# Fractions

> Edexcel International GCSE Mathematics A · 4MA1 2016
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-fractions/

This guide covers all fraction skills required for Edexcel IGCSE Maths A (4MA1) spec 1.2, including simplification, arithmetic, ordering, conversion, and problem solving with exam-aligned worked examples.

**Prerequisites:** [Understanding of common factors and multiples](https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-factors-multiples/)

## Learning objectives

- Simplify fractions to lowest terms using common factors
- Convert between mixed numbers and vulgar (improper) fractions
- Order fractions using common denominators
- Calculate fractions of quantities and express one number as a fraction of another
- Perform addition, subtraction, multiplication and division of fractions and mixed numbers
- Convert fractions to decimals and percentages, including recurring decimals
- Understand unit fractions as multiplicative inverses

## Equivalent Fractions, Simplification & Form Conversion

Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same non-zero number. To simplify a fraction to its lowest terms, cancel the highest common factor (HCF) of the numerator and denominator. You will also need to convert between mixed numbers (whole number + proper fraction) and vulgar (improper) fractions for arithmetic operations.

**Fraction Form Rules** — Always simplify fractions to lowest terms unless instructed otherwise. Convert mixed numbers to improper fractions before multiplication or division.

**Worked example:** a) Simplify $\frac{48}{72}$ to lowest terms. b) Convert $3\frac{2}{5}$ to an improper fraction. c) Convert $\frac{17}{4}$ to a mixed number.

1. Step 1a: Find the HCF of 48 and 72 = 24. Divide numerator and denominator by 24:

   $$\frac{48 \div 24}{72 \div 24} = \frac{2}{3}$$
2. Step 1b: For mixed to improper: (whole number × denominator) + numerator, keep the same denominator:

   $$3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{17}{5}$$
3. Step 1c: For improper to mixed: divide numerator by denominator. Quotient = whole number, remainder = new numerator:

   $$17 \div 4 = 4 \text{ remainder } 1 = 4\frac{1}{4}$$

> **Exam tip:** If you struggle to find the HCF, cancel smaller common factors repeatedly until no more common factors remain, e.g. cancel 2 first, then 3, then 4, until the fraction is simplified.

*Calculator:* allowed

## Ordering Fractions & Fraction of Quantities

To order fractions, convert all to equivalent fractions with a common denominator (preferably the lowest common denominator, LCD) then compare numerators. To calculate a fraction of a quantity, multiply the quantity by the numerator and divide by the denominator. To express one number as a fraction of another, write the first number as the numerator and the second as the denominator, then simplify.

**Worked example:** a) Order $\frac{3}{4}$, $\frac{5}{8}$, $\frac{2}{3}$ from smallest to largest. b) Calculate $\frac{3}{5}$ of £120. c) Express 18 as a fraction of 45, simplified.

1. Step 1a: LCD of 4, 8, 3 = 24. Convert all fractions to denominator 24:

   $$\frac{3}{4}=\frac{18}{24}, \frac{5}{8}=\frac{15}{24}, \frac{2}{3}=\frac{16}{24}$$
2. Order by numerator size:

   $$\frac{5}{8} < \frac{2}{3} < \frac{3}{4}$$
3. Step 1b: Multiply 120 by 3, divide by 5:

   $$\frac{3}{5} \times 120 = \frac{360}{5} = £72$$
4. Step 1c: Write 18 as numerator, 45 as denominator, cancel HCF = 9:

   $$\frac{18}{45} = \frac{2}{5}$$

> **Exam tip:** When ordering fractions, you can also convert them to decimals with your calculator to cross-check your answer quickly.

*Calculator:* allowed

## Adding & Subtracting Fractions & Mixed Numbers

You can only add or subtract fractions that have the same denominator. If denominators are different, find the LCD, convert all fractions to equivalent fractions with the LCD, then add or subtract only the numerators, keeping the denominator the same. For mixed numbers, you can either convert to improper fractions first, or add/subtract whole numbers and fractions separately, handling carries or borrows as needed.

**Worked example:** Calculate: a) $2\frac{1}{3} + 1\frac{3}{4}$ b) $5\frac{1}{2} - 2\frac{2}{3}$

1. Step 1a: Convert both mixed numbers to improper fractions:

   $$2\frac{1}{3} = \frac{7}{3}, 1\frac{3}{4} = \frac{7}{4}$$
2. Find LCD of 3 and 4 = 12, convert fractions:

   $$\frac{7}{3} = \frac{28}{12}, \frac{7}{4} = \frac{21}{12}$$
3. Add numerators, simplify, convert back to mixed number:

   $$\frac{28 + 21}{12} = \frac{49}{12} = 4\frac{1}{12}$$
4. Step 1b: Convert to improper fractions, LCD = 6:

   $$5\frac{1}{2} = \frac{11}{2} = \frac{33}{6}, 2\frac{2}{3} = \frac{8}{3} = \frac{16}{6}$$
5. Subtract numerators, simplify:

   $$\frac{33 - 16}{6} = \frac{17}{6} = 2\frac{5}{6}$$

> **Exam tip:** If adding mixed numbers and the sum of the fractions is greater than 1, carry over 1 to the whole number part. If subtracting and the first fraction is smaller than the second, borrow 1 from the whole number to increase the numerator of the first fraction.

