Exponential growth vs. linear growth

The common mistake

Treating growth of 5% a year as adding 5% of the starting amount every year, or modelling “doubles each week” as adding the start value each week.

Instead: Ask whether each step adds the same amount (linear, y = mx + b) or multiplies by the same factor (exponential, y = a · bˣ). A percentage increase of r% gives b = 1 + r/100; a decrease gives b = 1 − r/100. Check with a table: equal differences mean linear, equal ratios mean exponential.

Worked example with Guide me

$200 grows by 10% a year. How much after 3 years?

1. Sugar · Does it add the same amount or multiply by the same factor?
   Student · Multiply — 10% of a growing amount.
2. Sugar · What's the factor each year?
   Student · 1.10.
3. Sugar · So the model is?
   Student · 200 × 1.10³.

Answer: $266.20 (not $260, which would be adding $20 each year).

Quick answers

What is the most common mistake with exponential growth vs. linear growth?

Treating growth of 5% a year as adding 5% of the starting amount every year, or modelling “doubles each week” as adding the start value each week.

How do I get exponential growth vs. linear growth right?

Ask whether each step adds the same amount (linear, y = mx + b) or multiplies by the same factor (exponential, y = a · bˣ). A percentage increase of r% gives b = 1 + r/100; a decrease gives b = 1 − r/100. Check with a table: equal differences mean linear, equal ratios mean exponential.

Can you show a worked example?

$200 grows by 10% a year. How much after 3 years? $266.20 (not $260, which would be adding $20 each year).

Related concepts

• Exponent rules for multiplying powers