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 with a annual interest rate, then how much will you owe after 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 years, you will first of all owe the original amount, called the principal and denoted by , of . Thus . But you will also owe the interest on that amount. The interest rate is , which we write as a number in the form . (Remember that means out of a hundred, namely , which is as a decimal .)
The rate 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 years, the total interest is times the principal , namely
So when it comes time to pay back the money, you will need to pay back a total amount of . 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 is agreed on, then the interest is directly proportional to the time . The relation is
If you double the amount of time, the interest doubles etc. In this case the rate 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 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 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 months?
Q2 (M): You want to borrow from your friend. If she charges you $5 interest for a 4 week loan, what rate of annual interest is that?
A1. $25; $50
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