# Fractions, Decimals & the Four Operations

> Mathematics · CIE IGCSE 0580 2025-2027
> Source: https://www.owlsprep.com/study/cie-0580-u1-fractions-decimals-the-four-operations/

This guide covers fraction/decimal/percentage conversion, ordering, and four operations (including negatives and mixed numbers) for CIE IGCSE Maths 0580, plus Extended-only recurring decimal content aligned to the 2025–2027 syllabus.

**Prerequisites:** [Basic integer operations and negative number rules](https://www.owlsprep.com/study/cie-0580-u1-integers-negatives/); [BODMAS/BIDMAS order of operations](https://www.owlsprep.com/study/cie-0580-u1-order-of-operations/)

## Learning objectives

- Convert accurately between fractions, decimals and percentages
- Order numerical values using standard inequality symbols
- Perform four operations on fractions, decimals and negatives following BODMAS
- Apply operations to solve problems involving mixed numbers
- (Extended only) Use recurring decimal notation and convert recurring decimals to fractions

## Equivalence & Conversion Between Fractions, Decimals & Percentages

**Fraction-Decimal-Percentage Equivalence** — The property that the same numerical value can be represented as a fraction, decimal, or percentage, using the base rule $1 = 100\%$ for conversions.

**Worked example:** Convert $\frac{3}{8}$ to a decimal and a percentage.

1. Divide the numerator by the denominator to get a decimal:

   $$3 \div 8 = 0.375$$
2. Multiply the decimal by 100 and add a % sign to get a percentage:

   $$0.375 \times 100 = 37.5\%$$

Memorise common equivalents to save time in exams: $\frac{1}{4} = 0.25 = 25\%$, $\frac{1}{3} \approx 0.333 = 33.3\%$, $\frac{1}{2} = 0.5 = 50\%$.

> **Exam tip:** Always give fractions in their simplest form unless the question explicitly says otherwise to avoid losing easy marks.

## Ordering Fractions, Decimals & Integers

**Inequality Symbols for Ordering** — Symbols used to compare the size of two values: $=$ (equal to), $\neq$ (not equal to), $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), $\geq$ (greater than or equal to).

**Worked example:** Order the following values from smallest to largest: $0.6$, $\frac{2}{3}$, $-0.5$, $\frac{3}{4}$, $-1$.

1. Convert all fractions to decimals for easy comparison:

   $$\frac{2}{3} \approx 0.667, \frac{3}{4} = 0.75$$
2. List all values as decimals, sorting negative values first (larger absolute value = smaller number): $-1$, $-0.5$, $0.6$, $0.667$, $0.75$
3. Rewrite in original form: $-1$, $-0.5$, $0.6$, $\frac{2}{3}$, $\frac{3}{4}$

> **Exam tip:** Convert all values to decimals before ordering to avoid mistakes, especially with mixed positive and negative values.

## Four Operations on Fractions, Decimals & Negatives

Follow BODMAS order of operations at all times. For multiplication and division of mixed numbers, convert to improper fractions first. Remember negative number rules: same signs multiply/divide to positive, different signs multiply/divide to negative.

**Worked example:** Calculate $2\frac{1}{2} \times (-1\frac{3}{4}) + 3.2$.

1. Convert mixed numbers to improper fractions:

   $$2\frac{1}{2} = \frac{5}{2}, -1\frac{3}{4} = -\frac{7}{4}$$
2. Perform multiplication first (BODMAS):

   $$\frac{5}{2} \times -\frac{7}{4} = -\frac{35}{8} = -4.375$$
3. Add 3.2 to the result:

   $$-4.375 + 3.2 = -1.175 = -\frac{47}{40}$$

> **Exam tip:** For non-calculator papers, use fraction form for operations to avoid rounding errors, only converting to decimals if the question asks for a decimal answer.

## Extended Adds: Recurring Decimals

**Recurring Decimal** — A decimal that repeats a digit or sequence of digits infinitely, indicated by dots above the first and last digit of the repeating sequence, e.g., $0.\dot{3} = 0.333...$, $0.\dot{1}4\dot{7} = 0.147147147...$.

