Number

Overview

Fractions, decimals, and percentages are simply three different languages used to describe the exact same thing: a part of a whole. Whether you are dividing a pizza, calculating a discount in a sale, or interpreting probability, you are using these concepts. In GCSE Mathematics, this topic is foundational. Examiners test it directly in dedicated questions, but it also underpins almost every other area of the specification, from geometry and trigonometry to statistics and algebra.

Mastering this topic means you can fluently translate between these three forms, allowing you to choose the most efficient method for any given problem. A key focus for examiners, particularly at the Higher tier, is your ability to use multiplicative reasoning. Instead of calculating percentages in multiple steps, you will learn to use single multipliers, a skill that reduces errors and saves valuable exam time.

Key Concepts

Concept 1: Converting Fractions to Decimals and Percentages

To convert a fraction to a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, \frac{3}{8} means 3 \div 8, which equals 0.375. This is a terminating decimal. Once you have the decimal, converting to a percentage is as simple as multiplying by 100. So, 0.375 \times 100 = 37.5%.

Why it works: A fraction is a division operation waiting to happen. A percentage literally means "per hundred" (from the Latin per centum). By dividing to get a decimal (which is out of 1), and then multiplying by 100, you are scaling the value to be out of 100.

Example: Convert \frac{5}{8} to a percentage.
5 \div 8 = 0.625
0.625 \times 100 = 62.5%

Concept 2: Converting Decimals and Percentages to Fractions

To convert a terminating decimal to a fraction, write the decimal digits over the appropriate power of 10 (10 for one decimal place, 100 for two, 1000 for three), and then simplify fully. This simplification step is where many candidates lose marks.

To convert a percentage to a fraction, write the percentage value over 100 and simplify.

Example: Convert 45% to a fraction in its simplest form.
45% = \frac{45}{100}
Divide numerator and denominator by their highest common factor (5):
\frac{45 \div 5}{100 \div 5} = \frac{9}{20}

Concept 3: The Multiplier Method for Percentage Change

This is the most powerful tool in this topic. Instead of finding a percentage and adding or subtracting it from the original amount, you multiply the original amount by a single scaling factor called a multiplier.

• For a percentage increase of r%: The multiplier is 1 + \frac{r}{100}.
• For a percentage decrease of r%: The multiplier is 1 - \frac{r}{100}.

Why it works: If you increase a value by 20%, you are keeping the original 100% and adding 20%, giving you 120% of the original. 120% as a decimal is 1.2. Therefore, multiplying by 1.2 achieves the increase in one step.

Example: A TV costs £450. It is reduced by 15% in a sale. Calculate the sale price.
Multiplier = 1 - 0.15 = 0.85
Sale Price = £450 \times 0.85 = £382.50

Concept 4: Expressing One Quantity as a Fraction of Another

To express quantity A as a fraction of quantity B, write \frac{A}{B}. The critical rule here is that both quantities must be in the exact same units before you form the fraction. Once formed, the fraction must be simplified.

Example: Express 45 minutes as a fraction of 2 hours.
Convert 2 hours to minutes: 2 \times 60 = 120 minutes.
Form the fraction: \frac{45}{120}
Simplify by dividing by the highest common factor (15): \frac{3}{8}

Mathematical/Scientific Relationships

• Percentage Change Formula: \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 (Must memorise)
• Multiplier for Increase: 1 + \frac{r}{100} (Must memorise)
• Multiplier for Decrease: 1 - \frac{r}{100} (Must memorise)

Practical Applications

These skills are used daily in the real world. Calculating discounts during sales, working out interest rates on savings accounts or loans, determining profit margins in business, and understanding statistics in news reports all rely heavily on fractions, decimals, and percentages.

Listen to the full 17-minute revision podcast covering all core concepts, exam tips, and a quick-fire recall quiz.