Statistics — OCR GCSE Study Guide

## Overview  Fractions, decimals, and percentages are simply three different languages used to describe exactly the same mathematical concept: a proportion of a whole. Whether you are calculating a discount in a shop, interpreting data in a science experiment, or splitting a bill, fluency in converting between these three forms is an essential life skill and a cornerstone of GCSE Mathematics. Examiners test this topic rigorously across all papers. Questions range from straightforward one-mark conversions to complex, multi-step problem-solving scenarios involving percentage change and reverse percentages. Because this topic connects heavily to probability, ratio, and algebra, mastering FDP is one of the highest-yield investments you can make in your revision. ## Key Concepts ### Concept 1: Conversions Between Forms  The foundation of this topic is the ability to translate a value from one form to another. Examiners frequently test your ability to convert a fraction to a decimal, a decimal to a percentage, and so on. **Fraction to Decimal**: Divide the numerator (top number) by the denominator (bottom number). *Why this works*: A fraction line literally means "divided by". *Example*: $\frac{3}{8} = 3 \div 8 = 0.375$ **Decimal to Percentage**: Multiply by 100. *Why this works*: "Percent" means "per hundred". By multiplying by 100, you are finding how many hundredths the decimal represents. *Example*: $0.375 \times 100 = 37.5\%$ **Percentage to Fraction**: Write the percentage over 100 and simplify. *Why this works*: Again, percent means per hundred. $37.5\%$ is $37.5$ per $100$. To remove the decimal in the numerator, multiply top and bottom by 10 to get $\frac{375}{1000}$, then simplify by dividing by common factors to reach $\frac{3}{8}$. ### Concept 2: Percentage Multipliers  This is a crucial technique that examiners love to see because it demonstrates mathematical maturity. Instead of finding a percentage of an amount and then adding or subtracting it, you use a single multiplier. **Percentage Increase**: The multiplier is $1 + (\text{percentage as a decimal})$. *Example*: To increase £80 by 15%, the multiplier is $1 + 0.15 = 1.15$. The calculation is $80 \times 1.15 = £92$. **Percentage Decrease**: The multiplier is $1 - (\text{percentage as a decimal})$. *Example*: To decrease £80 by 20%, the multiplier is $1 - 0.20 = 0.80$. The calculation is $80 \times 0.80 = £64$. *Why this works*: The original amount represents 100% (or 1). A 15% increase means you want 115% of the original amount. 115% as a decimal is 1.15. ### Concept 3: Ordering Mixed Types When asked to order a list containing fractions, decimals, and percentages, the most reliable method is to convert all values to decimals first. Decimals are the easiest format to compare because they align neatly by place value. **Example**: Order $\frac{3}{5}$, $62\%$, and $0.615$ from smallest to largest. Step 1: Convert to decimals. $\frac{3}{5} = 3 \div 5 = 0.6$ $62\% = 62 \div 100 = 0.62$ $0.615$ is already a decimal. Step 2: Add placeholder zeros to make them the same length. $0.600$, $0.620$, $0.615$ Step 3: Order the decimals. $0.600 < 0.615 < 0.620$ Step 4: Write the final answer in the original forms. $\frac{3}{5}$, $0.615$, $62\%$ ### Concept 4: Recurring Decimals to Fractions (Higher Tier) A recurring decimal is a decimal that has a digit or group of digits that repeats infinitely. Higher tier candidates must know how to convert these into exact fractions using algebra. **Example**: Convert $0.\dot{4}\dot{5}$ to a fraction. Step 1: Let $x = 0.454545...$ Step 2: Because two digits recur, multiply by 100 to shift the decimal point past one repeating block. $100x = 45.454545...$ Step 3: Subtract the original equation from the new one. $100x = 45.454545...$ $-\quad x = 0.454545...$ $99x = 45$ Step 4: Solve for $x$ and simplify. $x = \frac{45}{99} = \frac{5}{11}$ ## Mathematical/Scientific Relationships **Percentage Multiplier Formula**: $$\text{New Value} = \text{Original Value} \times \text{Multiplier}$$ Where Multiplier = $1 \pm \left(\frac{\text{Percentage}}{100}\right)$ *Must memorise. Used for percentage increase/decrease and compound interest.* **Reverse Percentage Formula**: $$\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}$$ *Must memorise. Used when you are given the final amount after a percentage change and need to find the starting amount.* ## Practical Applications **Compound Interest**: Bank accounts use percentage multipliers repeatedly. If you invest £500 at 3% interest per year for 4 years, the calculation is simply $500 \times 1.03^4$. **Depreciation**: The value of cars decreases over time. If a £10,000 car depreciates by 15% each year, its value after 3 years is $10,000 \times 0.85^3$. **Retail Discounts**: Shops often advertise "20% off, plus an extra 10% off at the till". This is NOT a 30% discount! The multipliers are $0.8$ and $0.9$. The combined multiplier is $0.8 \times 0.9 = 0.72$, which represents a 28% discount. ## Audio Revision Listen to this 10-minute podcast to reinforce these concepts, hear common examiner pitfalls, and test your recall.