*Calculator:* allowed

## Multiplying & Dividing Fractions & Mixed Numbers

To multiply fractions: multiply the numerators together, multiply the denominators together, then simplify. You can cancel common factors between numerators and denominators before multiplying to make calculations easier. To divide fractions: keep the first fraction the same, change the division sign to multiplication, flip the second fraction (take its reciprocal), then multiply as normal. Unit fractions are multiplicative inverses: dividing by a whole number $n$ is the same as multiplying by $\frac{1}{n}$.

**Worked example:** Calculate: a) $2\frac{1}{2} \times 3\frac{3}{5}$ b) $4\frac{1}{3} \div 1\frac{1}{2}$

1. Step 1a: Convert mixed numbers to improper fractions:

   $$2\frac{1}{2} = \frac{5}{2}, 3\frac{3}{5} = \frac{18}{5}$$
2. Cancel common factors, multiply numerators and denominators:

   $$\frac{5}{2} \times \frac{18}{5} = \frac{1 \times 9}{1 \times 1} = 9$$
3. Step 1b: Convert mixed numbers to improper fractions, flip the second fraction, multiply:

   $$4\frac{1}{3} = \frac{13}{3}, 1\frac{1}{2} = \frac{3}{2}$$
4. $$\frac{13}{3} \div \frac{3}{2} = \frac{13}{3} \times \frac{2}{3} = \frac{26}{9} = 2\frac{8}{9}$$

> **Exam tip:** Never multiply the whole number and fraction parts of mixed numbers separately: always convert to improper fractions first to avoid errors.

*Calculator:* allowed

## Converting Fractions to Decimals & Percentages

To convert a fraction to a decimal, divide the numerator by the denominator. If the division repeats indefinitely, this is a recurring decimal, marked with a dot above the repeating digit(s). To convert a fraction to a percentage, multiply the decimal equivalent by 100 and add the % sign.

**Worked example:** Convert: a) $\frac{5}{8}$ to decimal and percentage. b) $\frac{7}{9}$ to exact decimal form.

1. Step 1a: Divide 5 by 8:

   $$5 \div 8 = 0.625$$
2. Multiply by 100 for percentage:

   $$0.625 \times 100 = 62.5\%$$
3. Step 1b: Divide 7 by 9, note the repeating 4 after the decimal:

   $$7 \div 9 = 0.\dot{7}$$

> **Exam tip:** If a question asks for an exact answer, use recurring dot notation for decimals instead of rounding, unless explicitly instructed to round.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to simplify fractions to lowest terms in final answers
  - Why it fails: Edexcel mark schemes often award 1 mark for correct simplification, so you lose easy marks even if your unsimplified fraction is correct
  - Correct: Always cancel the HCF of numerator and denominator before submitting your final answer
- **Wrong:** Multiplying or dividing mixed numbers without converting to improper fractions first
  - Why it fails: Multiplying whole number and fraction parts separately gives an incorrect result
  - Correct: Convert all mixed numbers to vulgar fractions first before multiplication or division
- **Wrong:** Adding or subtracting numerators directly without finding a common denominator
  - Why it fails: This is a common arithmetic error, e.g. $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$ is wrong
  - Correct: Find the lowest common denominator, adjust numerators, then add/subtract only numerators
- **Wrong:** Flipping the first fraction instead of the second when dividing
  - Why it fails: This reverses the operation and gives the reciprocal of the correct answer
  - Correct: Keep the first fraction the same, change division to multiplication, flip the second fraction then multiply
- **Wrong:** Using the wrong denominator when expressing one number as a fraction of another
  - Why it fails: For example, writing 5 as a fraction of 20 as $\frac{20}{5}$ instead of $\frac{5}{20}$ leads to lost marks
  - Correct: The number after 'of' is always the denominator, so the fraction is $\frac{first\thinspace number}{second\thinspace number}$, then simplify
- **Wrong:** Rounding recurring decimals when an exact fraction answer is requested
  - Why it fails: Rounding loses accuracy and violates exact answer requirements
  - Correct: Give the simplified exact fraction, or use recurring dot notation for decimals if allowed

## Cheatsheet

| Operation | Steps | Example |
| --- | --- | --- |
| Simplify fraction | Cancel HCF of numerator and denominator | $\frac{48}{72} = \frac{2}{3}$ |
| Mixed to improper | $\frac{(whole \times denominator) + numerator}{denominator}$ | $3\frac{2}{5} = \frac{17}{5}$ |
| Add/Subtract fractions | Find LCD, adjust numerators, add/subtract numerators | $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ |
| Multiply fractions | Multiply numerators, multiply denominators, cancel first if possible | $\frac{2}{3} \times \frac{9}{10} = \frac{3}{5}$ |
| Divide fractions | Keep first, flip second, multiply, simplify | $\frac{3}{4} \times \frac{2}{1} = \frac{3}{2}$ |
| Fraction to % | Divide numerator by denominator, multiply by 100 | $\frac{3}{8} = 37.5\%$ |

## What's next

Now that you have mastered core fraction skills, you are ready to apply these to other topics across the Edexcel IGCSE Maths A syllabus. Fraction operations are used extensively in ratio and proportion problems, percentage change calculations, and algebraic manipulation for higher tier students. You will also encounter fractions when working with probability, statistics, and geometry problems involving area and volume ratios. Make sure to practice past paper fraction questions to build speed and accuracy, as these skills are tested in almost every exam paper across both foundation and higher tiers. Remember to always simplify your answers and check your working using a calculator where permitted to avoid avoidable arithmetic errors.

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