Lesson 2 of 4

Growth rates.

About 20 minutes · 5 practice questions

Read the full article: Exponential growth and logarithms

Growth by so many per cent at every step is multiplying by a fixed factor: how to turn a percentage into a factor, why ten steps of ten per cent are more than a hundred per cent, how long it takes to double, and what happens when an amount shrinks by a percentage.

Step 1 of 5 · From per cent to factor

Step 1 of 5

From per cent to factor

In the Unit on numbers you saw that a percentage change is a multiplication: a $20\%$ ⁠ rise is multiplying by $1.2$ ⁠, because what is left is $100\% + 20\% = 120\%$ ⁠ of the amount. The growth factor is the number you multiply by at every step. Interest of $5\%$ ⁠ a year, a population growing by $2\%$ ⁠ a year or a price rising by $3\%$ ⁠ a year are all exponential growth, each with its own factor. Try a few values of Growth per step, and see how a small rate gives a gentle curve.

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Step 2 of 5

Ten steps of ten per cent

Growth per step is locked at $\times 1.1$ ⁠, that is $10\%$ ⁠ per step. Ten such steps are $1.1^{10}$ ⁠. Before you see the result, predict how much that is.

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Step 3 of 5

Doubling in five steps

Steps is locked at $5$ ⁠. Find the growth per step that doubles the amount in five steps, so that the last column is close to $2$ ⁠.

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Step 4 of 5

Doubling time

How many steps does it take an amount to double? That number is called the doubling time, and it does not depend on the amount you start from: getting from $1$ ⁠ to $2$ ⁠ takes as many steps as getting from $1000$ ⁠ to $2000$ ⁠. Move Steps and read the value above the last column.

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Step 5 of 5

When the amount shrinks

The same idea works for a fall. A $20\%$ ⁠ fall at each step leaves $80\%$ ⁠, so you multiply by $0.8$ ⁠ at every step, a number less than $1$ ⁠. Such an amount keeps shrinking: this is exponential decay. The board shows only growth, but you can read it backwards: $\frac{1}{0.8} = 1.25$ ⁠, so what rises $1.25$ ⁠-fold at every step falls by $20\%$ ⁠ at every step when you go backwards. The tall column is the starting value, and you reach the first column after five steps of falling.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Calculate

Savings grow by $5\%$ ⁠ each year. What are they multiplied by each year?

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Practice · Predict, then look

At every step the amount grows by $10\%$ ⁠. How many times bigger will it be after ten steps?

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Practice · Reach the target

Steps is locked at $5$ ⁠. Set Growth per step so that the amount doubles in five steps: the value above the last column must be between $1.9$ ⁠ and $2.1$ ⁠. Press Check.

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Practice · Calculate

Growth per step is locked at $\times 1.1$ ⁠. After how many whole steps has the amount first doubled, that is, passed $2$ ⁠?

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Practice · Choose

A car loses $20\%$ ⁠ of its value every year. What part of the original value is left after $5$ ⁠ years?