Basic Geometry — OCR GCSE Study Guide

 ## Overview Fractions, Decimals, and Percentages (FDP) form the bedrock of numerical reasoning in GCSE Mathematics. This topic is fundamentally about understanding that a single value or proportion can be represented in three distinct ways. Whether you are calculating a discount in a sale, determining the probability of an event, or comparing data sets, fluency in converting between these forms is absolutely critical. Examiners frequently test this topic in both isolated calculation questions and embedded within broader problem-solving contexts. You will encounter questions asking you to order mixed lists of values, calculate percentage changes using multipliers, and apply these skills to real-world financial scenarios. Mastery here is not just about learning rules; it is about developing an intuitive sense of proportion that will serve you throughout the entire syllabus. ## Key Concepts ### Concept 1: FDP Conversion The most essential skill is moving seamlessly between fractions, decimals, and percentages. Think of them as three different languages describing the exact same amount. * **Fraction to Decimal**: Divide the numerator (top) by the denominator (bottom). For example, to convert $\frac{3}{8}$, you calculate $3 \div 8 = 0.375$. * **Decimal to Percentage**: Multiply by $100$. This shifts the decimal point two places to the right. For example, $0.375 \times 100 = 37.5\%$. * **Percentage to Fraction**: Write the percentage over $100$ and simplify fully. For example, $45\%$ becomes $\frac{45}{100}$. Dividing both by their highest common factor ($5$) gives $\frac{9}{20}$.  **Examiner Tip**: A very common error is converting single-digit percentages incorrectly. Remember that $5\%$ is $0.05$, not $0.5$. Always divide by $100$ carefully. ### Concept 2: Fractions of Quantities When asked to find a fraction of an amount, you are essentially scaling that amount. To find $\frac{a}{b}$ of a quantity $Q$, you divide the quantity by the denominator $b$ (to find one part) and then multiply by the numerator $a$ (to find $a$ parts). **Example**: Find $\frac{4}{7}$ of $£350$. 1. Find $\frac{1}{7}$: $350 \div 7 = 50$ 2. Find $\frac{4}{7}$: $50 \times 4 = 200$ Final answer: $£200$ ### Concept 3: Percentage Multipliers This is the most efficient way to handle percentage increase and decrease, especially on calculator papers. Instead of calculating the percentage and then adding or subtracting it, you use a single multiplier. * **Percentage Increase**: Multiplier $= 1 + \text{decimal percentage}$. For a $15\%$ increase, the multiplier is $1 + 0.15 = 1.15$. * **Percentage Decrease**: Multiplier $= 1 - \text{decimal percentage}$. For a $20\%$ decrease, the multiplier is $1 - 0.20 = 0.80$.  **Example**: A car costs $£12,000$. It depreciates by $18\%$ in its first year. What is its new value? Multiplier $= 1 - 0.18 = 0.82$ New Value $= 12000 \times 0.82 = £9,840$ ### Concept 4: Ordering Mixed Types When asked to order a list containing fractions, decimals, and percentages, the most reliable method is to convert all of them into decimals first. Once they are all in the same format, compare their place values, order them, and then write your final answer using the *original* forms given in the question. **Example**: Order from smallest to largest: $0.62$, $60\%$, $\frac{5}{8}$, $\frac{2}{3}$. 1. Convert to decimals: $0.62 = 0.62$, $60\% = 0.60$, $\frac{5}{8} = 0.625$, $\frac{2}{3} = 0.666...$ 2. Order decimals: $0.60$, $0.62$, $0.625$, $0.666...$ 3. Original forms: $60\%$, $0.62$, $\frac{5}{8}$, $\frac{2}{3}$ ### Concept 5: Recurring Decimals to Fractions (Higher Tier Only) A recurring decimal has a repeating pattern of digits. To convert it to a fraction, we use an algebraic method to eliminate the recurring part. **Example**: Convert $0.\dot{4}\dot{5}$ to a fraction in its simplest form. 1. Let $x = 0.454545...$ 2. Multiply by $100$ (because 2 digits recur): $100x = 45.454545...$ 3. Subtract the original equation: $100x - x = 45.454545... - 0.454545...$ 4. $99x = 45$ 5. $x = \frac{45}{99}$ 6. Simplify (divide by 9): $x = \frac{5}{11}$ ## Mathematical Relationships * **Fraction to Decimal**: $Decimal = \frac{\text{Numerator}}{\text{Denominator}}$ * **Decimal to Percentage**: $Percentage = Decimal \times 100$ * **Percentage Increase Multiplier**: $M = 1 + \frac{P}{100}$ * **Percentage Decrease Multiplier**: $M = 1 - \frac{P}{100}$ * **New Value**: $\text{New Value} = \text{Original Value} \times \text{Multiplier}$ ## Practical Applications Understanding FDP is crucial for everyday financial literacy. * **Retail and Sales**: Calculating discounts during sales requires percentage decrease multipliers. * **Banking and Finance**: Understanding interest rates on savings accounts or loans relies on percentage increase multipliers (and compound interest, which builds on this). * **Data Analysis**: Interpreting statistics in news articles often involves converting raw data (fractions) into percentages for easier comparison.