It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

# 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.

## Introduction

Question: Would you rather receive $10,000 a day for a month or one penny ($0.01) that doubles every day for a month?

You might choose the $10,000 at first, but you'll see that with the magic of compounding, the doubling penny option would earn you a whopping$10,737,418 while the $10,000 every day would only make you$310,000 richer. This means that the penny would earn almost 35X the amount that the $10,000 did! ## Terminology & Formulas Interest that is earned on both the initial principal amount and the interest accrued in previous periods during the investment/loan term is called compound interest. In other words, it can be seen as "interest on interest" - hence the name compound interest. Unlike simple interest, compound interest grows exponentially (while simple interest grows linearly), making it a common method of earning interest when investing or borrowing money. To calculate compound interest, we have to consider the original amount of money, called principal (present value), the time (period over which the money is being used), the length and number of compounding periods (details below), and the (compound) interest rate. Taking these factors into account, we can use the following formula to calculate the maturity/future value of an amount earning compound interest: $$S=P(1+i)^n$$, where: • $$S$$ is the maturity/future value, • $$P$$ is the principal/present value, • $$i$$ is the periodic rate of interest, and • $$n$$ is the total number of compounding periods. Tip: We can also rearrange the above formula to solve for $$P$$, the principal (present value): $$P=S(1+i)^{-n}$$ Compounding Periods The following table shows common compounding frequencies/periods, their durations, how many times each period occurs per year, and how to calculate the interest rates per compounding period given the annual interest rate: ## Application Examples The following examples will demonstrate how we can use the formulas introduced in the previous section. Example: Marina is deciding between two investment opportunities for the next six years. Investment A pays 8.4% p.a. compounded bi-weekly and Investment B pays 7.5% p.a. compounded monthly. If she has$25,000 to invest, which investment option should she choose and how much more interest will she earn for that option?

Solution:

First, we can identify the information that is given in the problem. For both options, we have:

$$P=25,000, t=6$$ years

 Investment A Investment B Given information: $$i=\frac{0.084}{26}, n=26 \times 6 = 156$$ Calculation: $$S=25000(1+\frac{0.084}{26})^{156}=41349.63$$ Maturity Value: $$41,349.63$$ Interest Earned: $$41,349.63-25,000=$$ $$16,349.63$$ Given information: $$i=\frac{0.075}{12}=0.00625,n=12 \times 6=72$$ Calculation: $$S=25000(1+0.00625)^{72}=39152.94$$ Maturity Value: $$39,152.94$$ Interest Earned: $$39,152.94-25,000=$$ $$14,152.94$$

Comparing the two maturity values we get for both options, we find that Marina should choose Investment A, and she will earn $$16,349.63-14,152.94=$$ $$2196.69$$ more in interest with this option.

Example: What amount must be invested at 4.75% compounded quarterly for 6 years to reach a maturity value of $15,000? Solution: Since we're looking for the amount invested, we are determining the principal amount or $$P$$. First, we can identify the information that is given in the problem: $$S=15,000, i=\frac{0.0475}{4}=0.011875, n=6\times 4=24$$ Substituting these values into the compound interest formula gives: $$15000=P(1+0.011875)^{24}$$ Then, we can isolate for $$P$$: $$P=\frac{15000}{(1+0.011875)^{24}}$$ $$=11299.17$$ So, $$11, 299.17$$ must be invested to reach the required maturity value. Example: Debts of$1250 due 6 months from today and \$4725 due 4 years from today are to be settled by a single payment 1.5 years from today. If money is worth 8.75% compounded bi-annually, what is the size of the single equivalent payment?

Solution:

See the video below for the solution!

Tip: This solution will use the ideas of equivalent values and setting focal dates (To review these concepts, see Equivalent Values).