---
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: 2026-09-12
---

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# 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 years pass before a value drops below a target

## Lesson: Interest earns its own interest

Compound change repeats the same multiplier every step, and the interest earned starts earning too. Growing 3% a year means times 1.03 each year, on the new bigger balance every time. Simple interest is different: it pays only on the original amount, so the yearly addition never grows.

**Example.** 2000 pounds at 3% a year for 5 years needs one calculation: 2000 times 1.03^5. The power counts the years, and the answer is about 2318.55 pounds. In general, start times multiplier to the number of steps. Raising first and multiplying once beats five separate steps.

Decay works the same way with a multiplier below 1. A car losing 15% a year keeps 0.85 each year, so after 4 years it is worth 18000 times 0.85^4, about 9396 pounds. The power still counts the steps, and the shrinking balance shrinks each yearly loss too.

**Tip.** To find how many steps pass a target, try whole powers one at a time. An 800 g sample losing 20% an hour shrinks by times 0.8 each hour, and hour 7 is the first below 200 g. Stop at the first crossing, not after it.

**Recap.** Raise one multiplier to the number of steps, and trial powers to find when a target falls.

## 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)