2.11

UNSW Sydney Interest charges can take a bite out of earnings.

Principle, interest and amounts

An important direct proportionality in our every day life is the amount of interest we get from our bank deposits.

In this step we will review basic terminology associated to simple interest.

Borrowing and interest rates

If you borrow $\normalsize{\800}$ with a $\normalsize{2\%}$ annual interest rate, then how much will you owe after $\normalsize{3}$ years? The answer depends on whether we are talking about simple interest or compound interest. Let’s here consider the simpler case of simple interest.

After $\normalsize{3}$ years, you will first of all owe the original amount, called the principal and denoted by $\normalsize{P}$, of $\normalsize{\800}$. Thus $\normalsize{P=800}$. But you will also owe the interest on that amount. The interest rate is $\normalsize{2\%}$, which we write as a number in the form $\normalsize{r=0.02}$. (Remember that $\normalsize{1\%}$ means $\normalsize{1}$ out of a hundred, namely $\normalsize{1/100}$, which is as a decimal $\normalsize{0.01}$.)

The rate $\normalsize{r}$ is in this case an annual rate, that is the amount you pay per year, and since you have borrowed the money for a period of $\normalsize{t=3}$ years, the total interest $\normalsize{I}$ is $\normalsize{rt=0.02 \times 3=0.06}$ times the principal $\normalsize{P=800}$, namely

So when it comes time to pay back the money, you will need to pay back a total amount $\normalsize{A}$ of $\normalsize{\848}$. Using symbols, this is the total

which we can simplify to

This is our basic formula.

Interest rate as a direct proportionality

There is a direct proportionality at work here. Once the interest rate $\normalsize{r}$ is agreed on, then the interest $\normalsize{I}$ is directly proportional to the time $\normalsize{t}$. The relation is

If you double the amount of time, the interest doubles etc. In this case the rate $\normalsize{r}$ is playing exactly the same role of a constant of proportionality, or as the slope of a hill.

It is one of the delights of mathematics to realize that sometimes quite different situations are governed by the same mathematical relationships. So the understanding that we gain in one area can be directly applied to another.

While this example involved a loan, the same relation is involved if you lend someone money.

Q1 (E): You agree to give $\normalsize{\2000}$ to a friend for a start-up. He promises to pay you back once he gets proper funding from a bank, and agrees on an (annual) rate of $\normalsize{5\%}$ for a certain number of months. If he pays you back your money in three months, how much interest will you have earned? How about if he pays you back in $\normalsize{6}$ months?

A2. 13%