3, 0, 15)

Operation — a mathematical process: addition (+), subtraction (−), multiplication (×) or division (÷)

Proper fraction — a fraction where the numerator is smaller than the denominator (e.g., 3/7)

Improper fraction — a fraction where the numerator is greater than or equal to the denominator (e.g., 11/4)

Mixed number — a whole number combined with a proper fraction (e.g., 2 3/4)

Reciprocal — the multiplicative inverse of a number; for a fraction a/b, the reciprocal is b/a

Order of operations (BIDMAS/BODMAS) — the hierarchy for calculating expressions: Brackets, Indices (or Orders), Division and Multiplication (left to right), Addition and Subtraction (left to right)

Lowest common multiple (LCM) — the smallest positive integer divisible by two or more numbers, essential for adding and subtracting fractions with different denominators

Core concepts

Operations with integers

Addition and subtraction with negative numbers

Understanding sign rules forms the basis of integer arithmetic tested throughout CIE IGCSE Mathematics:

• Adding a negative number is equivalent to subtraction: 7 + (−3) = 7 − 3 = 4
• Subtracting a negative number is equivalent to addition: 5 − (−8) = 5 + 8 = 13
• Two negatives in sequence become positive: −(−6) = +6
• Use a number line mentally: moving right adds, moving left subtracts

Multiplication and division with integers

Sign rules for multiplication and division:

• Positive × Positive = Positive: 4 × 5 = 20
• Negative × Negative = Positive: (−6) × (−3) = 18
• Positive × Negative = Negative: 7 × (−2) = −14
• Negative × Positive = Negative: (−9) × 4 = −36

The same rules apply to division:

• 20 ÷ 4 = 5
• (−15) ÷ (−3) = 5
• (−24) ÷ 6 = −4
• 18 ÷ (−9) = −2

Operations with fractions

Addition and subtraction of fractions

CIE IGCSE Mathematics Paper 1 frequently tests fraction operations without a calculator. The essential process:

1. Convert mixed numbers to improper fractions if necessary
2. Find a common denominator (preferably the lowest common multiple)
3. Convert each fraction to equivalent fractions with this common denominator
4. Add or subtract the numerators only
5. Simplify the result and convert back to a mixed number if appropriate

Example: Calculate 2/3 + 5/6

• Common denominator: LCM of 3 and 6 is 6
• 2/3 = 4/6
• 4/6 + 5/6 = 9/6 = 3/2 = 1 1/2

Example: Calculate 3 1/4 − 1 5/8

• Convert: 3 1/4 = 13/4 and 1 5/8 = 13/8
• Common denominator: 8
• 13/4 = 26/8
• 26/8 − 13/8 = 13/8 = 1 5/8

Multiplication of fractions

The process is simpler than addition:

1. Convert mixed numbers to improper fractions
2. Multiply numerators together
3. Multiply denominators together
4. Simplify by cancelling common factors (can be done before or after multiplication)

Example: 2/5 × 3/7 = 6/35

Example: 2 2/3 × 1 1/4 = 8/3 × 5/4 = 40/12 = 10/3 = 3 1/3

Cancelling before multiplying often simplifies calculation:

3/8 × 4/9 = (3×4)/(8×9) but cancel first: 3/8 × 4/9 = 1/2 × 1/3 = 1/6

Division of fractions

Division by a fraction means multiplication by its reciprocal:

1. Convert mixed numbers to improper fractions
2. Change ÷ to × and flip (invert) the second fraction
3. Proceed as with multiplication
4. Simplify the result

Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8

Example: 2 1/3 ÷ 1 3/4 = 7/3 ÷ 7/4 = 7/3 × 4/7 = 4/3 = 1 1/3

Operations with decimals

Addition and subtraction of decimals

Critical for non-calculator papers:

• Align decimal points vertically
• Add placeholder zeros if necessary to make columns equal
• Add or subtract as normal integers, keeping the decimal point aligned

Example: 23.47 + 8.9

  23.47
+  8.90
-------
  32.37

Example: 15.6 − 7.83

  15.60
-  7.83
-------
   7.77

Multiplication of decimals

1. Ignore decimal points initially and multiply as whole numbers
2. Count total decimal places in both original numbers
3. Place the decimal point in the answer so it has that many decimal places

Example: 3.2 × 1.5

• 32 × 15 = 480
• Total decimal places: 1 + 1 = 2
• Answer: 4.80 = 4.8

Example: 0.06 × 0.4

• 6 × 4 = 24
• Total decimal places: 2 + 1 = 3
• Answer: 0.024

Division of decimals

Two main scenarios in CIE IGCSE examinations:

Method 1: Division by a whole number

• Divide normally, keeping the decimal point aligned

Example: 14.7 ÷ 3 = 4.9

Method 2: Division by a decimal

• Multiply both numbers by 10, 100, or 1000 to make the divisor a whole number
• Then divide

Example: 8.4 ÷ 0.7

• Multiply both by 10: 84 ÷ 7 = 12

Example: 3.75 ÷ 0.15

• Multiply both by 100: 375 ÷ 15 = 25

Converting between fractions and decimals

Essential for choosing the most efficient calculation method:

Fraction to decimal: Divide the numerator by the denominator

• 3/8 = 3 ÷ 8 = 0.375
• 2/3 = 2 ÷ 3 = 0.666... = 0.6̇ (recurring decimal)

Decimal to fraction:

1. Write the decimal as a fraction over the appropriate power of 10
2. Simplify

Example: 0.45 = 45/100 = 9/20

Example: 0.125 = 125/1000 = 1/8

Combined operations and order of operations

CIE IGCSE questions often test multiple operations in one expression. Apply BIDMAS/BODMAS strictly:

Example: 12 − 3 × 2 + 8 ÷ 4

• Multiplication first: 3 × 2 = 6
• Division next: 8 ÷ 4 = 2
• Left to right for addition/subtraction: 12 − 6 + 2 = 8

Example with fractions: 1/2 + 1/3 × 3/4

• Multiplication first: 1/3 × 3/4 = 3/12 = 1/4
• Then addition: 1/2 + 1/4 = 2/4 + 1/4 = 3/4

Worked examples

Example 1 (Paper 1 style — non-calculator)

Calculate 3 3/4 − 1 5/8 + 2 1/2

Solution:

Convert to improper fractions:

• 3 3/4 = 15/4
• 1 5/8 = 13/8
• 2 1/2 = 5/2

Find common denominator (LCM of 4, 8, 2 is 8):

• 15/4 = 30/8
• 13/8 = 13/8
• 5/2 = 20/8

Calculate: 30/8 − 13/8 + 20/8 = 37/8

Convert to mixed number: 37/8 = 4 5/8

Answer: 4 5/8 [3 marks]

Example 2 (Paper 1 style — non-calculator)

Calculate (−5) × 3 − (−2) × (−4)

Solution:

Apply multiplication rules for negative numbers:

• (−5) × 3 = −15
• (−2) × (−4) = +8

Then subtract: −15 − 8 = −23

Answer: −23 [2 marks]

Example 3 (Mixed operations)

A recipe requires 2 2/3 cups of flour. Sarah wants to make 1 1/2 times the recipe. How much flour does she need? Give your answer as a mixed number.

Solution:

Calculate 2 2/3 × 1 1/2

Convert to improper fractions:

• 2 2/3 = 8/3
• 1 1/2 = 3/2

Multiply: 8/3 × 3/2 = 24/6 = 4

Answer: 4 cups [3 marks]

Common mistakes and how to avoid them

• Mistake: Adding or subtracting fractions by adding/subtracting both numerators and denominators (e.g., 1/2 + 1/3 = 2/5). Correction: Always find a common denominator first; only add or subtract the numerators.

• Mistake: When dividing fractions, flipping the first fraction instead of the second (e.g., 3/4 ÷ 2/3 = 4/3 × 2/3). Correction: Keep the first fraction unchanged, change ÷ to ×, and flip only the second fraction (reciprocal).

• Mistake: Misplacing the decimal point in multiplication (e.g., 0.3 × 0.2 = 0.6). Correction: Count total decimal places: 0.3 × 0.2 should have 2 decimal places, so 0.06 not 0.6.

• Mistake: Applying BIDMAS incorrectly, particularly doing addition before multiplication (e.g., 2 + 3 × 4 = 20). Correction: Always complete multiplication and division before addition and subtraction: 2 + 3 × 4 = 2 + 12 = 14.

• Mistake: Sign errors with integers, especially double negatives (e.g., 5 − (−3) = 2). Correction: Subtracting a negative becomes addition: 5 − (−3) = 5 + 3 = 8.

• Mistake: Forgetting to convert mixed numbers to improper fractions before multiplying or dividing. Correction: Always convert first: 2 1/2 × 3 must become 5/2 × 3 = 15/2 = 7 1/2.

Exam technique for four operations with integers, fractions and decimals

• Paper 1 (non-calculator) tests fraction and decimal operations extensively. Show clear working in stages—examiners award method marks even if the final answer is incorrect. Write conversions explicitly (e.g., show 2 3/4 = 11/4).

• Command words: "Calculate" requires a numerical answer with working; "evaluate" means the same; "simplify" means reduce fractions to lowest terms or combine like terms; "express as a mixed number" or "express as an improper fraction" specifies the required form.

• Mark allocation patterns: 1 mark for simple single operations, 2-3 marks for multi-step fraction problems, 1 mark often given for correct method even with arithmetic errors. Always show intermediate steps to capture method marks.

• Accuracy requirements: Unless stated otherwise, fraction answers should be in simplest form, and recurring decimals should use dot notation (e.g., 0.3̇). Check the question for specific instructions about answer format.

Quick revision summary

Master integer sign rules: negative × negative = positive, negative × positive = negative. For fractions, find common denominators to add/subtract; multiply straight across; divide by flipping the second fraction. Decimal operations require careful alignment of decimal points (addition/subtraction) or counting decimal places (multiplication). Always convert mixed numbers to improper fractions before multiplying or dividing. Apply BIDMAS/BODMAS strictly in combined operations. Show all working on Paper 1 non-calculator questions to secure method marks.