Fractions, Decimals & the Four Operations

Mathematics· 1.4, 1.5, 1.6· 25 min read

1. Equivalence & Conversion Between Fractions, Decimals & Percentages★★☆☆☆⏱ 6 min

📘 Definition

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.

2. Ordering Fractions, Decimals & Integers★★☆☆☆⏱ 5 min

📘 Definition

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.

3. Four Operations on Fractions, Decimals & Negatives★★★☆☆⏱ 8 min

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.

4. Extended Adds: Recurring Decimals★★★★☆Extended only⏱ 6 min

📘 Definition

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 $1 0^{n}$ before subtracting to fully eliminate the repeating sequence.

5. Common Pitfalls

Wrong move:

Forgetting to convert mixed numbers to improper fractions before multiplying/dividing, multiplying whole number and fraction parts separately.

Why:

This method produces mathematically incorrect results, as mixed numbers are sums not products of their parts.

Correct move:

Always convert mixed numbers to improper fractions first, perform operations, then convert back to mixed number if required.

Wrong move:

Ordering negative values as if they are positive, e.g., writing $- 0.2 > - 0.1$ .

Why:

Negative numbers with larger absolute values are smaller, as they sit further left on the number line.

Correct move:

Convert all values to decimals first, sort negative values by reverse absolute size, then sort positive values by ascending size.

Wrong move:

Performing addition/subtraction before multiplication/division, ignoring BODMAS.

Why:

Order of operations is explicitly tested in IGCSE 0580, and incorrect order leads to wrong answers even if individual calculations are correct.

Correct move:

Always follow BODMAS: Brackets first, then Orders, Division/Multiplication left to right, then Addition/Subtraction left to right.

Wrong move:

(Extended) Using the wrong multiplier when converting recurring decimals to fractions, e.g., multiplying by 10 for a 2-digit repeating sequence.

Why:

The multiplier must shift the decimal past one full repeating sequence to eliminate all recurring digits when subtracting.

Correct move:

Count the number of repeating digits first, multiply by $1 0^{n}$ where $n$ is the number of repeating digits, then subtract the original $x$ value.

Wrong move:

Leaving fractions unsimplified in final answers.

Why:

CIE examiners deduct 1 mark per question for unsimplified fractions unless the question explicitly states simplification is not required.

Correct move:

Divide numerator and denominator by their highest common factor (HCF) to write all fractions in lowest terms.

6. Quick Reference 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

7. Frequently Asked

Do I need to simplify fractions in my final answers?

Yes, CIE examiners deduct marks for unsimplified fractions unless the question explicitly states you do not need to reduce them. Always divide numerator and denominator by their highest common factor.

How do I write recurring decimals in my working?

Place a single dot over a single repeating digit, or dots over the first and last digit of a repeating sequence, e.g., $0. \dot{3}$ for 0.333... or $0. \dot{1} 4 \dot{2}$ for 0.142142142...

Can I use a calculator for all fraction operations?

Only for calculator papers (Papers 3 and 4). For non-calculator papers (Papers 1 and 2) you will need to perform all fraction operations manually.

Going deeper

study_guideNegative Number Rules Revision
study_guideBODMAS Order of Operations

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.