**Worked example:** Convert $0.\dot{2}\dot{7}$ to a fraction in its simplest form.

1. Let $x$ equal the recurring decimal:

   $$x = 0.272727...$$
2. Multiply by 100 (10 raised to the number of repeating digits, which is 2) to shift the decimal past one full repeating sequence:

   $$100x = 27.272727...$$
3. Subtract the original $x$ value to eliminate the repeating part:

   $$100x - x = 27 \implies 99x = 27$$
4. Rearrange for $x$ and simplify:

   $$x = \frac{27}{99} = \frac{3}{11}$$

> **Exam tip:** For recurring decimals with $n$ repeating digits, multiply by $10^n$ before subtracting to fully eliminate the repeating sequence.

## Common pitfalls

- **Wrong:** Forgetting to convert mixed numbers to improper fractions before multiplying/dividing, multiplying whole number and fraction parts separately.
  - Why it fails: This method produces mathematically incorrect results, as mixed numbers are sums not products of their parts.
  - Correct: Always convert mixed numbers to improper fractions first, perform operations, then convert back to mixed number if required.
- **Wrong:** Ordering negative values as if they are positive, e.g., writing $-0.2 > -0.1$.
  - Why it fails: Negative numbers with larger absolute values are smaller, as they sit further left on the number line.
  - Correct: Convert all values to decimals first, sort negative values by reverse absolute size, then sort positive values by ascending size.
- **Wrong:** Performing addition/subtraction before multiplication/division, ignoring BODMAS.
  - Why it fails: Order of operations is explicitly tested in IGCSE 0580, and incorrect order leads to wrong answers even if individual calculations are correct.
  - Correct: Always follow BODMAS: Brackets first, then Orders, Division/Multiplication left to right, then Addition/Subtraction left to right.
- **Wrong:** (Extended) Using the wrong multiplier when converting recurring decimals to fractions, e.g., multiplying by 10 for a 2-digit repeating sequence.
  - Why it fails: The multiplier must shift the decimal past one full repeating sequence to eliminate all recurring digits when subtracting.
  - Correct: Count the number of repeating digits first, multiply by $10^n$ where $n$ is the number of repeating digits, then subtract the original $x$ value.
- **Wrong:** Leaving fractions unsimplified in final answers.
  - Why it fails: CIE examiners deduct 1 mark per question for unsimplified fractions unless the question explicitly states simplification is not required.
  - Correct: Divide numerator and denominator by their highest common factor (HCF) to write all fractions in lowest terms.

## Cheatsheet

| Task | Core Rule | Extended Add-On |
| --- | --- | --- |
| Fraction → Decimal | Divide numerator by denominator | Use recurring dot notation for infinite repeating sequences |
| Decimal → Percentage | Multiply by 100, add % sign | No additional rule |
| Order values | Convert all to decimals, use inequality symbols | Same rule applies for recurring decimals |
| Add/Subtract fractions | Find common denominator, add/subtract numerators | Same rule applies |
| Multiply fractions | Multiply numerators, multiply denominators, simplify | Same rule applies |
| Divide fractions | Multiply by reciprocal of the second fraction | Same rule applies |
| Decimal → Fraction | Write over 10/100/1000 etc., simplify | Use elimination method for recurring decimals |

## What's next

Mastering fractions, decimals and their operations is the foundation for almost all other topics in CIE IGCSE Maths 0580, from percentages and ratio to algebra and geometry. Next, you should build on this knowledge to learn how to apply these operations to real-world problems involving percentages, such as interest and profit/loss calculations, which are frequently tested in both Core and Extended papers. Extended students should also practice recurring decimal conversion problems regularly, as these often appear as 2-3 mark questions in Paper 2 and Paper 4. If you struggled with order of operations or negative number rules, revisit those foundational topics first to avoid errors in more complex problems later.

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