SAMPLE. Financial arithmetic

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1 C H A P T E R 6 Financial arithmetic How do we determine the new price when discounts or increases are applied? How do we determine the percentage discount or increase applied, given the old and new prices? How do we determine the old price, given the new price and the percentage discount or increase? What do we mean by simple interest, and how is it calculated? What do we mean by compound interest, and how is it calculated? How do we calculate appreciation and depreciation of assets? What is hire-purchase, and how is the interest rate determined? 6.1 Percentage change In many financial situations you will need to calculate percentages. Suppose that the price of an item is discounted, or marked down, by10%. This means that you will pay a reduced price. The amount of the reduction or discount is: and Discount = 10% of original price = 0.10 original price New price = % of old price 10% of old price = 90% of old price = 0.90 old price In general, if r% discount is applied: Discount = r original price New price = original price discount = ( r) original price 224

2 Chapter 6 Financial arithmetic 225 Example 1 Calculating the discount and the new price a How much is saved if a 10% discount is offered on an item marked $50.00? b What is the new discounted price of this item? Solution a Evaluate the discount. b Evaluate the new price by either: subtracting the discount from the original price, or directly calculating 90% of the original price. Discount = 10 $50.00 = $5.00 New price = original price discount = $50.00 $5.00 = $45.00 New price = $ 90 $50.00 = $45.00 Sometimes, prices are increased, or marked up. Ifaprice is increased by 10%: and Increase = 10% of original price = 0.10 original price New price = % of old price + 10% of old price = 110% of old price = 1.10 old price In general, if r % increase is applied: Increase = r original price ( + r) New price = original price + increase = original price Example 2 Calculating the increase and the new price a How much is added if a 10% increase is applied to an item marked $50.00? b What is the new increased price of this item? Solution a Evaluate the increase. b Evaluate the new price by either: adding the increase to the original price, or directly calculating 110% of the original price. Increase = = $5.00 New price = original price + increase = = $55.00 New price = = $55.00

3 226 Essential Standard General Mathematics Calculating the percentage change Given the original price and the new price of an item, we can work out the percentage change. To do this, the amount of the decrease or increase is determined and then converted to a percentage of the original price. Example 3 Percentage discount = discount original price 1 % Percentage increase = increase original price 1 % Calculating the percentage discount or increase a If the price of an item is reduced from $50 to $45, what percentage discount has been applied? b If the price of an item is increased from $50 to $55, what percentage increase has been applied? Solution a 1 Determine the amount of the discount. 2 Express this amount as a percentage of the original price. b 1 Determine the amount of the increase. 2 Express this amount as a percentage of the original price. Calculating the original price Discount = original price new price = = $5.00 Percentage discount = = 10% 1 Increase = new price original price = = $5.00 Percentage increase = = 10% 1 Sometimes we are given the new price and the percentage increase or decrease (r%), and asked to determine the original price. Since we know that: ( r) for a discount: New price = original price ( + r) for an increase: New price = original price we can rearrange these formulas to give rules for determining the original price, as follows: When r % discount has been applied: Original price = new price ( r) When r % increase has been applied: Original price = new price ( + r)

4 Chapter 6 Financial arithmetic 227 Example 4 Calculating the original price Suppose that Cate has a $50 gift voucher from her favourite shop. a If the store has a 10% off sale, what is the original value of the goods she can now purchase? b If the store raises its prices by 10%, what is the original value of the goods she can now purchase? Solution a Substitute new price = 50 and r = 10 into the formula for a r% discount. b Substitute new price = 50 and r = 10 into the formula for a r% increase. Exercise 6A Original price = = $55.56 Original price = = $ Calculate the amount of the discount in each of the following, to the nearest whole cent. a 24% discount on $360 b 72% discount on $250 c 6% discount on $9.60 d 9% discount on $812 e 12% discount on $86.90 f 19% discount on $38 g 2.6% discount on $900 h 14.2% discount on $ Calculate the amount of the increase in each of the following, to the nearest whole cent. a 1.08% increase on $ b 15.9% increase on $4760 c % increase on $1520 d 12 1 % increase on $ e 9 1 % increase on $ f 2.8% increase on $ g 0.2% increase on $ Calculate the following as percentages, correct to 2 decimal places. a $19.56 of $400 b 60c/ of $2 c $6.82 of $24 d $24 of $2600 e 30c/ of 90c/ f $1.50 of $13.50 g 38c/ of $10.50 h 25c/ of $186 4 Calculate the new increased price for each of the following. a $260 marked up by 12% b $580 marked up by 8% c $42.50 marked up by 60% d $5400 marked up by 17% e $42.80 marked up by 35% f $9850 marked up by 8% g $258 marked up by 14.2% h $1960 marked up by 17.4% 5 Calculate the new discounted price for each of the following. a $2050 discounted by 9% b $11.60 discounted by 4% c $154 discounted by 82% d $ discounted by 3% e $980 discounted by 13.5% f $2860 discounted by 8%

5 228 Essential Standard General Mathematics g $ discounted by 22.7% h $147 discounted by 2.8% i $674 discounted by 12% 6 Find the original prices of the items that have been marked down as follows. a Marked down by 10%, now priced $54.00 b Marked down by 25%, now priced $37.50 c Marked down by 30%, now priced $50.00 d Marked down by 12.5%, now priced $ Find the original prices of the items that have been marked up as follows. a Marked up by 20%, now priced $15.96 b Marked up by 12.5%, now priced $70.00 c Marked up by 5%, now priced $ d Marked up by 2.5%, now priced $ Mikki has a card that entitles her to a 7.5% discount at the store where she works. How much will she pay for boots marked at $230? 9 The price per litre of petrol is $1.15 on Friday. When Rafik goes to fill up his car, he finds that the price has increased by 2.3%. If his car holds 50 L of petrol, how much will he pay to fill the tank? 6.2 Simple interest When you borrow money, you have to pay for the use of that money. When you invest money, someone else will pay you for the use of your money. The amount you pay when you borrow, or the amount you are paid when you invest, is called interest. There are many different ways of calculating interest. The simplest of all is called, rather obviously, simple interest. Simple interest is a fixed percentage of the amount invested or borrowed and is calculated on the original amount. Suppose we invest $0 in a bank account that pays simple interest at the rate of 5% per annum. This means that, for each year we leave the money in the account, interest of 5% of the original amount will be paid to us. That is, Interest, I = $0 5 = $50 If the money is left in the account for several years, the interest will be paid each year. To calculate simple interest we need to know: the amount of the investment, the interest rate and the length of time for which the money is invested.

