Fractions, decimals and percentages

1. Overview

Fractions, decimals, and percentages are different ways to represent the same value, expressing a part of a whole. The ability to fluently convert between these forms is crucial for success in the IGCSE Mathematics exam. Many problems require you to manipulate numbers in different formats to compare values, perform calculations, and arrive at accurate final answers. Mastering these conversions and operations is therefore essential.

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Key Definitions

• Proper Fraction: A fraction where the numerator (top) is smaller than the denominator (bottom), representing a value less than 1 (e.g., $\frac{3}{4}$).
• Improper Fraction: A fraction where the numerator is greater than or equal to the denominator, representing a value of 1 or more (e.g., $\frac{7}{5}$).
• Mixed Number: A number consisting of a whole number and a proper fraction (e.g., $2\frac{1}{3}$).
• Terminating Decimal: A decimal that has a finite number of digits (e.g., $0.625$).
• Recurring Decimal: A decimal with a digit or a pattern of digits that repeats infinitely (e.g., $0.333...$ or $0.1\dot{4}\dot{6}$).
• Percentage: A fraction or ratio expressed as a fraction of 100, denoted by the symbol % (e.g., $45% = \frac{45}{100}$).

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Core Content

3.1 Types of Fractions and Simplification

To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). Always check if your final answer can be simplified further.

Worked example 1 — Simplifying a fraction

Question: Simplify the fraction $\frac{42}{56}$ to its simplest form.

1. Identify the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
2. Identify the factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
3. Identify the highest common factor (HCF) of 42 and 56: 14
4. Divide both the numerator and the denominator by the HCF: $$\frac{42 \div 14}{56 \div 14} = \frac{3}{4}$$ Final Answer: $\frac{3}{4}$

3.2 Converting Mixed Numbers to Improper Fractions

Method: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Worked example 2 — Mixed to improper fraction

Question: Convert the mixed number $5\frac{3}{8}$ to an improper fraction.

1. Multiply the whole number (5) by the denominator (8): $5 \times 8 = 40$
2. Add the numerator (3) to the result: $40 + 3 = 43$
3. Place the result (43) over the original denominator (8): $\frac{43}{8}$ Final Answer: $\frac{43}{8}$

3.3 Shading Fractions of Shapes

To find the fraction shaded: 1. Count total squares (12). 2. Count shaded squares (3). 3. Write as fraction: $\frac{3}{12}$. 4. Simplify: $\frac{1}{4}$.

Worked example 3 — Shading a fraction of a shape

Question: A rectangle is divided into 20 equal squares. How many squares must be shaded to represent $\frac{2}{5}$ of the rectangle?

1. Calculate the number of squares that represent $\frac{1}{5}$ of the rectangle: $\frac{1}{5} \times 20 = 4 \text{ squares}$
2. Multiply the result by the numerator (2) to find the number of squares that represent $\frac{2}{5}$: $4 \times 2 = 8 \text{ squares}$ Final Answer: 8 squares

3.4 Converting Between Forms

• Fraction to Decimal: Divide the numerator by the denominator.
  • Example: $\frac{3}{8} = 3 \div 8 = 0.375$
• Decimal to Percentage: Multiply the decimal by 100.
  • Example: $0.07 \times 100 = 7%$
• Percentage to Fraction: Put the number over 100 and simplify.
  • Example: $65% = \frac{65}{100} = \frac{13}{20}$

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Extended Content (Extended Only)

4.1 Converting Recurring Decimals to Fractions

The notation $\dot{x}$ means the digit $x$ repeats. If there are dots over two digits, like $0.\dot{1}\dot{4}$, the whole sequence repeats ($0.141414...$). Understanding this notation is crucial for correctly converting recurring decimals to fractions.

Worked example 4 — Recurring decimal to fraction

Question: Convert the recurring decimal $0.\dot{2}\dot{7}$ to a fraction in its simplest form.

