Interest, Growth and Decay

Number

The model

A quantity changing by a fixed percentage each period grows or decays exponentially
The same relationship covers both directions: B = A × kⁿ, where A is the starting amount, B the amount after n periods, and k the multiplier
A value of k above 1 gives growth, and a value between 0 and 1 gives decay
k is never negative, since a multiplier below zero would flip the sign of the quantity
The period need not be a year: questions use hours, days or minutes just as readily

Where it appears

Growth	Decay
Population size	Radioactive mass
Bacterial cultures	A cooling liquid
An investment compounding	An asset losing value

Rearranging the model

Where the starting amount is unknown, divide the final amount by kⁿ
Where the multiplier is unknown, divide the final amount by the starting amount and take the nth root
Where the number of periods is unknown, test whole-number values until the two sides agree, since n counts completed periods

Exam tip

Match the multiplier to the period the rate is quoted for: a monthly rate needs the number of months. These papers never use the word "depreciation" — they describe a value falling by a percentage each year instead. Let the context fix the form, since a count of people or organisms has to be a whole number.

Worked example

Exponential growth of a population

A colony of 500 bacteria grows by 8% every hour. Find the population after 10 hours.

Solution:

Growth of a fixed percentage each period is exponential, so the same multiplier is applied once per period
Growing by 8% leaves 108% of the population, so the multiplier is 1.08
The rate is per hour and the time is in hours, so the power is 10 — always match the power to the period the rate is quoted for
Population = 500 × 1.08¹⁰
1.08¹⁰ = 2.158925…, so the population has slightly more than doubled
500 × 2.158925… = 1079.46
Bacteria are counted in whole organisms, so the answer is about 1079
The context fixes the form of the answer here: 1079.46 bacteria is not a possible population, and rounding is decided by the situation rather than by the halfway rule