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# Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

## Terminology & Formulas

Usually, whenever we borrow, loan or invest money, we often also have to pay interest on it. The interest that is earned on solely the original amount of money being used is called simple interest.

To calculate the amount of simple interest, we have to consider the original amount of money, called principal (present value), the time (period over which the money is being used), and the (simple) interest rate. Taking these factors into account, we can use the following formula to calculate simple interest:

$$I=Prt$$,

where:

• $$I$$ is the total amount of interest earned,
• $$P$$ is the principal/present value,
• $$r$$ is the interest rate per time period $$t$$, and
• $$t$$ is the time.

Tips:

• Interest rates are usually annual simple interest rates, so you may have to convert the time period from days, months, or weeks into years for the appropriate $$t$$ value to use in the above formula.
• Additionally, interest rates are often expressed as a percentage (%), so don't forget to convert it into a decimal (by dividing by 100) before using it in the formula (and vice versa if we're determining the interest rate in percent when it's given as a decimal).

The previous formula only calculates the total simple interest earned, but we can also determine the total amount of money accumulated, often called maturity/future value using this formula:

$$S=P+I$$

We can use the first formula introduced, $$I=Prt$$, and substitute it into this one, which gives us the formula:

$$S=P+Prt=P(1+rt)$$,

and rearranging this formula to solve for P gives us:

$$P=\frac{S}{(1+rt)}$$

## Application Examples

The following examples will demonstrate how we can use the formulas introduced in the previous section.

Example: Courtney borrowed $1200 from her bank at 7.5% simple interest for 40 weeks. Determine the amount due at the end of the 40 weeks (rounded to two decimal places). Solution: Since we want to determine the final amount due, we use the formula: $$S=P(1+rt)$$ Next, we can identify the information given in the problem: $$P=1200,\ r=\frac{7.5}{100}=0.075,\ t=\frac{40}{52}=\frac{10}{13}$$ *Note: The $$t$$ value is calculated as $$\frac{40}{52}$$ since the given interest rate is the annual/yearly rate, but we we're given the time in weeks. So to convert from weeks to years, we divide by $$52$$ (the number of weeks in a year). Now that we have the appropriate values for the principal, interest rate, and time, we can substitute them into the formula and evaluate the final amount due: $$S=1200(1+0.075(\frac{10}{13}))=1269.23$$ So, the amount due is $$1269.23$$. Example: Jamie lends a friend$500 for fifteen days. When his friend pays it back, he gets an extra $13 as a token of appreciation. What is the annual rate of simple interest that Jamie earned (rounded to two decimal places)? Solution: First, we can identify the information that is given in the problem: $$P=500,\ t=\frac{15}{365},\ S=500+13=513$$ *Note: The $$t$$ value is calculated as $$\frac{15}{365}$$ since the interest rate we want to find is the annual rate, but we we're given the time in days. So to convert from days to years, we divide by $$365$$ (the number of days in a year). Since we want to determine the annual rate of simple interest, $$r$$, and we're given $$P$$, $$t$$, and $$S$$, we use the formula: $$S=P(1+rt)$$ and rearrange it to find $$r$$: $$513=500(1+r(\frac{15}{365}))$$ $$\frac{513}{500}=1+r(\frac{15}{365})$$ $$\frac{513}{500}-1=r(\frac{15}{365})$$ $$\frac{\frac{513}{500}-1}{\frac{15}{365}}=r$$ $$r=0.6327$$ So, Jamie earned $$63.27$$% per annum. Tip: This example demonstrates that we can rearrange the general formula $$S=P(1+rt)$$ into $$r=\frac{\frac{S}{P}-1}{t}=\frac{S-P}{Pt}$$, then use this rearranged formula to solve for $$r$$. Example: Samantha's insurance company gives her the choice to collect$20,000 now or \$20,750 seven months from now. If money is worth 4.5% simple interest, which option is better for Samantha and by how much (in terms of today's dollar)?

Solution:

See the below video for the solution.