Arithmetic governs how we count objects, compare quantities, and perform precise calculations with numbers. It covers everything from simple counting to the exact procedures used in budgeting and measurement. You meet it when you shop, split a bill, or follow a recipe.
Why it matters
Two concrete cases illustrate its reach. If a shirt costs £32 and is discounted by 25%, the sale price is £24 because 25% of 32 is 8 and 32 − 8 = 24. If you earn £2,500 a month and want to save 12%, you save £300; miscalculating the percentage can leave you short of your goal. These are everyday decisions where accurate arithmetic keeps outcomes predictable.
How to approach this
Begin with real-world tasks: counting money, measuring ingredients, checking prices. Plan a few weeks of steady practice, then broaden to fractions, decimals, and percentages. The common sticking point is aligning decimal places and avoiding rounding errors.
The learning path
1. Natural numbers and counting
Starting from counting objects, you learn to express how many items are in a group, which is described in Natural numbers and counting. How many objects are in a group of seven?
2. Integers and negative numbers
Integers extend counting to include zero and negative values; see Integers and negative numbers. How would you represent a debt of 5 pounds, or a temperature of −3°C?
3. The four operations
The four basic arithmetic operations are addition, subtraction, multiplication, and division; see The four operations. If you have 7 apples and give away 2, how many remain?
4. Properties of operations
Properties of operations are rules such as commutativity (order doesn’t matter for addition and multiplication), associativity (grouping doesn’t affect the result), and distributivity (multiplication distributes over addition); see Properties of operations. If you add 3 + 5 + 2, does the order matter?
5. Fractions
Fractions are numbers written as a numerator over a denominator; the numerator is the top part and the denominator the bottom, and the fraction represents part of a whole divided into equal parts; see Fractions. If you split a cake into 8 equal pieces and eat 3, what fraction remains?
6. Decimals
Decimals are fractions written in base 10 using a decimal point; the digits to the right of the point indicate tenths, hundredths, thousandths, and so on; see Decimals. They give exact representations of many fractions and are the standard form in money and measurement; how many pence are in £3.45?
7. Percentages
Percentages are fractions with denominator 100, written with a percent sign; 50% means 50 out of 100 and equals 0.50 in decimal form; see Percentages. If a price of £40 is discounted by 25%, what is the final price?
8. Ratios and proportions
A ratio compares two quantities by division; a proportion is an equation that two ratios are equal; see Ratios and proportions. If a recipe calls for 2 parts flour to 1 part water, how much water for 6 parts flour?
9. Estimation and mental math
Estimation is getting a quick, reasonable answer rather than an exact one; mental math is performing calculations in your head; see Estimation and mental math. If you have 57 + 38, what’s the quick sum?
10. Applications and problem solving
Applied arithmetic in real tasks such as shopping, cooking, budgeting, and interpreting data; uses the four operations plus fractions, decimals, and percentages; see Applications and problem solving. If you have £50 and items totalling £33.75, what remains?
Where this leads
After mastering this path, you can handle budget planning, pricing, and basic data tasks with confidence. You will be prepared to tackle algebra and geometry, where quantities and their relationships are described with the same arithmetic language.