6 Chapter 6 Financial arithmetic 229 Example 5 Calculating simple interest How much interest will be earned if we invest $0 at 5% interest per annum for 3 years? Solution 1 Calculate the interest for the first year. 2 Calculate the interest for the second year. 3 Calculate the interest for the third year. 4 Calculate the total interest. Interest = 0 5 = $50 Interest = 0 5 = $50 Interest = 0 5 = $50 Interest for 3 years = = $150 The same principles apply when the simple interest is applied to a loan rather than an investment. The simple interest formula Since the amount of simple interest earned is the same every year, we can apply a general rule. amount invested interest rate (per annum) length of time (in years) Interest = or I = P r t = Prt where the amount invested or borrowed ($P) isknown as the principal, r% isthe interest rate per annum and t is the time in years. How does this relationship look graphically? Suppose we were to borrow $0 at 5% per annum simple interest for a period of years. A plot of interest against time is shown. Interest ($) Year From this graph we can see that the relationship is linear. The amount of interest paid is directly proportional to the time for which the money is borrowed or invested. The slope, or gradient, of a line that could be drawn through these points is numerically equal to the interest rate. To determine the amount of the investment, the interest is added to the amount invested. Amount of the investment A = P + I = P + Prt where $P is the amount invested or borrowed, r% isthe interest rate per annum and t is the time in years.

7 230 Essential Standard General Mathematics If the money is invested for more or less than 1 year, the amount of interest payable is proportional to the length of time for which it is invested. Example 6 Calculating simple interest for periods other than one year Calculate the amount of simple interest that will be paid on an investment of $5000 at 10% simple interest per annum for 3 years and 6 months. Solution Apply the formula with P = $5000, r = 10% and t = 3.5 (since 3 years and 6 months is equal to 3.5 years). Example 7 I = Prt 10 = = $1750 Calculating the total amount borrowed or invested Find the total amount owed on a loan of $ at 8% per annum simple interest at the end of 2years. Solution 1 Apply the formula with P = $16 000, r = 8% and t = 2tofind the interest. 2 Find the total owed by adding the interest to the principal. I = Prt 8 = = $2560 A = P + I = = $ The graphics calculator enables us to investigate simple interest problems using both the tables and graphing facilities of the calculator. How to solve simple interest problems using the TI-Nspire CAS How much interest is earned if $ is invested at 8.25% simple interest for 10 years? Show that the graph of simple interest earned is linear. Steps 1 Substitute P = $ and r = 8.25% in the formula for simple interest. I = Prt t = = 825t

8 Chapter 6 Financial arithmetic Start a new document (by pressing / + N) and select 3:Add Lists & Spreadsheet. Name the lists time (to represent time in years) and interest. Enter the data values 1 10 into the list named time,asshown. Note: You can also use the sequence command to do this. 3 Place the cursor in the grey formula cell in the list named interest and type in = 825 time. Note: You can also use the key and paste time from the variable list. Press to display the values. enter By scrolling down the table (using ) we can see that the interest amount of $8250 will be earned after 10 years. 4 Press /5:Add Data & Statistics and plot the graph as shown. a To connect the data points Move the cursor to the graphing area and press / + b. Select 2:Connect Data Points. b To display a value Place the cursor on the data point and hold. Press before moving to another data point. From the plot we can see that the graph of the amount of simple interest earned is linear. The slope of the graph is equal to the interest paid each year. Note: You can also graph this example in the Graphs & Geometry application.

9 232 Essential Standard General Mathematics How to solve simple interest problems using the ClassPad How much interest is earned if $ is invested at 8.25% simple interest for 10 years? Show that the graph of simple interest earned is linear. Steps 1 Substitute P = $ and r = 8.25% in the formula for simple interest. I = Prt t = = 825t 2 To form a table of values, open the Sequence ( ) application. Select the Explicit tab and move the cursor to the box opposite a n E: and type 825n. The n is found in the toolbar ( ). Press E to confirm your entry (which is indicated by a tick in the square to the left of a n E:). 3 To display the terms of the sequence in table format, tap the # icon. By scrolling down the table, it can be seen that the interest amount of $8250 will have been earned after 10 years. Note: Tap the Sequence TableInput (8) icon in the toolbar and adjust the Start and End values if more values are required.

10 Chapter 6 Financial arithmetic To graph the sequence of simple interest values, select the Sequence Grapher ( ) icon from the toolbar. Note: To define the graph window scale, select the View Window (6) icon from the toolbar and set the values as shown. Tap OK to confirm your settings. Leave the xdot and ydot settings as they are. These values control the trace increment of the cursor. 5 From the menu bar, select Analysis and then Trace. This will place a marker on the graph at the first value in the table. Use the cursor arrow keys ( and )to move from one table value to the next. From the plot we can see that the graph of the amount of simple interest earned is linear. The slope of the graph is equal to the interest paid each year. Interest paid to bank accounts One very useful application of simple interest is in the calculation of the interest earned on a bank account. When we keep money in the bank, interest is paid. The amount of interest paid depends on: the rate of interest the bank is paying; and the amount on which the interest is calculated. Generally, banks will pay interest on the minimum monthly balance, which is the lowest amount the account contains in each calendar month. When this principle is used, we will assume that all months are of equal length, as illustrated in the next example.

