Loans, Savings and Financial Math

Money decisions are maths decisions

Borrowing, saving and budgeting all come down to numbers. The people who understand the maths pay less to borrow, earn more on savings, and avoid debt traps. This lesson turns everyday money questions into calculations you can actually do.

It builds on percentages and interest. If multipliers and compound interest are new, read compound interest and exponential growth first, and percentages in real life for the basics.

The total cost of borrowing

When you borrow, you pay back more than you received. The extra is the cost of borrowing:

total cost of borrowing = total amount repaid − amount borrowed

To find the total repaid for a loan with regular payments:

total repaid = monthly payment × number of payments

Worked example 1: You borrow £1000 and repay it in 24 monthly instalments of £49.50. Find the total cost of borrowing.

1. total repaid = 49.50 × 24 = £1188.
2. cost of borrowing = 1188 − 1000 = £188.

So the loan really cost £188 on top of the £1000 — that is the price of borrowing.

Worked example 2: A phone costs £600 outright, or £25 per month for 30 months. How much more do you pay on the instalment plan?

1. instalment total = 25 × 30 = £750.
2. extra cost = 750 − 600 = £150.
3. Paying monthly costs £150 more.

APR: comparing loans fairly

Different loans quote interest in different ways, so they are hard to compare. The APR (Annual Percentage Rate) solves this: it is the true yearly cost of borrowing, including interest and compulsory fees, as a single percentage. A lower APR is usually cheaper.

Worked example 3: Two lenders offer to lend £2000 for one year.

Lender	APR	Approx. interest for one year
A	8%	2000 × 0.08 = £160
B	12%	2000 × 0.12 = £240

Lender A charges £80 less for the same loan, simply because its APR is lower. Always compare APRs, not headline "low monthly payment" adverts.

Why a longer term can cost more

Spreading a loan over more months lowers each payment but stretches the borrowing — so you pay interest for longer.

Worked example 4: £3000 is borrowed. Compare two repayment plans.

1. Plan 1: £140/month for 24 months → 140 × 24 = £3360. Cost = 3360 − 3000 = £360.
2. Plan 2: £85/month for 48 months → 85 × 48 = £4080. Cost = 4080 − 3000 = £1080.
3. The longer plan has smaller payments but costs £720 more in total.

Lesson: smaller monthly payments are not automatically a better deal.

Credit cards and minimum payments

Credit cards charge compound interest on any unpaid balance, often around 20% APR. Paying only the minimum lets the debt grow.

Worked example 5: You owe £500 on a card at 20% APR and pay nothing for a year. Estimate the new balance.

1. Multiplier for +20% is 1.20.
2. 500 × 1.20 = £600.
3. The debt grows by £100 in a year just from interest — and faster if you keep spending.

Savings: making interest work for you

Borrowing pays interest out; saving earns interest in. With compound interest, savings grow faster the longer they are left, because you earn interest on your interest.

Worked example 6: You save £1000 at 4% compound interest per year. Find the balance after 5 years.

1. balance = 1000 × 1.04⁵.
2. 1.04⁵ ≈ 1.21665.
3. balance ≈ £1216.65, so the interest earned is about £216.65.

Worked example 7: Two savers each put away £2000 at 5% compound interest. Amir leaves it for 10 years; Beth for 20 years. Compare.

1. Amir: 2000 × 1.05¹⁰ ≈ 2000 × 1.6289 = £3257.79.
2. Beth: 2000 × 1.05²⁰ ≈ 2000 × 2.6533 = £5306.60.
3. Doubling the time more than doubles the growth — Beth earns far more. Time is the saver's biggest advantage.

Budgeting: income versus spending

A budget compares money coming in with money going out:

balance = income − total spending

A positive balance can be saved; a negative balance means you are heading into debt.

Worked example 8: A student earns £480 a month and spends £180 on rent, £120 on food, £60 on transport and £40 on subscriptions. How much can they save?

1. total spending = 180 + 120 + 60 + 40 = £400.
2. balance = 480 − 400 = £80.
3. They can save £80 a month — about £960 a year before interest.

To budget by percentage, use percentage of an amount: saving 20% of £480 is 0.20 × 480 = £96.

Why APR and total cost matter more than the monthly payment: Adverts highlight a small monthly figure because it feels affordable. But the real price is the total you repay minus what you borrowed. A low monthly payment over a long term, at a high APR, can quietly cost far more than a slightly higher payment over a short term. Comparing APR and total cost reveals the true price.

Where financial maths is used

Everyone uses it: choosing a phone contract, a student loan, a car finance deal or a mortgage; deciding which savings account or ISA to open; building a monthly budget; and judging whether "0% for 12 months" is genuinely a good offer. Banks, accountants and financial advisers do exactly these calculations professionally — and knowing them yourself protects you from expensive mistakes.

Practice activity

1. A loan is repaid in 36 payments of £45. Find the total repaid.
2. You borrow £800 and repay £980 in total. Find the cost of borrowing.
3. £1500 is saved at 3% compound interest for 4 years. Find the balance.
4. A laptop is £900 outright or £40/month for 27 months. How much extra does the plan cost?
5. Income £550, spending £470. How much can be saved each month?

Answers:

1. 45 × 36 = £1620.
2. 980 − 800 = £180.
3. 1500 × 1.03⁴ = 1500 × 1.12550881 ≈ £1688.26.
4. 40 × 27 = 1080; 1080 − 900 = £180 extra.
5. 550 − 470 = £80.

Summary

Smart money decisions are calculations. The total cost of borrowing is total repaid − amount borrowed, and total repaid = payment × number of payments. APR lets you compare loans fairly as a yearly cost — usually the lower the better — and a longer term often costs more overall. Savings grow by compound interest, so starting early matters enormously. A budget is income − spending, and only the positive balance can safely fund repayments or savings. Look past the monthly figure to the APR and total cost to find the real price.