Simple Interest 💡

students, imagine borrowing money to buy a laptop or saving money in a bank account. If the extra money added is always based on the original amount, not on earlier interest, you are working with simple interest. This lesson explains how simple interest works, why it matters in real life, and how it fits into the IB Mathematics: Applications and Interpretation SL topic of Number and Algebra.

What you will learn

By the end of this lesson, you should be able to:

explain the key words used in simple interest,
use simple interest formulas accurately,
solve real-world problems involving borrowing and saving,
connect simple interest to algebra, sequences, and numerical modelling,
use technology to check and interpret results.

Simple interest is one of the clearest examples of how mathematics models money in everyday life 💰. It is also a good introduction to algebraic reasoning because the same formula can be rearranged to find different unknowns.

1. What is simple interest?

Simple interest is interest calculated only on the original principal, not on accumulated interest. The principal is the starting amount of money, written as $P$. The interest rate is usually written as $r$ and is often given as a percentage per year. The time is written as $t$ and is usually measured in years.

If money earns simple interest, the interest grows at a constant rate. That means the increase each year is the same amount. This makes simple interest a linear model.

The main formula is:

$$I = Prt$$

where:

$I$ is the interest earned or paid,
$P$ is the principal,
$r$ is the annual interest rate as a decimal,
$t$ is the time in years.

The total amount after interest is:

$$A = P + I$$

Substituting $I = Prt$ gives:

$$A = P(1 + rt)$$

This formula is very useful because it combines the principal and the interest in one expression.

Example 1

Suppose students deposits $500$ into a savings account that pays simple interest at a rate of $4\%$ per year for $3$ years.

First convert the percentage to a decimal:

$$r = 0.04$$

Then apply the formula:

$$I = Prt = 500(0.04)(3)$$

$$I = 60$$

So the interest earned is $60$.

The total amount is:

$$A = 500 + 60 = 560$$

After $3$ years, the account has $560$.

2. Key terminology and units

Understanding the vocabulary is important in IB Maths because word problems often hide the mathematics inside everyday language.

Principal $P$: the starting amount of money.
Rate $r$: the interest rate per time period, usually per year.
Time $t$: how long the money is borrowed or invested.
Interest $I$: the extra money earned or paid.
Amount $A$: the total value after interest is added.

A common mistake is mixing up a percentage and a decimal. For example, $6\%$ must be written as $0.06$ before using the formula. Another common issue is the time unit. If the rate is per year, then time must be measured in years.

Example 2

A loan of $1200$ is taken for $9$ months at simple interest of $5\%$ per year.

Because the rate is per year, the time must be written in years:

$$t = \frac{9}{12} = \frac{3}{4}$$

Now calculate the interest:

$$I = 1200(0.05)\left(\frac{3}{4}\right)$$

$$I = 45$$

The total amount to repay is:

$$A = 1200 + 45 = 1245$$

This example shows why unit conversion matters in financial mathematics.

3. Simple interest as a linear model

In Number and Algebra, simple interest is a strong example of a linear relationship. If $P$ and $r$ are fixed, then the interest $I$ changes directly with time $t$.

From

$$I = Prt$$

we can see that $I$ has the form

$$I = mt$$

where $m = Pr$. This means the graph of interest against time is a straight line through the origin.

A straight-line model is useful because it helps predict future values and compare different financial situations. For example, if an investment earns $20$ dollars per year in simple interest, then after $1$ year it earns $20$, after $2$ years it earns $40$, and after $5$ years it earns $100$.

This is different from compound interest, where interest is earned on both the principal and previously earned interest. Simple interest does not grow faster over time in that way; it increases steadily.

Example 3

A savings plan earns simple interest of $75$ each year. The interest after $t$ years is

$$I = 75t$$

This is a linear expression in $t$. If $t = 4$, then

$$I = 75(4) = 300$$

The graph of $I$ against $t$ has gradient $75$ and passes through $(0,0)$.

That gradient represents the yearly interest earned.

4. Rearranging the formula

IB Mathematics often asks students to solve for different variables. The formula

$$I = Prt$$

can be rearranged to find any one unknown if the other three are known.

To find the principal:

$$P = \frac{I}{rt}$$

To find the rate:

$$r = \frac{I}{Pt}$$

To find time:

$$t = \frac{I}{Pr}$$

This is a key algebra skill because it shows how formulas can be manipulated to answer different questions.

Example 4

A person earns $90$ in simple interest from a $600$ investment over $3$ years. Find the rate.

Use

$$r = \frac{I}{Pt}$$

Substitute the values:

$$r = \frac{90}{600 \times 3}$$

$$r = \frac{90}{1800} = 0.05$$

So the rate is

$$5\%$$

This kind of problem appears often in finance-based questions because the formula can be rearranged in several ways.

5. Simple interest in real life

Simple interest is used in situations where the interest is calculated only on the original amount. This can happen in short-term loans, some informal lending situations, or simplified classroom models. It is also used as a comparison tool because it is easy to calculate and interpret.

For example, if two banks offer different interest plans, simple interest can help compare them quickly. Suppose one bank offers $3\%$ simple interest and another offers $2.5\%$ compound interest. The best choice depends on the time period. For a short time, simple interest may be easier to estimate. For a long time, compound interest usually grows more.

students, this shows the importance of interpretation. In IB Mathematics, the goal is not only to calculate answers but also to understand what those answers mean in context.

Example 5

You borrow $2000$ at $7\%$ simple interest for $2$ years.

First, calculate the interest:

$$I = 2000(0.07)(2)$$

$$I = 280$$

Then calculate the amount:

$$A = 2000 + 280 = 2280$$

This means you repay $2280$ in total.

If you know the total amount and the original amount, you can also work backward to check whether a loan offer is reasonable.

6. Technology and checking answers

Technology-supported interpretation is part of the course, so calculators and spreadsheet software can help verify simple interest calculations. For instance, a spreadsheet can be set up with columns for $P$, $r$, $t$, $I$, and $A$. This helps detect patterns and reduces calculation errors.

You can also use technology to graph the function

$$I = Prt$$

for fixed values of $P$ and $r$. The graph should be a straight line. If your graph is not linear, you may have entered a value incorrectly.

Technology is helpful, but it should not replace understanding. You still need to know why the formula works and what each variable means.

Conclusion

Simple interest is a basic but powerful idea in Number and Algebra. It uses the formula

$$I = Prt$$

and the amount formula

$$A = P(1 + rt)$$

to model financial situations with steady growth. It connects algebra, percentages, linear functions, and real-world financial reasoning. For IB Mathematics: Applications and Interpretation SL, simple interest is important because it builds the skills needed to analyse numerical models, interpret context, and communicate mathematical meaning clearly.

students, if you can explain simple interest, calculate it accurately, and interpret results in context, you have a strong foundation for more advanced financial mathematics 📘.

Study Notes

Simple interest is calculated only on the principal $P$.
The core formula is $I = Prt$.
The total amount is $A = P + I$, or $A = P(1 + rt)$.
Convert percentages to decimals before substituting into formulas.
Match the time unit to the rate unit, such as years for annual rates.
Simple interest is a linear model because interest increases by a constant amount each time period.
Rearranging formulas is an important algebra skill: $P = \frac{I}{rt}$, $r = \frac{I}{Pt}$, and $t = \frac{I}{Pr}$.
In graphs, $I$ against $t$ is a straight line through the origin.
Technology can help check calculations and explore patterns, but understanding the model is still essential.