11 234 Essential Standard General Mathematics Example 8 Calculating interest paid to a bank account The table shows the entries in Tom s bank account. Date Transaction Debit Credit Total 30 June Pay July Cash July Cash August If the bank pays interest at a rate of 3% per annum on the minimum monthly balance, find the interest payable for the month of July. Solution 1 Determine the minimum monthly balance for July. 2 Determine the interest payable for July. Exercise 6B The minimum balance in the account for July was $ I = Prt = = $0.88 or 88 cents 1 Calculate the amount of simple interest for each of the following: a b c d e f g h i Principal Interest rate Time $400 5% 4years $750 8% 5years $ % 8years $ % 3years $ % 15 years 4 $865 15% years $599 10% 6 months $ % 9 months $ % years

12 2 Calculate the amount to be repaid for each of the following: a b c d e Principal Interest rate Time $500 5% 4years $ % 3years $ % 6 months $ % 8 months $ % 250 days Chapter 6 Financial arithmetic A sum of $ was invested in a fixed term account for 3 years. Calculate: a the simple interest earned if the rate of interest is 6.5% per annum. b the total value of the investment at the end of 3 years. 4 A personal loan of $ is taken out for a period of 5 years, at a simple interest rate of 12% per annum. Find the amount still owing after 1 year, if payments of $ are made every month of that year. 5 A loan of $1200 is taken out at a simple interest rate of 14.5% per annum. How much is owing after 3 months? 6 A company invests $ in the short-term money market at 11% per annum simple interest. How much interest is earned by this investment in 30 days? Give your answer to the nearest cent. 7 Abuilding society offers the following interest rates for its cash management accounts. Interest rate (per annum) on term (months) Balance 1 <3 3 <6 6 <12 12 <24 24 <36 $ $ % 3.35% 3.85% 4.35% 4.85% $ $ % 3.50% 4.00% 4.50% 5.00% $ 000 $ % 3.90% 4.40% 4.90% 5.40% $ and over 4.00% 4.50% 5.00% 5.50% 6.00% Using this table, find the interest earned by each of the following investments. Give your answers to the nearest cent. a $ for 2 months b $ for 6 months c $ for 18 months d $ for 2 years e $ for 8 months f $ for 23 months 8 An account at a bank is paid interest of 4% per annum on the minimum monthly balance, credited to the account at the beginning of the next month. In October, the following transactions took place: 7 October $0 withdrawn 12 October $500 deposited

13 236 Essential Standard General Mathematics If the opening balance for October was $5000: a what was the balance of the account at the end of October? b how much interest was paid for the month? 9 The minimum monthly balances for three consecutive months are: $ $ $ How much interest is earned over the 3-month period if it is calculated on the minimum monthly balance at a rate of 3.5% per annum? 10 The bank statement below shows transactions for a savings account that earns simple interest at a rate of 4.5% per annum on the minimum monthly balance. Date Transaction Debit Credit Balance 1 March March Cash March Cash April 0.00 How much interest was earned in March? 11 The bank statement below shows transactions over a 3-month period for a savings account that earns simple interest at a rate of 3.75% per annum on the minimum monthly balance. Date Transaction Debit Credit Balance 1 March April Cash May Cash June a What were the minimum monthly balances in March, April and May? b How much was earned over this 3-month period? 6.3 Rearranging the simple interest formula The formula for simple interest can be rearranged to find any one of the variables when the values of the other three variables are known. Then either the formula for simple interest, or the 7 facility of a graphics calculator, may be used to determine the unknown value. Calculating the interest rate To find the interest rate per annum, r%, given the values of P, I and t: r = I Pt where $P is the principal, $I is the amount of interest and t is the time in years.

14 Chapter 6 Financial arithmetic 237 Example 9 Calculating the interest rate Find the rate of simple interest if a principal of $8000 increases to $ in 4 years. Solution 1 Find the interest earned on the investment. 2 Apply the formula with P = $8000, I = $3040 and t = 4tofind the value of r. 3 Since the unit of time was years, the interest rate can be written as the interest per annum. Calculating the time period Interest, I = = $3040 r = I 3040 = Pt = 9.5% Interest rate = 9.5% per annum To find the number of years or term of an investment, t years, given the values of P, I and r: t = I Pr where $P is the principal, $I is the amount of interest and r%isthe interest rate per annum. Example 10 Calculating the time period of a loan or investment Find the length of time it would take for $5000 invested at an interest rate of 12% per annum to earn $1800 interest. Solution Apply the appropriate formula with P = $5000, I = $1800 and r = 12 to find the value of t. Calculating the principal t = I 1800 = Pr = 3years To find the value of the principal, $P, given the values of I, r and t: P = I rt where $I is the amount of the interest, r% isthe interest rate per annum and t is the time in years. To find the value of the principal, $P, given the values of A, r and t: A P = ( 1 + rt ) where $A is the amount of the investment, r% isthe interest rate per annum and t is the time in years.

15 238 Essential Standard General Mathematics Example 11 Calculating the principal of a loan or investment a Find the amount that should be invested in order to earn $1500 interest over 3 years at an interest rate of 5% per annum. b Find the amount that should be invested at an interest rate of 5% per annum if you require $ in 4 years time. Solution a Since we are given the value of the interest, I, we will use the first formula with I = $1500, r = 5 and t = 3tofind the principal, P. b Here we are not given the value of the interest, I, but the value of the total investment, A. We will use the second formula with A = $15 600, r = 5 and t = 4tofind the principal, P. Exercise 6C P = I 1500 = rt 5 3 = $ A P = ( 1 + rt ) = ( ) = = $ Calculate the time taken for $2000 to earn $975 at 7.5% simple interest. 2 Calculate the principal that earns $ in 10 years at 4.25% simple interest. 3 Calculate the rate of simple interest if a principal of $5000 amounts to $6500 in years. 4 Calculate the principal that earns $780 in days at 6.25% per annum simple interest. 5 Calculate the annual rate of simple interest if a principal of $500 amounts to $550 in 8 months. 6 Calculate the time in days for $760 to earn $35 at 4 3 % simple interest. 4 7 Calculate the answers to complete the following table. Principal Rate Time Simple interest Total investment $600 6% 5years a b $ % c $ d $1290 e 6 months $45.15 f g 10% 4 months $150 h $3600 i 200 days $98.63 j $ % k l $ m 7 1 % 4 6 months $52.50 n 8 If Geoff invests $ at 10% per annum simple interest until he has $42 000, for how many years will he need to invest the money?

