Maths for Personal Finance

Everyone has a tax-free personal allowance. Income above this is taxed in bands at increasing rates — only the income within each band is taxed at that band's rate (this is why it's called a "progressive" system). For high earners, the personal allowance itself is gradually reduced once income passes a threshold.

National Insurance is calculated separately from income tax, also in bands, and funds state benefits and the NHS.

Simple interest is calculated only on the original amount each year. Compound interest is calculated on the original amount plus any interest already earned, so it grows faster over time: $$A = P(1+r)^n$$ where $P$ is the amount invested, $r$ is the interest rate (as a decimal), and $n$ is the number of years.

AER (Annual Equivalent Rate) lets you compare savings accounts that compound interest at different frequencies (monthly, daily, etc.) on a like-for-like annual basis.

APR (Annual Percentage Rate) is the standard measure for comparing the true cost of borrowing (loans, credit cards, mortgages) — it includes interest and standard charges, expressed as a yearly rate. A lower APR means cheaper borrowing overall, even if the headline monthly rate looks similar.

Inflation reduces the real purchasing power of money over time — an amount that stays the same in cash terms is worth less in real terms each year inflation is positive. Exchange rates convert between currencies; watch carefully which direction you're converting (multiply or divide) and round sensibly for the context (e.g. currency to 2 d.p.).