Percentage change

Discount:

Example: a How much is saved if a 25% discount is offered on an item marked $8.00? Discount = 25×8 = 200 = $2 100 100

b What is the new discounted price of this item? New price = original price - discount = 8 - 2 =$6 or New price = (100 - r) × original price = 75×8 = $6 100 100

Increase:

Example: a If petrol prices increase by 10%, what is the amount of the increase when the price is 99.0 cents per litre? Increase = 10 × 99 = 9.9 cents 100b What is the new price per litre for petrol? New price = 99 + 9.9 = 108.9 cents per litre

Given the original price and the new price of an item, we can work out the percentage change.

Example: a If the price of a book is reduced from $25 to $20, what percentage discount has been applied? percentage discount = discount × 100 = 5 × 100 =20 % original price 25

b If the price of a book is increased from $20 to $25, what percentage increase has been applied? percentage increse = increase × 100 = 5 × 100 =25 % original price 20

Sometimes we are given the new price and the percentage increase or decrease (r%), and asked to determine the original price. We can do this by using the above formulas for new price and rearange them to find the original price .

Example:Suppose that Steve has a $60 gift voucher from his favourite shop. a If the store has a ‘25% off’ sale, what is the original value of the goods he can now buy? New price = original price ×100 - r 100 60 = original price × 75 100 original price = 100 ×60 = $80 75

b If the store raises its prices by 25%, what is the original value of the goods he can now buy?

new price = original price × 100 +r 100 60 = original price × 125 100original price = 100 × 60 = $48 125

Simple interest

The price of borrowing or lending money is called interest. The simple interest is the interest that is calculated only by the initial amount of investment or loan and is not calculated according to the balance. The following formula is used for simple interest:

From the graph we can see that the relationship of ‘time’ and ‘simple interest’ is linear. The slope or gradient of a line which could be drawn through these

points is numericaly equal to the interest rate.

To determine the amount of the investment, the interest earned is added to the amount initially invested.

Example:How much interest would be due on a loan of $5000 at 10% per annum for six months? I = Prt = 5000 × 10 × 0.5 = 2000 = $250 100 100 100

Example:Find the total amount owed on a loan of $10 000 at 12% per annum simple interest at the end of two years.

I = Prt = 10000 × 12 × 2 = 240000 = $2400 100 100 100A = P+I =10 000 + 2400 = $12 400

Using the simple interest formula we can find the Principal, the Interest Rate and the Period.

Example: Find the length of time it would take for $50 000 invested at an interest rate of 8% per annum to earn $10 000 interest.I = Prt 100 t = 100 I = 100 × 10 000 = 25 years Pr 50 000 × 8

Example: Find the amount that should be invested in order to earn $1350 interest over 3 years at an interest rate of 4.5% per annum. I = Prt 100 P = 100 I = 100 × 1350 = $10 000 rt 4.5 × 3

Example: Find the rate of simple interest charged per annum if a loan of $20 000 incurs interest of $12 000 after eight years. I = Prt 100 r = 100 I = 100 × 12 000 = 7.5% per annum Pt 20 000 × 8

Compound interest

Compound interest is different from simple interest because is calculated according to the new balance and not according to the initial amount.The relationship of the ‘compount interest’ against ‘time’ is not linear and is clearly shown on the graph below:

To find the compount interest we can use the ‘Compount Interest Formula’ or the ‘TVM’ solver which is on the calculator menu of your CAS.

To find the amount of interest earned, we need to subtract the initial investment (P) from the final amount(A). I = A - P

Example:a Determine the amount of money accumulated after four years if $10000 is invested at an interest rate of 9% per annum, compounded annually, giving your answer to the nearest dollar. Using the formula:A = P (1+ r/n )t

100A = 10000 ( 1 + 9/100)4 =10000×1.4116 =$14116 to the nearest dollar

The interpretation of each function of ‘finance solver is as follows:N = number of periodsI% = interest PV=initial amount (invested or borrowed)PMT= repayments (not applicable)FV = final amountPpY = payments in a yearCpY = the times that the interest is compounded in a yearNote:We enter +when we receive money (loans) and - when we give money (repayments,investements)

Using the TVM

b Determine the amount of interest earned. I = A - PI = 14116 - 10000 = $ 4116

Example:a Determine the amount accrued if $2700 is invested at an interest rate of 6% per annum for a period of two years and interest is compounded monthly.

Using the formula:A = P (1+ r/n )nt

100A = 2700 ( 1 + 6/12)12×2 = $3043.33 100

Example: How much money must you deposit at 7% per annum compound interest, compounding yearly, if you require $10 000 in three years’ time? Give your answer to the nearest dollar.

Using the finance solver