Mathematics
Growth Rate
Quick fact
A 10% annual growth rate means a quantity doubles every 7.2 years, revealing the power of compounding.
Why this is interesting
A tiny change in growth rate—say, from 1% to 2% per year—can double a population or investment in half the time. How does such a small number create such a huge effect?
Read the full explanation
Understanding Growth Rate
Growth rate tells us how fast something grows relative to its current size. Imagine a plant that grows 2 inches each day: that's a constant absolute growth rate. But if it grows 10% of its height each day, the actual inches increase over time—that's relative growth. Most interesting growth, like money in a bank or a city's population, follows a relative (percentage) rate. To compare growth across different starting sizes, we use percentage growth rate. For example, a town of 100 people adding 10 people per year grows at 10% annually, while a city of 10,000 adding 100 grows at only 1%—even though both add 10 people per year. The growth rate captures the intensity of change relative to the starting point.
A deeper explanation
The underlying principle is exponential change: when a quantity grows at a constant relative rate, its size over time follows the formula N(t) = N₀ e^(rt), where r is the growth rate and t is time. This arises because the change at each instant is proportional to the current amount. This simple equation explains phenomena from bacterial colonies (doubling every hour) to compound interest (money growing over decades). The concept of doubling time—roughly 70 divided by the percentage growth rate—lets you quickly estimate how fast something will double. Growth rate matters because it turns small differences into massive outcomes over long periods: an economy growing at 3% versus 1% becomes vastly different after a century. Understanding growth rate helps you evaluate claims about growth, make financial decisions, and grasp the dynamics of natural and human systems.