Compound Interest and Repeated Percentage Change
Applying a percentage change again and again multiplies rather than adds, so the multiplier raised to a power gives the answer in one step. The same calculation covers savings growing, a debt growing, and a value depreciating.
What a learner can do afterwards
- Turn a percentage rise or fall into a multiplier and use a power of it for n periods
- Show that a 10% rise followed by a 10% fall does not return to the starting value, and say why
- Compare the total repaid on a debt at simple and at compound interest over the same term
1 · Read
Turn each percentage move into one multiplier. A rise of 5 percent becomes 1.05, and a fall of 20 percent becomes 0.8. Multiply the amount once and the whole step is done.
Repeat a change by raising the multiplier to a power. A 500 pound deposit at 5 percent for 2 years is 500 times 1.05 squared, which is 551.25 pounds. The power counts the periods.
Equal rises and falls never cancel. A 200 pound jacket rises 10 percent to 220, then falls 10 percent to 198, because the cut applies to the new price. Simple interest pays on the original sum only, while compound pays on the growing total.
Debt shows the gap fast. Borrow 300 pounds for 2 years at 10 percent: compound asks 363 back while simple asks 360. Small rate gaps explode over many years, so starting early wins by miles.
Rise or fall becomes a multiplier, repeats become a power, and interest on interest is why the total outruns simple adding.
More to read
7.2 Time Value of Money (TVM) Basics
Money today is worth more than the same sum years from now, because cash in hand can grow. That idea is called the time value of money. If you know the interest rate, you can push any amount forward to its future value or pull a future payout back to what it is worth today. Every loan, savings plan and investment choice runs on this one comparison.
3.4 Interest Rates
Interest rates are the price of borrowing money, usually quoted per year. Part of the rate rewards the lender for waiting, and part covers the risk that the money comes back late or never. When prices in the shops rise, lenders also demand extra to make up for weaker pounds later. Reading any rate means asking which of these pieces sits inside it.
6.1 Exponential Functions
An exponential function multiplies by the same factor each step, which is exactly how compound balances grow. The base of the function is your multiplier: above 1 for growth, below 1 for decay. Small differences in the base explode over many steps, so a slightly better rate wins by miles given time. Sketching the curve shows why starting early beats saving bigger but later.
6.7 Exponential and Logarithmic Models
Doubling times, half lives and investment horizons all come from solving exponential equations for the time. Logs undo the power and hand you the number of periods directly. The same maths prices a phone contract, sizes a pension pot and dates a fossil.
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2 · Watch
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Where it sits
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Optional connections
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.