GCSE Mathematics: Number, Algebra and Ratio

TL;DR

This topic covers foundational skills in handling numbers, using letters for unknown values, and understanding how quantities relate to each other. Mastering these areas is crucial because they're the building blocks for nearly all other GCSE maths. You'll use these skills repeatedly, so understanding them well now will save you a lot of trouble later.

1. The Mental Model

Think of numbers as your basic tools, algebra as using those tools to solve puzzles where some pieces are missing, and ratio as comparing how much of one tool you have compared to another. It's all about understanding quantities and their relationships.

2. The Core Material

Number

Number skills are fundamental. You need to be confident with different types of numbers (integers, fractions, decimals, percentages), and basic operations (+, -, ×, ÷).

Types of Numbers

• Integers: Whole numbers, positive or negative (e.g., -3, 0, 5).
• Rational Numbers: Can be written as a fraction $p/q$ where p and q are integers and q is not zero (e.g., 0.5 is 1/2, 3 is 3/1).
• Irrational Numbers: Cannot be written as a simple fraction (e.g., $\pi$, $\sqrt{2}$). Their decimal representation goes on forever without repeating.

Key Operations and Concepts

• BODMAS/PEMDAS: The order of operations (Brackets/Parentheses, Orders/Exponents, Division/Multiplication, Addition/Subtraction).
• Fractions: Adding, subtracting, multiplying, and dividing. Remember you need a common denominator for adding/subtracting.
• Decimals: Converting to/from fractions and percentages. Performing operations.
• Percentages: "Per hundred." Used for calculating discounts, interest, etc. To find x% of y, calculate $(x/100) \times y$.
• Standard Form: Used for very large or very small numbers (e.g., $3 \times 10^5$). Always has one non-zero digit before the decimal point.

Algebra

Algebra is essentially generalised arithmetic. You use letters (variables) to represent unknown numbers or quantities.

Basic Algebra

• Expressions: Combinations of numbers, variables, and operations (e.g., $3x + 5$).
• Equations: Statements that two expressions are equal (e.g., $3x + 5 = 11$). Your goal is usually to find the value of the unknown.
• Formulae: Equations that express a relationship between several variables (e.g., $A = \pi r^2$).
• Simplifying Expressions: Collecting "like terms" (e.g., $2x + 3y + x - y = 3x + 2y$).
• Expanding Brackets: Multiplying everything inside the bracket by what's outside (e.g., $2(x+3) = 2x+6$).
• Factorising: The reverse of expanding – putting expressions into brackets by finding common factors (e.g., $3x+6 = 3(x+2)$).

Solving Equations

To solve an equation, you need to isolate the variable. Whatever you do to one side of the equation, you must do to the other to keep it balanced.

Ratio

Ratio compares how much of one thing there is compared to another. It tells you the relative sizes of parts.

Understanding Ratios

• Notation: Written with a colon, e.g., $2:3$. This means for every 2 parts of the first item, there are 3 parts of the second.
• Simplifying Ratios: Just like fractions, you can simplify ratios by dividing all parts by a common factor (e.g., $10:15 = 2:3$).
• Sharing in a Ratio: If you need to share a total amount in a given ratio, first find the total number of parts. Then divide the total amount by the total parts to find the value of one part. Finally, multiply the value of one part by the number of parts for each item.
• Direct Proportion: When two quantities increase or decrease at the same rate. If A is directly proportional to B, then $A = kB$ for some constant $k$.

graph TD
    A["Problem Involving Numbers, Algebra, or Ratio"] --> B{"Is it about 'how many' or 'how much'?"}
    B -- Yes, a specific amount --> C["NUMBER: Use arithmetic, percentages, fractions, decimals."]
    B -- Yes, an unknown amount --> D["ALGEBRA: Use variables, expressions, equations."]
    B -- Yes, a comparison or division --> E["RATIO: Compare quantities, share proportionally."]

    C --> F["Calculate solution using operations, BODMAS, standard form."]
    D --> G["Formulate equation(s), simplify, solve for unknown variable(s)."]
    E --> H["Simplify ratio, find total parts, calculate shares/proportions."]

    F --> I["Check answer in context."]
    G --> I
    H --> I

3. Worked Example

Let's say you have £60 to share between two friends, Sarah and Tom, in the ratio 2:3. How much money does each person get?

1. Find the total number of parts: The ratio is 2:3, so total parts = $2 + 3 = 5$ parts.
2. Find the value of one part: Divide the total amount by the total number of parts: $£60 / 5 = £12$ per part.
3. Calculate each person's share:
   • Sarah gets 2 parts: $2 \times £12 = £24$.
   • Tom gets 3 parts: $3 \times £12 = £36$.
4. Check your answer: £24 + £36 = £60. This matches the original amount, so the division is correct.

4. Key Takeaways

• Always follow the order of operations (BODMAS/PEMDAS) when calculating.
• In algebra, whatever you do to one side of an equation, you must do to the other.
• Ratios can be simplified just like fractions by dividing all parts by a common factor.
• Understand the difference between an expression (no equals sign) and an equation (has an equals sign).
• Percentages are "out of 100" and can be easily converted to decimals for calculations.

Common Mistakes to Avoid

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• Forgetting BODMAS/PEMDAS, especially with negative numbers or exponents.
• Not collecting like terms correctly in algebraic expressions (e.g., adding x to y).
• Adding or subtracting fractions without a common denominator.
• Not finding the total number of parts when sharing an amount in a given ratio.
• Confusing expanding brackets with factorising – they're inverse operations.

5. Now Try It

You have a recipe that requires flour and sugar in a ratio of 5:2 by weight. If you use 350g of flour, how much sugar do you need? What if you wanted to make a larger batch using a total of 1.4kg of flour and sugar combined – how much of each ingredient would you need then? Show your working clearly. You've got 15 minutes!