16 Chapter 6 Financial arithmetic Josh decides to put $5000 into an investment account that is paying 5.0% per annum simple interest. If he leaves the money there until it doubles, how long will this take, to the nearest month? 10 A personal loan of $ over a period of 3 years costs $500 per month to repay. a How much money will be repaid in total? b What equivalent rate of simple interest is being charged over the 3 years? 11 To buy her first car, Cassie took out a loan for $4000. She paid it back over a period of 2years, and this cost her $1450 in interest. What simple interest rate was she charged? 12 Over a period of 4 years, an investment earnt $913.50, at a simple interest rate of 5.25% per annum. What was the original amount deposited? 13 How long does it take for $2400 invested at 12% per annum simple interest to earn $360 interest? 6.4 Compound interest We have seen that simple interest is calculated on the original amount borrowed or invested. A more common form of interest, called compound interest, calculates the interest each time period on a sum of money to which the previous amount of interest has been added. The interest is said to compound. Consider, for example, $250 invested at 10% per annum, where the interest is added to the account each year. In the first year: Interest = $250 10% 1 = $25.00 so at the end of the first year the amount of money in the account is $250 + $25 = $275 In the second year: Interest = $275 10% 1 = $27.50 so at the end of the second year the amount of money in the account is In the third year: $275 + $27.50 = $ Interest = $ % 1 = $30.25 so at the end of the third year the amount of money in the account is And so on. $ $30.25 = $332.75

17 240 Essential Standard General Mathematics Continuing in this way, we can build a table of values. For comparison, the amount of the money if invested at 10% per annum simple interest is also shown. Amount of investment ($) Year (n) 10% simple interest 10% compound interest From the table, we can see that after the first month, the compound interest is higher, and this advantage to the investor becomes more obvious as time increases. The difference between the two investment strategies is even clearer when viewed graphically. Amount of investment ($) Year Compound interest at 10% per year Simple interest at 10% per year From the graph we see that the growth in the investment over time when interest is compounded is clearly not linear, rather, it curves upwards and away from the simple interest graph at an ever increasing rate. The compound interest formula Calculation of compound interest is very tedious if carried out for many years. We can, however, soon see a pattern to the calculations. In the previous example, the principal was increased by ( 10% at the end of each year. The multiplying factor to increase a quantity by 10% is ) = 1.1.

18 Thus, the amount accrued at the end of: 1st year = $ = $275 2nd year = $ = $250 (1.1) 2 = $ rd year = $ = $250 (1.1) 3 = $ nth year = $250 (1.1) n Chapter 6 Financial arithmetic 241 It can be seen that a formula for calculating the amount of an investment earning compound interest can be written as follows. In general, the amount of the investment is given by ( A = P 1 + r ) t where $A = final amount $P = initial amount (principal) r% = interest rate per annum t = number of years That is, if an amount $P is invested at r% per annum compound interest for t years, it will grow to $A. To find the amount of interest earned, we need to subtract the initial investment from the final amount. The interest ($I ) that would result from investing $P at r% per annum, compounded annually for t years, is given by: ( I = A P = P 1 + r ) t P where $A is the amount of the investment after t years. Example 12 Calculating the amount of the investment and interest a Determine, to the nearest dollar, the amount of money accumulated after 3 years if $2000 is invested at an interest rate of 8% per annum, compounded annually. b Determine the amount of interest earned. Solution a Substitute P = $2000, t = 3, r = 8 into the formula giving the amount of the investment. b Subtract the principal from this amount to determine the interest earned. ( A = P 1 + r ) t ( = ) 3 = $2519 to the nearest dollar I = A P = = $519 Another way of determining compound interest is to enter the appropriate formula into a graphics calculator, and examine the interest earned using both the tables and graphing facility of the calculator.

19 242 Essential Standard General Mathematics How to investigate compound interest problems using the TI-Nspire CAS a Determine, to the nearest dollar, the amount of money accumulated after 3 years if $2000 is invested at an interest rate of 8% per annum, compounded annually. b Determine the amount of interest earned. c Show that the graph of the amount of money accumulated curves up wards. Steps 1 Substitute P = $2000 and r = 8 into the formula for compound interest. 2 Start a new document (by pressing / + N) and select 3:Add Lists & Spreadsheet. Name the lists time (to represent time in years) and amount. Enter the data values 1 10 into the list named time,asshown. Note: You can also use the sequence command to do this. 3 Place the cursor in the grey formula cell in the list named amount and type in = 2000 (1 + 8 ) time Note: You can also use the key and paste time from the variable list. Press enter to display the values as shown. By scrolling down the table we can see that the a amount of money accumulated after 3years is $ b interest earned = $ $2000 = $ Press /5:Add Data & Statistics and plot the graph as shown. Notes: 1 To connect the data points: Press / + b and select 2:Connect Data Points. 2 To display a value: Place the cursor on the data point and hold. Press before moving to another data point. 3 You can use / + b and select 1:Zoom/1:Window Settings and set the Ymin to 0, ifyou prefer. c From the plot we see that, for compound interest, the graph of amount of money accumulated curves upwards with time. ( A = ) t

