Algebra — OCR GCSE Study Guide

## Overview  Fractions, decimals, and percentages form the bedrock of numerical and algebraic reasoning in GCSE Mathematics. This topic is not just an isolated chapter; it is a fundamental toolkit that connects to probability, statistics, ratio, and algebra. Examiners frequently test your fluency in moving between these three forms, often within the same multi-step question. Whether you are calculating compound interest, comparing probabilities, or simplifying algebraic fractions, mastering these conversions is non-negotiable for achieving top grades. Typical exam questions range from simple 1-mark conversions in the Foundation tier to complex 4-mark compound interest or recurring decimal problems in the Higher tier. Candidates who confidently use percentage multipliers tend to save valuable time and make fewer calculation errors than those relying on traditional, multi-step methods. Listen to the audio guide below for a comprehensive walk-through of the core concepts, complete with exam tips and a quick-fire recall quiz.  ## Key Concepts ### Concept 1: The Holy Trinity of Equivalence Fractions, decimals, and percentages are simply three different languages used to describe the exact same proportion of a whole. Understanding how to translate between them fluidly is where many marks are gained or lost.  **Fraction to Decimal**: Divide the numerator (top) by the denominator (bottom). *Example*: $$\frac{3}{8} = 3 \div 8 = 0.375$$ **Decimal to Percentage**: Multiply by 100 (move the decimal point two places to the right). *Example*: $$0.375 \times 100 = 37.5\%$$ **Percentage to Decimal**: Divide by 100 (move the decimal point two places to the left). *Example*: $$42\% \div 100 = 0.42$$ **Percentage to Fraction**: Write the percentage over 100, then simplify to its lowest terms. *Example*: $$65\% = \frac{65}{100} = \frac{13}{20}$$ ### Concept 2: Percentage Multipliers A percentage multiplier is a single decimal number that applies a percentage increase or decrease in one efficient calculation. Examiners heavily reward the use of multipliers because it demonstrates higher-order mathematical fluency and reduces the risk of arithmetic errors.  **For an Increase**: The multiplier is $$1 + \text{the decimal equivalent}$$. *Example*: A 15% increase means you retain the original 100% and add 15%, giving 115%. As a decimal, this is 1.15. To increase £80 by 15%, calculate $$80 \times 1.15 = £92$$. **For a Decrease**: The multiplier is $$1 - \text{the decimal equivalent}$$. *Example*: A 20% decrease means you subtract 20% from 100%, leaving 80%. As a decimal, this is 0.80. To decrease £80 by 20%, calculate $$80 \times 0.80 = £64$$. ### Concept 3: Compound Interest and Depreciation Compound interest involves applying a percentage change repeatedly over multiple time periods. The percentage multiplier is raised to the power of the number of periods. **Formula**: $$A = P \times (1 \pm \frac{r}{100})^n$$ Where: - **A** = Final Amount - **P** = Principal (starting amount) - **r** = Percentage rate - **n** = Number of time periods (e.g., years) *Example*: £500 invested at 3% compound interest for 4 years. Calculation: $$500 \times 1.03^4 = £562.75$$ ### Concept 4: Recurring Decimals to Fractions (Higher Tier Only) Some decimals repeat infinitely, such as $$0.333...$$ ($$0.\dot{3}$$). Converting these to exact fractions requires an algebraic approach that examiners test specifically. **Method**: 1. Set $$x$$ equal to the recurring decimal. 2. Multiply both sides by a power of 10 ($$10^1$$, $$10^2$$, etc.) to shift the decimal point past the first repeating block. 3. Subtract the original equation to eliminate the recurring part. 4. Solve for $$x$$ and simplify the resulting fraction. ## Mathematical Relationships | Conversion | Operation | Example | | :--- | :--- | :--- | | Fraction → Decimal | Numerator ÷ Denominator | $$\frac{4}{5} = 4 \div 5 = 0.8$$ | | Decimal → Percentage | Multiply by 100 | $$0.8 \times 100 = 80\%$$ | | Percentage → Decimal | Divide by 100 | $$80\% \div 100 = 0.8$$ | | Percentage → Fraction | Write over 100, simplify | $$80\% = \frac{80}{100} = \frac{4}{5}$$ | ## Practical Applications Understanding these concepts is crucial for personal finance, such as calculating the final cost of a sale item (percentage decrease), determining the return on a savings account (compound interest), or comparing mortgage rates. In science, you will frequently convert between these forms when expressing experimental yields, calculating percentage error, or dealing with probabilities in genetics.