Year 11: How to solve exponential word problems

In Year 11 Mathematics in Australia, exponential functions are used to model situations where a quantity changes by the same factor in each time period. Common examples include:

Money in a bank account earning compound interest
Population growth
Radioactive decay and half-life
Anything that “doubles every 5 years” or “decreases by 8% each hour”

The challenge is usually not the calculation, but turning the words into an exponential equation. This tutorial gives you a clear “how to solve it” method.

Overall strategy

Understand the situation and identify what is changing.
Build an exponential model.
Substitute values or solve for the unknown.
Interpret and check your answer.

Stage 1: understand the problem

Step 1: read and highlight key information

As you read the question, highlight:

The quantity that is changing (money, people, mass, etc.).
The initial amount (starting value).
The growth or decay rate (often a percentage).
The time period for that rate (per year, per hour, per month, and so on).
What the question is asking for (final amount, time, rate, or initial amount).

Step 2: decide if the situation is exponential

You have an exponential situation if:

The quantity multiplies by the same factor every equal time step.
There are words like “grows by 5% per year” or “loses 12% of its mass each hour”.
The question mentions doubling time or half-life.

Stage 2: build an exponential model

Step 3: choose variables

Common choices:

$A$ for the amount after time $t$
$A_{0}$ or $P$ for the initial amount
$t$ for time
$r$ for the growth or decay rate as a decimal (for example, 7% becomes 0.07)

Step 4: write the exponential equation

The standard exponential models you will use are:

Compound interest / percentage growth
$A = P \times {(1 + r)}^{t}$ for growth
$A = P \times {(1 - r)}^{t}$ for decay
General exponential model
$A = A_{0} \times b^{t}$ , where $b$ is the growth factor per time unit.
Half-life model
If the half-life is $h$ units of time, then
$A = A_{0} \times {(1 / 2)}^{t / h}$

Substitute the information from the question into the model, leaving the unknown as a variable.

Stage 3: calculate or solve

Step 5: if the unknown is the final amount

If you are asked “what is the amount after 6 years?” or similar, you can usually just substitute into the model and use your calculator.

Example

A population of bacteria is 500 at first and grows by 12% per hour. What will the population be after 5 hours?

Model: $A = A_{0} \times {(1 + r)}^{t}$
Here $A_{0} = 500$ , $r = 0.12$ , $t = 5$ .

So $A = 500 \times {(1.12)}^{5}$ .

Evaluate ${(1.12)}^{5}$ on your calculator, then multiply by 500.

Step 6: if the unknown is the time or rate

If you are asked “how long will it take...” or “what rate is needed...” then the unknown is in the exponent. You will usually need logarithms to solve it. The basic idea is:

First, isolate the exponential expression (for example, $b^{t}$ ).
Then take logarithms of both sides and use the power rule to bring the exponent down.

You can see the full method for this in the logarithms tutorial below.

Stage 4: interpret and check

Step 7: write your answer in words with units

Turn your final number into a sentence, including units and any rounding the question asks for.

For example: “After 5 hours, the population will be approximately 880 bacteria.”

Step 8: does the answer make sense?

Ask yourself:

Has the amount gone up if it was growth, or down if it was decay?
Does the size of the answer feel reasonable given the rate and time?
If you roughly estimate the factor (for example, doubling or halving), is your answer in the right ballpark?

Summary checklist

Did you identify the initial amount, rate, and time correctly?
Did you choose the right exponential model and write it correctly?
Did you substitute the values carefully and use your calculator correctly?
Did you write your answer with correct units and context?

With practice, exponential word problems become very routine. Focus on spotting the rate, the time period, and the initial amount, and the rest will follow.