20 Chapter 6 Financial arithmetic 243 How to investigate compound interest problems using the ClassPad a Determine, to the nearest dollar, the amount of money accumulated after 3 years if $2000 is invested at an interest rate of 8% per annum, compounded annually. b Determine the amount of interest earned. c Show that the graph of the amount of money accumulated curves up wards. Steps 1 Substitute P = $2000 and r = 8 into the formula for compound interest. 2 To form a table of values, open the Sequence ( ) application. Select the Explicit tab and move the cursor to the box opposite a n E: and type 2000 (1 + 8/) n. The n is found in the toolbar ( ). Press E to confirm your entry (which is indicated by a tick in the square to the left of a n E:). 3 Tap # from the toolbar to view a table of values. By scrolling down the table we can see that the a amount of money accumulated after 3 years is $ b interest earned = $ $2000 = $ To graph the sequence of simple interest values, select the Sequence Grapher ( ) icon from the toolbar. Note: To define the graph window scale, select the View Window (6) icon from the toolbar and set the values as shown. Tap OK to confirm your settings. c From the plot we see that, for compound interest, the graph of amount of money accumulated curves upwards with time. ( A = ) t

21 244 Essential Standard General Mathematics How would our answer to Example 12 change if, instead of the interest being added to the account at the end of each year (called compounded annually), the interest is added every 3 months (called compounded quarterly)? Compounding 8% quarterly means that the interest rate for the 3-month period is reduced to 2%. However, the number of times the interest is added increases from three to 12, since the interest is added four times each year. This situation is shown in the table below. Amount of investment ($) Year Compounding annually Compounding quarterly The final entry in the table may be determined by evaluating: $2000 ( ) 12 = $ Thus, we need to modify the formula stated previously to take into account situations in which interest is compounded, or adjusted, other than annually. In general: ( Amount of the investment, A = P 1 + r/n where: $A = amount of the investment after t years $P = initial amount (principal) r% = interest rate per annum n = number of times per year interest is compounded t = number of years. Example 13 Interest compounding monthly a Determine the amount reached if $5000 is invested at an interest rate of 12% per annum for a period of 2 years and interest is compounded monthly. b Determine the amount of interest earned. ) nt

22 Solution a Substitute P = $5000, r = 12, n = 12 and t = 2 into the formula giving the amount of the investment. b Subtract the principal from this final amount to determine the interest earned. Calculating the principal Chapter 6 Financial arithmetic 245 ( A = P 1 + r /n ) nt ( = /12 ) 12 2 = 5000 (1.01) 24 = $6349 to the nearest dollar I = A P = $6349 $5000 = $1349 Sometimes we are interested in calculating the amount of money that should be invested, the principal P,toreach a particular amount in the investment A. The formula for compound interest can be rearranged as follows. In general, the principal is given by A P = ( 1 + r/n ) nt where: $A = amount of the investment after t years $P = initial amount (principal) r% = interest rate per annum n = number of times per year interest is compounded t = number of years. Example 14 Calculating the principal How much money should be invested at 8% per annum compound interest, compounding monthly, if $5000 is needed in 2 years time? Solution Substitute A = $5000, r = 8, n = 12 and t = 2 into the formula giving the principle. Exercise 6D A $5000 P = ( 1 + r /n ) nt = ( 1 + 8/12 ) 12 2 = $4263 to the nearest dollar 1 Calculate the final amount if $3500 is invested at 5% compound interest per annum for 5years. 2 Calculate the amount of compound interest earned from investing $7000 at 8% per annum for 4 years.

23 246 Essential Standard General Mathematics 3 Find the total amount needed to repay a loan of $1250 at 7 1 % compound interest per 2 annum over 3 years. 4 Calculate the difference between the simple interest and the compound interest on a loan of $2000 at 7% per annum over 5 years. 5 Calculate the amount of money that would need to be invested at 6 3 % per annum 4 compound interest to achieve a sum of $ in 4 years. 6 Calculate the interest paid on a loan of $2750 at 11% per annum, compounded quarterly for 4years. 7 Calculate the final amount to be paid on a loan of $ at 12% per annum for 5 years, for each of the following conditions. a Simple interest b Compound interest calculated annually c Compound interest calculated quarterly d Compound interest calculated monthly e Compound interest calculated daily 8 A bank offers various investment possibilities to a customer wishing to invest $ for 12 years. a Calculate the final amount for each of the following. i Simple interest at 13% per annum ii Compound interest at 12% per annum, calculated annually iii Compound interest at 11% per annum, calculated quarterly iv Compound interest at 10% per annum, calculated monthly v Compound interest at 9% per annum, calculated daily b Which investment would you recommend? 9 Suppose that $ is invested at 4.2% per annum, compounding monthly. How much interest is paid in the fourth year of the investment? 10 Joe buys a skateboard costing $530 on his credit card, knowing that he will not be able to pay it off for 6 months. If he is charged 18% per annum interest for 6 months, compounded monthly, how much will the skateboard end up costing him? (Assume that no payments are made within this time, and give your answer to the nearest cent.) 11 How much money must you invest at 12.5% per annum, compounded monthly, if you know that you will need $ in 3 years time? Give your answer to the nearest $10.

24 Chapter 6 Financial arithmetic Sue s parents open an investment account when she is born, in which they deposit $5000. They intend to withdraw the money on her 18th birthday. If the account pays 8.0% per annum, compounded quarterly, for the first 10 years, and then 6.0% per annum, compounded monthly, for the rest of the term of the investment, how much will Sue receive? Give your answer to the nearest $ Flat rate depreciation Depreciation and book value In the previous sections we looked at the ways in which investments grow or appreciate. Itis also possible, of course, for the value of investments to decrease. For example, many of the items purchased by businesses, such as motor vehicles and office equipment, decrease in value, or depreciate, and this is allowed for by accountants when estimating the value of a business. Depreciation is also important for tax purposes, where it is allowed as a deductible expense for a business. The government has prescribed the maximum rates of depreciation allowed on various types of equipment. The longer an item can be expected to last, the lower the rate of depreciation. Depreciation is an estimate of the annual reduction in the value of items, caused by such things as age and wear and tear. The book value of an item is its value at any given time. Book value = purchase price depreciation Once the item has reached the stage where it can no longer be used profitably by the company, it is sold off. The price that the item is expected to fetch at this point is called the scrap value of the item. The amount of time for which the item is in use is called the effective life of that item. Two common methods for calculating depreciation will be considered here: flat rate depreciation reducing balance depreciation. Flat rate depreciation Using this method the equipment or assets are depreciated in value by a fixed amount each year, normally a percentage of the original value. This is an equivalent, but opposite, situation to simple interest. We have already established a formula for simple interest. This formula can also be applied to the calculation of flat rate depreciation. purchase price depreciation rate (per annum) length of time (in years) Depreciation = or D = Prt where $D is the flat rate depreciation of the item after t years, $P is the purchase price of the item, r%isthe flat rate of depreciation per annum and t is the time in years.