1. Let $x = 0.272727...$
2. Multiply both sides of the equation by 100 (since two digits repeat) to shift the decimal point two places to the right: $100x = 27.272727...$
3. Subtract the original equation ($x = 0.272727...$) from the new equation ($100x = 27.272727...$) to eliminate the repeating decimal part: $100x - x = 27.272727... - 0.272727...$ $99x = 27$
4. Solve for $x$ by dividing both sides by 99: $x = \frac{27}{99}$
5. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 9: $x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}$ Final Answer: $\frac{3}{11}$

Worked example 5 — Recurring decimal to fraction (more complex)

Question: Convert the recurring decimal $0.2\dot{3}$ to a fraction in its simplest form.

1. Let $x = 0.233333...$
2. Multiply both sides by 10 to move the non-repeating digit to the left of the decimal point: $10x = 2.33333...$
3. Multiply both sides by 10 again to move one repeating cycle past the decimal point: $100x = 23.33333...$
4. Subtract the equation in step 2 from the equation in step 3 to eliminate the repeating decimal: $100x - 10x = 23.33333... - 2.33333...$ $90x = 21$
5. Solve for $x$: $x = \frac{21}{90}$
6. Simplify the fraction by dividing both numerator and denominator by their HCF, which is 3: $x = \frac{21 \div 3}{90 \div 3} = \frac{7}{30}$ Final Answer: $\frac{7}{30}$

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Key Equations

Concept	Formula	Notes
Percentage of Amount	$\frac{\text{Percentage}}{100} \times \text{Amount}$	No units (result takes units of amount)
Percentage Change	$\frac{\text{Change}}{\text{Original}} \times 100$	Always divide by the original value
Fraction to Percentage	$(\frac{\text{Numerator}}{\text{Denominator}}) \times 100$	Result is in %

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Common Mistakes to Avoid

• ❌ Wrong: Forgetting to simplify a fraction to its lowest terms. Leaving an answer as $\frac{15}{25}$ when the question asks for "simplest form."
• ✅ Right: Always reduce to lowest terms by dividing by the HCF: $\frac{15}{25} = \frac{3}{5}$.
• ❌ Wrong: Incorrectly subtracting mixed numbers, leading to negative fraction errors. For example, trying to do $4\frac{2}{5} - 1\frac{4}{5}$ by subtracting the whole numbers and fractions separately without borrowing.
• ✅ Right: Convert both mixed numbers to improper fractions first: $4\frac{2}{5} - 1\frac{4}{5} = \frac{22}{5} - \frac{9}{5} = \frac{13}{5} = 2\frac{3}{5}$.
• ❌ Wrong: Misinterpreting recurring decimal notation. Assuming $0.3\dot{1}$ means $0.313131...$
• ✅ Right: Only the digit with the dot repeats: $0.3\dot{1}$ means $0.311111...$
• ❌ Wrong: Prematurely rounding a fraction to a decimal during a multi-step calculation.
• ✅ Right: Keep the exact fraction throughout the calculation and only round the very final answer if a decimal is requested and rounding is specified.
• ❌ Wrong: Failing to provide the answer in the format requested by the question (e.g., giving a decimal when a fraction is required).
• ✅ Right: Always double-check the question's wording to confirm your answer is in the correct format (fraction, decimal, percentage, mixed number, simplest form, etc.).

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Exam Tips

• Command Words:
  • "Show that": You must write down every single step of the conversion. Do not skip the middle step, even if it seems obvious.
  • "Write in its simplest form": This is a instruction to simplify fractions or ratios completely.
• Calculator vs. Non-Calculator:
  • On non-calculator papers, look for common denominators that are multiples of each other (e.g., 4, 8, 16).
  • On calculator papers, use the fraction button ($a \frac{b}{c}$ or $\frac{\square}{\square}$) to ensure accuracy, but still write down your working.
• Recurring Decimals: If a question involves recurring decimals, usually worth 2-3 marks, you must show the algebraic method (letting $x = ...$) to get full marks.
• Final Form: If Question 4 asks for a fraction, do not provide $0.75$. If it asks for a mixed number, do not leave it as $\frac{7}{4}$. Always check the required format in the last sentence of the question.