Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
What multiplier would you use to increase a value by:
a) \(10\%\)
b) \(25\%\)
c) \(8\%\)
d) \(12\%\) followed by a decrease of \(12\%\)
Solutions
Get Ready
Questions
Maria puts \($400\) in a bank account that earns \(5\%\) per annum interest compounded annually. If she leaves the money in the account for \(4\) years, how much money does she have when she withdraws it?
Solutions
Notes
Compound interest is when you earn interest which is added to your account, and you then earn interest on your interest from previous periods.
Sometimes interest is compounded more frequently than once per year.
In this case, the per annum interest rate is split equally between each period in the year.
The amount of time between each compound is known as the compounding period.
Find the amount of money that Maria would have received if the bank paid \(5\%\) per annum compounded:
a) half yearly
b) quarterly (every 3 months)
c) monthly
We can formalise this idea into the following formula for compound interest: $$FV=PV\times\left(1+\frac{r}{100k}\right)^{nk}$$where
\(PV\) is the principal value - the amount invested at the beginning
\(FV\) is the final value - the amount at the end of the investment
\(r\) is the rate - the percentage interest rate per annum
\(k\) is the number of compounding periods
The value of \(k\) is determined as follows:
“compounded annually” means \(k=1\);
“compounded semi-annually” means \(k=2\);
“compounded quarterly” means \(k=4\);
“compounded monthly” means \(k=12\);
“compounded daily” means \(k=365\).
Examples and Your Turns
Example
You invest \($1000\) at an interest rate of \(4\%\) per annum compounded monthly. If you invest for \(8\) years, calculate how much money you have at the end of the investment.
Your Turn
You want to invest \($1000\) at \(6\%\) per annum, compounded quarterly. How long will it take for this investment to increase to \($2000\)?
Your Turn
You want to invest \($1000\). What annual interest rate is needed to make this investment grow to \($2000\) in \(10\) years, if the interest is compounded quarterly?
Your Turn
You invest a lump sum for \(10\) years at an annual interest rate of \(2.5\%\) compounded monthly. How much did you invest if you have \($2000\) when the investment matures?
Notes
Most goods devalue over time. The older it is, the less it is worth. This is known as depreciation.
There are some exceptions to this general rule.
The formula for depreciation is $$FV=PV\times\left(1-\frac{r}{100k}\right)^{nk}$$
Notice that the only difference is that we subtract the interest rate instead of adding it.
Examples and Your Turns
Example
An industrial dishwasher was purchased for \($2400\) and depreciated by \(15\%\) each year.
Find its value after \(6\) years.
Your Turn
A car was bought for \($25,000\) 8 years ago. It sells today for \($11,500\). Assuming a constant rate of depreciation, determine the annual rate at which it depreciated.
Notes
You can use the graphical display calculator to perform financial calculations with compound interest.
The symbols in your GDC have the following meaning:
\(n\) - number of years
\(I\%\) - interest rate as a percentage
\(PV\) - principal (starting) value
\(PMT\) - payments (not a part of A&A course, will always be \(0\))
\(FV\) - final value
\(P/Y\) - number of payments per year (set to \(1\))
\(C/Y\) - number of compounding periods per year (value of \(k\))
It is important to note that when investing PV is negative as you are giving your money to the bank, so you no longer have it. In this case FV is positive as you are receiving the money back at the end of the investment.
If you are taking a loan, then PV is positive as you have some new money right now. FV will be negative in this case as you are paying the money back to the bank.
The most important thing is that PV and FV must have different signs.
If using this to answer a question, you write out the values of the variables above as your working.
Examples and Your Turns
Example
Sally invests \($15,000\) in an account that pays \(4.25\%\) p.a. interest compounded monthly. How much is her investment worth after \(5\) years?
-
Input the following in the GDC Finance package $$n=5 \\ I\%=4.25 \\ PV=-15000 \\ PMT=0 \\ FV=0 \\ PpY=1 \\ CpY=12$$
Choose the button to calculate FV which gives $$FV=18544.53$$
Your Turn
Helena is investing money in a savings account paying \(5.2\%\) p.a. interest compounded quarterly. How much does she need to invest now in order to collect \($5000\) at the end of \(3\) years?
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Input the following in the GDC Finance package $$n=3 \\ I\%=5.2 \\ PV=0 \\ PMT=0 \\ FV=5000 \\ PpY=1 \\ CpY=4$$
Choose the button to calculate PV which gives $$PV=4282.10$$
Your Turn
For how long must Magnus invest \($4000\) at \(6.45\%\) p.a. compounded semi-annually, for it to amount to \($10,000\)?
-
Input the following in the GDC Finance package $$n=0 \\ I\%=6.45 \\ PV=-4000 \\ PMT=0 \\ FV=10000 \\ PpY=1 \\ CpY=2$$
Choose the button to calculate \(n\) which gives $$n=14.43$$
Your Turn
Iman deposits \($5000\) in an account that compounds interest monthly. \(2.5\) years later, the account has a balance of \($6000\). What is the annual rate of interest that has been paid?
-
Input the following in the GDC Finance package $$n=2.5 \\ I\%=0 \\ PV=-5000 \\ PMT=0 \\ FV=6000 \\ PpY=1 \\ CpY=12$$
Choose the button to calculate \(I\) which gives $$I=7.32\%$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.