25 248 Essential Standard General Mathematics To determine the book value of the item, the amount of depreciation is subtracted from the purchase price. The book value is given by V = P D = P Prt where $V is the book value of the item after t years, $P is the purchase price of the item, r% is the flat rate of depreciation per annum and t is the time in years. If the item is kept for longer than the original estimated life, the flat rate method will sooner or later give the item a book value of zero.the item is then said to be written off. As with simple interest, the formulas for flat rate depreciation and book value can be rearranged to find the value of any one of the variables, when the values of other variables are known. Example 15 Determining the flat rate depreciation and book value Michael purchases a new car for $ If it decreases in value by 10% of the purchase price each year: a what is the amount of the annual depreciation? b what is the amount of the depreciation after 4 years? c what is its book value after 4 years? Solution a The annual depreciation is 10% of the purchase price. b Substitute P = $24 000, r = 10 and t = 4 into the formula for flat rate depreciation. c The book value at the end of 4 years is the cost of the item less the depreciation Annual depreciation = = $2 400 D = Prt = = $9600 After 4 years V = = $14400 Because the depreciation occurs at a constant rate, a negative straight line relationship exists between the book value and the time over which the depreciation occurs. For this reason, this form of depreciation is also sometimes called straight-line depreciation. The graphics calculator can be set up to generate a table for flat rate depreciation over time.

26 Chapter 6 Financial arithmetic 249 How to determine flat rate depreciation and book value using the TI-Nspire CAS Michael purchases a new car for $ If it decreases in value by 10% of the purchase price each year: a What is the amount of the annual depreciation? b What is the amount of the depreciation after 4 years? c What is its book value after 4 years? Steps 1 Substitute P = $ and r = 10 into the formulae for depreciation and book value under flat rate depreciation. 2 Start a new document (by pressing / + N) and select 3:Add Lists &Spreadsheet. Name the lists time, depreciation, and book value. Hint: Use / + for the underscore or just write as bookvalue. 3 Enter the data values 1 10 into the list time. 4 Move the cursor to the grey formula cell of the list depreciation and type in = ( time)/ Press enter to calculate the values for depreciation. Move cursor to the grey formula cell of the list book value and type in = ( time)/ Press enter to calculate the values for book value. Note: An alternative formula to use to calculate the list book value would be = depreciation Hint: You can use the key todisplay the variable list rather than retyping the list names t D = = 2400t t V = Write your answers. a After 1 year, depreciation = $ 2400 b After 4 years, depreciation = $ 9600 c After 1 year, book value = $14400

27 250 Essential Standard General Mathematics How to determine flat rate depreciation and book value using the ClassPad Michael purchases a new car for $ If it decreases in value by 10% of the purchase price each year: a What is the amount of the annual depreciation? b What is the amount of the depreciation after 4 years? c What is its book value after 4 years? Steps 1 Substitute P = $ and r = 10 into the formulae for depreciation and book value under flat rate depreciation. 2 To form a table of values, open the Sequence ( ) application. Select the Explicit tab. Opposite an E: type in 2400n b n E: type in n t D = = 2400t t V = The n is found in the toolbar ( ). Press E to confirm your entries (which is indicated by ticks in the squares to the left of a n E: and b n E:). 3 Tap # from the toolbar to view the table of values. 4 Write your answers. a After 1 year, depreciation = $2400 b After 4 years, depreciation = $9600 c After 1 year, book value = $14400 Example 16 Flat rate depreciation and book value Afactory manager assesses that a particular machine, purchased for $30 000, has a useful life of 10 years. Its scrap value is estimated as $6000. a What is the annual amount of depreciation? b What is the flat rate of depreciation? c Draw a graph of the book value of the machine against time.

28 Chapter 6 Financial arithmetic 251 Solution a 1 Work out the total depreciation over the 10-year period. 2 Determine how much this is per year. b To find the flat rate of depreciation, the amount of the annual depreciation is expressed as a percentage of the original cost. c To draw a graph of the book value against time: 1 Substitute P = $ and r = 8 into the formula for flat rate depreciation (book value (V)). 2 Use a graphics calculator to plot V (book value) against t (time). Total depreciation = = $ Depreciation per year = = $ r = = 8% t V = = t From Example 16, we can see that the amount by which an item is depreciated each year using flat rate depreciation is constant. Thus, we may write: Flat rate book value after t years = purchase price depreciation per year t When calculations need to be repeated several times, it is useful to generate these as a table on the graphics calculator. This can be achieved by entering the equation for the flat rate book value as above, and then requesting a table. Exercise 6E 1 Afarmer who purchased a tractor for $ estimates that it will have an effective life of 5 years, when its value will be $5500. a What is the amount of the annual depreciation? b Find its book value after 3 years.

