--- title: "Compound Growth and Decay" description: "Apply a repeated percentage change as a multiplier raised to a power, for savings, depreciation and populations, and find how many steps it takes to pass a target." canonical: https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf source: https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Compound Growth and Decay Apply a repeated percentage change as a multiplier raised to a power, for savings, depreciation and populations, and find how many steps it takes to pass a target. Subject: Mathematics · Area: Ratio & Proportion · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf ## Ready when they can - Use the multiplier to the power n for n years of growth - Tell compound growth from simple interest in a word problem - Find by trial how many whole steps pass before a value first crosses a target ## Lesson: Interest earns its own interest Compound change repeats the same multiplier every step. With compound interest, the interest earned starts earning interest too. For 3% yearly growth, multiply by 1.03 each year, using the new bigger balance every time. Simple interest is different: it pays only on the original amount, so the yearly addition never grows. **Example.** 1500 pounds at 4% a year for 6 years needs one calculation: 1500 times 1.04^6. The power counts the years, and the answer is about 1897.98 pounds. In general, value = start times multiplier to the number of steps. Calculate the power first, then multiply once instead of taking six separate steps. Decay works the same way with a multiplier below 1. A van losing 20% a year keeps 0.80 each year, so after 3 years a 12000 pound van is worth 12000 times 0.8^3 = 6144 pounds. The power still counts the steps, and each yearly loss shrinks because it comes off a smaller value. **Example.** To find how many steps it takes to pass a target, try whole powers in order. A 640 g sample losing 25% an hour is multiplied by 0.75 each hour. After 4 hours it is 640 times 0.75^4 = 202.5 g, still above 200 g. After 5 hours it is about 151.9 g. So hour 5 is the first below 200 g. Stop at the first crossing. **Recap.** Raise one multiplier to the number of steps, and try whole powers in order to find when a value passes a target. ## Practice 20 questions on this page, each with its working shown. ## Needs first - [Percentage Change with Multipliers](https://lightmysky.com/learn/mathematics/percentage-change-with-multipliers-mt_uqclfF4OBR) ## Opens up - [The exponential function and the number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ) - [Recognising cubic, reciprocal and exponential graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9) - [The Logistic Equation and Saturating Growth](https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L) - [Reading Orders and Half-Lives from Rate Graphs](https://lightmysky.com/learn/science/reading-orders-and-half-lives-from-rate-graphs-mt_gCToNUUYll) - [Geometric Sequences and Common Ratios](https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY)