What is the difference between simple and compound interest?

Simple interest is always worked out on the original amount only. Compound interest is worked out on the original amount plus any interest already earned, so the balance grows faster over time.

Why is budgeting useful?

A budget shows where your money goes and whether you are spending less than you earn. It helps you avoid debt, plan for big purchases, and build savings.

Is interest always good?

It depends on which side you are on. Interest is good when you are saving or investing because your money grows. It works against you when you borrow, because you repay more than you took.

Budgeting and Interest | Math

Why money math matters

Money is one of the most useful places to apply mathematics. Whether you are saving for a phone, paying back a loan, or just trying to make your allowance last the month, the same few calculations come up again and again. This lesson covers two big ones: budgeting (planning what comes in and goes out) and interest (how money grows when saved or grows against you when borrowed).

The interest formulas rely on confident percentage work, so if percentages still feel awkward, review Percentages Made Easy before going further.

Building a budget

A budget is simply a plan that compares the money coming in (income) with the money going out (expenses). The key formula could not be simpler:

$$ \text{Balance} = \text{Income} - \text{Expenses} $$

If the balance is positive, you have money left to save. If it is negative, you are spending more than you earn — a warning sign.

Worked example 1. A student earns £600 a month from a part-time job. Here is a sample monthly budget:

Category	Amount
Income (job)	+£600
Rent / board	−£250
Food	−£140
Transport	−£60
Phone	−£25
Fun / eating out	−£65
Total expenses	−£540
Balance left to save	+£60

The balance is £600 − £540 = £60, so this person can save £60 each month. Over a year that is 12 × £60 = £720 saved, before any interest is added.

The 50/30/20 guideline

A popular rule of thumb splits income into needs (50%), wants (30%), and savings (20%). On £600 that would be £300 for needs, £180 for wants, and £120 for savings. It is only a guide, but it shows how percentages turn a vague plan into concrete targets.

Simple interest

When you save money in a bank, the bank pays you interest — a reward for letting them hold your money. The simplest kind is simple interest, which is always calculated on the original amount (the principal) only.

$$ I = P \times r \times t $$

where P is the principal, r is the interest rate as a decimal, and t is the time in years.

Worked example 2. You save £500 at 4% per year simple interest for 3 years.

I = P × r × t = 500 × 0.04 × 3 = £60.
Total amount after 3 years = £500 + £60 = £560.

With simple interest the amount earned is the same every year (here £20 per year), because it is always based on the original £500.

Compound interest

Compound interest is more powerful — and more common in real life. Each period the interest is added to the balance, and the next lot of interest is calculated on the new, larger amount. You earn "interest on your interest."

The fastest way to calculate it is with a multiplier, exactly like adding a percentage:

$$ A = P(1 + r)^t $$

where A is the final amount, P the principal, r the rate as a decimal, and t the number of periods.

Worked example 3. £1000 is saved at 10% compound interest for 2 years.

Doing it year by year shows the idea clearly:

Year 1: £1000 × 1.10 = £1100.
Year 2: £1100 × 1.10 = £1210.

Using the formula directly: A = 1000 × (1.10)² = 1000 × 1.21 = £1210. The interest earned is £210.

Simple vs compound: the difference grows

Over a short time the two are close, but compound interest pulls ahead as the years pass, because the balance keeps getting bigger. Compare £1000 at 10% for several years:

Years	Simple interest total	Compound interest total
1	£1100	£1100
2	£1200	£1210
3	£1300	£1331
5	£1500	£1610.51
10	£2000	£2593.74

After 10 years compound interest has earned almost £600 more than simple interest, from the very same starting amount and rate. This is why financial advisers say to start saving early — time is what lets compounding work.

Interest can work against you

The same formulas apply to borrowing. When you take out a loan or carry a balance on a credit card, you pay the interest, and it usually compounds. A £500 debt at 20% interest left unpaid for 2 years grows to 500 × (1.20)² = £720. Understanding this is the difference between borrowing wisely and falling into a debt trap.

Practice activity

Try these with a calculator. Round money to the nearest penny.

Income is £450 and expenses are £415. What is the balance?
Find the simple interest on £800 at 3% for 4 years.
What is the total amount when £1200 is saved at 5% simple interest for 3 years?
£2000 is invested at 6% compound interest for 2 years. Find the total amount.
Using the 50/30/20 rule, how much of a £900 monthly income should go to savings?

Answers: 1) £35 2) I = 800 × 0.03 × 4 = £96 3) interest £180, total £1380 4) 2000 × (1.06)² = £2247.20 5) 20% of £900 = £180.

Why this matters

Budgeting and interest are the foundations of financial literacy — skills you will use for the rest of your life. A budget keeps you in control of your money and stops small overspends from becoming big problems. Understanding interest tells you which savings account genuinely grows your money and warns you how quickly a loan or credit card balance can balloon. The maths is straightforward: balance equals income minus expenses, simple interest is P × r × t, and compound interest is P(1 + r)^t. Master these three formulas now, and you will make smarter money decisions for decades.

Budgeting and Interest

Key takeaways