29 252 Essential Standard General Mathematics 2 Geoffrey s new motorbike, which he purchased for $8500, decreases in value by 12% of the purchase price each year. a What is the amount of the annual depreciation? b What is its book value after 2 years? 3 A machine purchased for $ decreases in value by 13.5% of the purchase price each year. What is its book value after 5 years? 4 Some office furniture is purchased for $1950, and it is estimated that it will be used for 6years. If at the end of 6 years the furniture is worth $546, use the straight line method to: a calculate the amount of annual depreciation b calculate the depreciation rate as a percentage of the cost price. 5 The cooking and catering equipment purchased for a newly opened restaurant cost $ It is estimated that it will have a book value of $ after 8 years. To allow the restaurant owner to prepare a budget: a calculate the annual depreciation rate as a percentage of the cost price b calculate the book value of the equipment after 6 years c draw a graph to show book value against time. 6 A publisher buys a new phone system. The cost is $12 500, and it is depreciated by 9% per year by the flat rate method. a Calculate the annual depreciation of the equipment. b Calculate the book value of the equipment after 5 years. c Draw a graph to show book value against time. d Use the graph to estimate the number of years it would take for the equipment to be written off; that is, for the book value to be zero. 7 A tractor purchased for $ has a useful life of 8 years. Its scrap value is estimated as $ a What is the annual amount of depreciation? b What is the flat rate of depreciation? c Draw a graph of the book value of the tractor over time. 8 Adeveloper installs a new air-conditioning system into an office block she is building. The cost is $ , and its value depreciates by 12.5% per year by the flat rate method. a Calculate the annual depreciation of the system. b Calculate the book value of the system after 4 years. c Draw a graph to show book value against time. d Estimate the number of years it would take for the book value of the system to be zero.

30 Chapter 6 Financial arithmetic Reducing balance depreciation In reducing balance depreciation, the annual depreciation is not a fixed amount but is calculated as a percentage of the book value at the beginning of each year. The amount of depreciation is largest in the first year, becoming smaller in each subsequent year. Reducing balance depreciation is the equivalent, but opposite, situation to compound interest, where an investment increases by a constant percentage each year. We can apply some of our knowledge of compound interest to this situation. Earlier we established the following formula: The amount of money ($A) that would result from investing $P at r% per annum, compounded annually for a period of t years, is: ( A = P 1 + r ) t Extending this to reducing balance depreciation we can say Book value is given by ( V = P 1 r ) t where: $V = book value of the item after t years $P = purchase price of the item r% = reducing balance depreciation rate per annum, compounded annually t = number of years. That is, the book value of an item that has a purchase price of $P and depreciates at r% per annum, compounded annually, will reduce to $V after t years. To find the amount of depreciation, we need to subtract the book value from the purchase price. The amount of depreciation resulting from depreciating an item with a puchase price of $P at r% per annum, compounded annually for t years, is given by: Depreciation is given by ( D = P V = P P 1 r ) t where $V is the book value of the item after t years. Example 17 Determining book value The factory manager in Example 16 decided that it was better to depreciate the machine, purchased for $30 000, using the reducing balance method. If he depreciates the machine at 15% per annum, what is its book value after 4 years? Solution Substitute P = $30 000, r = 10 and t = 4 into the formula for book value with reducing balance depreciation. ( V = P 1 r ) t ( = $ ) 4 = $ to the nearest dollar

31 254 Essential Standard General Mathematics The graphics calculator can be set up to generate a table for reducing balance depreciation over time. How to determine reducing balance depreciation and book value using a TI-Nspire CAS The factory manager in Example 16 decided that it was better to depreciate the machine, purchased for $30 000, using the reducing balance method. If he depreciates the machine at 15% per annum, what is its book value after 4 years? By how much has it depreciated in value? Draw a graph of book value against time for 10 years. Steps 1 Substitute P = $ and r = 15 into the formulae for book value and depreciation under reducing balance depreciation. 2 Start a new document (by pressing / + N) and select /3:Add Lists & Spreadsheet. Name the lists time, book value, and depreciation. Hint: Use / + for the underscore or just write as bookvalue. 3 Enter the data values 1 10 into the list time. 4 To calculate book value a Move the cursor to the grey formula cell of the list book value and type in = (1 15 ) time b Press enter to list the values for book value. From this list, we see that, after 4 years, the book value of the machine is $15 660, to the nearest dollar. 5 To determine depreciation a Move the cursor to the grey formula cell of the list depreciation and type in = (1 15/) time Note: An alternative formula to use to calculate the list depreciation would be = book value. b Press enter to list the values for depreciation. From this list, we see that, after 4 years, the machine depreciates in value by $14 340, to the nearest dollar. ) t ( V = ( D = ) t

32 Chapter 6 Financial arithmetic Use the Data & Statistics application to construct a graph of book value against time. The graph shows that book value decreases in a non-linear way with time. How to determine reducing balance depreciation and book value using a ClassPad The factory manager in Example 16 decided that it was better to depreciate the machine, purchased for $30 000, using the reducing balance method. If he depreciates the machine at 15% per annum, what is its book value after 4 years? By how much has it depreciated in value? Draw a graph of book value against time for 10 years. Steps 1 Substitute P = $ and r = 15 into the formulae for book value and depreciation under reducing balance depreciation. 2 To form a table of values, open the Sequence ( ) application. Select the Explicit tab. Opposite an E: type in (1 15/) n and press E. b n E: type in a n E and press E. Note: To obtain a n E, tap n,a n in the menu bar and select a n E (i.e. the depreciation value). ) t ( V = ( D = ) t

33 256 Essential Standard General Mathematics 3 Tap # from the toolbar to view the table of values. Note: Tap table values for a higher degree of accuracy. A more accurate value is then displayed at the bottom of the screen. From the table we see that, after 4years, the a book value of the machine is $15 660, to the nearest dollar b machine depreciates in value by $14 340, to the nearest dollar 4 To graph the sequences of book and depreciation values, select the Sequence Grapher ( ) icon from the toolbar. Note: To define the graph window scale select the View Window (6) icon from the toolbar and set the values as shown. Tap OK to confirm your settings. The graph shows that book value decreases in a non-linear way with time. How do the two methods of depreciation compare? The graph below shows book value plotted against time for both the flat rate method and the reducing balance method, to allow comparison, using the values from Examples 16 and 17. It can be clearly seen that reducing balance depreciation is not linear. This is not unexpected, since reducing balance depreciation is the equivalent but opposite situation to compound interest. Book value ($) Straight line depreciation at 8% per year Reducing balance depreciation at 15% per year Years From the graph we can also see that, although the rate of depreciation is higher for reducing balance depreciation (15% compared to 8% for flat rate depreciation), the amount of 12

