Money - The Mathematics of Everyday Finance

Money is the medium we use to exchange goods and services. Without it, every trade would require finding someone who has exactly what you want and wants exactly what you have – a system called barter that quickly becomes impractical. Money solves this by acting as a universally accepted store of value that anyone can use at any time.

Some of the earliest known forms of standardised money were cowrie shells, used across Africa, Asia, and Oceania for thousands of years, and metal coins, which first appeared in the kingdom of Lydia (modern-day Turkey) around 600 BCE under King Alyattes. The Lydian innovation – stamping a standard weight of electrum (a natural gold-silver alloy) with an official mark – solved barter's biggest weakness by guaranteeing a coin's value without requiring buyer and seller to weigh and test the metal themselves each time. Paper money followed much later, pioneered in China during the Tang and Song dynasties (7th–11th centuries CE), where merchants found carrying promissory notes far easier than hauling strings of heavy coins over long trade routes – the same convenience logic that today drives the shift toward digital and contactless payments.

What Money Does

Money serves three key purposes in any economy:
Medium of exchange – it is accepted in payment for goods and services.
Store of value – it holds its worth over time so you can save now and spend later.
Unit of account – it gives us a standard way to measure and compare the value of things.

Types of Money

Type	Description	Examples
Coins	Metal discs issued by governments	1p, 5p, £1, 50 cents
Banknotes	Paper or polymer notes	£5, £10, $20, €50
Bank deposits	Money held electronically in accounts	Current account balance
Digital / card payments	Electronic transfers between accounts	Debit card, bank transfer
Cryptocurrency	Decentralised digital tokens	Bitcoin, Ethereum

Income and Expenditure

Income is money you receive – wages, salary, freelance earnings, benefits, or interest on savings.
Expenditure is money you spend – rent, food, transport, entertainment, and bills.
The golden rule of personal finance: spend less than you earn.

Profit and Loss

Profit = Selling Price − Cost Price (when selling price is higher)
Loss = Cost Price − Selling Price (when cost price is higher)
Profit percentage = (Profit ÷ Cost Price) × 100
Loss percentage = (Loss ÷ Cost Price) × 100

Worked Examples

A trader buys a jacket for £40 and sells it for £55. Find the profit and the profit percentage.

Profit = £55 − £40 = £15.
Profit % = (15 / 40) × 100 = 37.5%.

A phone bought for £300 is sold for £255. Find the loss and the loss percentage.

Loss = £300 − £255 = £45.
Loss % = (45 / 300) × 100 = 15%.

A shopkeeper wants to make a 20% profit on an item that cost £85. What selling price should they set?

Profit = 20% of £85 = 0.20 × 85 = £17.
Selling price = £85 + £17 = £102.

Value Added Tax (VAT)

VAT is a tax added to the selling price of most goods and services. In the UK the standard VAT rate is 20%.
Price including VAT = Original price × 1.20
To find the original price from a VAT-inclusive price: divide by 1.20.

A laptop costs £750 before VAT. Find the price including 20% VAT.

Price with VAT = £750 × 1.20 = £900.

A bill is £216 including 20% VAT. Find the price before VAT.

Original price = £216 ÷ 1.20 = £180.

Discounts and Sale Prices

A discount is a reduction on the original price.
Sale price = Original price × (1 − discount rate)
Example: 30% off £60 → £60 × 0.70 = £42.

Common Mistakes

Mistake	Correction
Calculating profit % using selling price as the base	Profit % is always calculated on the cost price
Adding VAT to a price that already includes VAT	Check whether the given price is ex-VAT or inc-VAT before calculating
Subtracting discount % directly from price without converting to decimal	Convert the percentage to a multiplier first, e.g. 25% off = × 0.75

Key Takeaways

Money acts as a medium of exchange, store of value, and unit of account.
Profit = Selling Price − Cost Price; Profit % is based on cost price.
VAT (20% standard UK rate): multiply by 1.20 to add, divide by 1.20 to remove.
Discount: multiply by (1 − rate) to find the sale price.

Practice: Profit, Loss and VAT

Profit and Loss Percentage