MISSION · Number sense
Simple and Compound Interest
The difference one word makes to a savings account
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Simple interest is paid on the original amount forever; compound interest is paid on the amount as it currently stands, including interest already earned.
Simple interest
\[I = P \times r \times t\]The same amount every period, calculated on the principal \(P\) alone. £1,000 at 5% simple interest earns £50 a year, every year. After 10 years that’s £500, and the balance is £1,500.
It’s a straight line. Plot the balance against time and you get constant growth, because the thing being multiplied never changes.
Compound interest
\[A = P(1 + r)^t\]Now the interest joins the balance, and next year’s interest is calculated on the new, larger total. £1,000 at 5% earns £50 in year one. In year two, 5% of £1,050 is £52.50. In year three, 5% of £1,102.50 is £55.13.
After 10 years the balance is £1,628.89 — nearly £129 more than simple interest gave. After 30 years the gap is enormous: £2,500 simple against £4,322 compound.
This is exponential growth, and the shape is the point. The curve starts almost indistinguishable from the straight line and then leaves it behind entirely.
How often it compounds matters
Interest added monthly compounds twelve times a year, each time on a slightly larger balance:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]where \(n\) is the number of compounding periods per year. £1,000 at 5% for one year gives £1,050 compounded annually and £1,051.16 compounded monthly. Small over a year, meaningful over thirty.
This is why loans advertise an APR — a single annual figure that accounts for the compounding frequency, so two products can actually be compared. A credit card quoting “2% monthly” is not 24% a year; compounded, it’s \(1.02^{12} - 1 \approx 26.8\%\).
The rule of 72
To estimate how long money takes to double at compound interest, divide 72 by the percentage rate.
At 6%, doubling takes about \(72 \div 6 = 12\) years. At 8%, about 9. It’s an approximation and a good one for ordinary rates, and it makes the cost of a small rate difference visible without a calculator.
It cuts both ways
Compound interest is the most powerful force in a savings account and the most punishing one in a debt. The same formula that turns £1,000 into £4,322 over thirty years turns an unpaid credit card balance into something far larger than what was spent.
Whether it works for you or against you depends entirely on which side of the loan you are on.
Common mix-ups
- Using simple interest for a savings account. Almost all real accounts compound.
- Multiplying the annual rate by the years for compound interest. It’s a power, not a product.
- Forgetting to divide the rate when compounding more often than annually.
- Reading a monthly rate as a twelfth of the annual one. Compounding makes the annual figure higher.
Check yourself
£2,000 is invested at 4% for 3 years. Find the balance under simple and compound interest.
Simple: \(2000 + 2000(0.04)(3) = £2{,}240\). Compound: \(2000(1.04)^3 = £2{,}249.73\) — about £10 more, and the gap widens every year.
LEVEL 3 · REFLECT
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