34 Chapter 6 Financial arithmetic 257 depreciation reduces from year to year until eventually the two lines cross. The points have been joined to show this. Exercise 6F 1 Anew car was purchased for $ It is estimated that a new car depreciates by 30% during its first year after registration and then by 20% in each subsequent year, calculated on the previous year s value. What would you expect its book value to be after 3 years? 2 A small business bought a new computer system for $3500. If the tax office allows a reducing balance annual depreciation rate of 40%, calculate the book value of the system at the end of 5 years. 3 A boat is purchased for $35 750, and depreciated on a reducing balance basis at an annual depreciation rate of 23%. Calculate, to the nearest dollar, the book value of the boat at the end of 6 years. 4 A stereo system purchased for $1500 incurs 12% per annum reducing balance depreciation. a Find, to the nearest dollar, the book value after 7 years. b What is the total depreciation after 7 years? c Draw a graph to show book value against time. 5 Awashing machine purchased for $768 incurs 30% per annum reducing balance depreciation. a Calculate the book value of the washing machine after 4 years. b Draw a graph to show book value against time. 6 A hire-car company can claim a 40% reducing balance depreciation on minibuses designed to carry nine or more people. If the hire-car company purchases a new 12-seater minibus for $45 000, find, to the nearest dollar: a the book value of the minibus after 7 years b the total depreciation after 7 years. 7 Anew computer system, purchased by an engineering company, has an initial value of $ a Calculate the value of the system after 3 years if the annual depreciation rate is 30% using the reducing balance method. b Calculate the value of the system after 3 years if the annual depreciation rate is 15% using the flat rate method. 8 For tax purposes, a taxi owner-driver can claim 40% depreciation for his vehicle using the reducing balance method or 17% depreciation using the flat rate method. a For each method, prepare a table showing the book value for the first 5 years of the life of a taxi originally purchased for $ b Use your tables to decide which method gives the greatest cumulative depreciation after 5 years.

35 258 Essential Standard General Mathematics 6.7 Hire-purchase One way of purchasing goods when you have insufficient cash available is to enter into a hire-purchase agreement. This means the purchaser agrees to hire the item from the seller and to make periodic payments of an agreed amount. At the end of the period of the agreement, the item is owned by the purchaser. If the purchaser stops making payments at any stage of the agreement, the item is returned to the vendor and no money is refunded to the purchaser. We are interested in being able to calculate the interest rate being charged in these contracts, as it is not always stated explicitly. Flat interest rate If we calculate the total interest paid as a proportion of the original debt, and express this as an annual rate, this is called the flat rate of interest. The interest rate is exactly the same as the simple interest rate, but is generally called by this name in the hire-purchase context. Earlier in this chapter we established that, for simple interest: r = I Pt where r%isthe interest rate per annum, $P is the principal, $I is the amount of interest and t is the time in years. In the case of hire-purchase, we will define the formula as follows: Flat rate of interest per annum, r f = I Pt where $I is the total interest paid, $P is the principal owing after the deposit has been deducted and t is the number of years. Example 18 Calculating the interest and flat rate when t = 1 A student buys a notebook computer priced at $1850. She pays a deposit of $370 and repays the loan in 12 monthly instalments of $ each. a How much interest is paid on the computer? b What is the flat rate of interest, correct to 2 decimal places? Solution a 1 First, we need to determine how much the student has paid in total. 2 Determine the interest paid. This is the difference between the total paid and the purchase price. Total paid = deposit + repayments = = $2068 Interest paid = = $218

36 Chapter 6 Financial arithmetic 259 b 1 Apply the appropriate formula with P = $1480 (since only $1480 is owing after the deposit is paid), I = $218 and t = 1 (since the computer is paid off in 1 year) to find the value of r f. 2 Since the unit of time is years, the interest rate can be written as the interest rate per annum. r f = I Pt 218 = = Interest rate = 14.73% per annum The formula for r f given above gives the flat rate of interest over the period of the contract. Since the contract in Example 18 is for 1 year, the flat rate of interest calculated is also the flat rate of interest per annum. The distinction should be clear in the next example, in which the contract is not for 1 year. Example 19 Calculating the interest and flat rate when t 1 A hire-purchase contract for a sound system, priced at $1400, requires Josh to pay a deposit of $400 and then make six monthly payments of $185. a How much interest does he pay? b What is the flat rate of interest per annum that this represents? Solution a 1 Determine how much Josh pays in total. 2 Determine the interest paid. This is the difference between the total paid and the purchase price. b 1 Apply the appropriate formula with P = $0 (since only $0 is owing after the deposit is paid), I = $110 and t = 0.5 (since the sound system is paid off in 6 months) to find the value of r f. 2 Since the unit of time is years, the interest rate can be written as the interest rate per annum. Total paid = deposit + repayments = = $1510 Interest paid = = $110 r f = I 110 = Pt = 22% Interest rate = 22% per annum

37 260 Essential Standard General Mathematics Exercise 6G 1 The cash price of a tennis racquet is $330. To buy it on hire-purchase requires a deposit of $30 and 12 equal monthly instalments of $28. Calculate: a the total cost of buying the racquet by hire-purchase b the extra cost of buying by hire-purchase. 2 A bicycle has a marked price of $300. It can be bought through hire-purchase with a deposit of $60 and 10% interest on the outstanding balance, to be repaid in 10 monthly instalments. Calculate: a the amount of each monthly instalment b the total cost of buying the bicycle by hire-purchase. 3 A hire-purchase agreement offers hi-fi equipment, with a marked price of $897, for $87 deposit and $46.80 a month payable over 2 years. Calculate: a the total hire-purchase price b the amount of interest charged. 4 A second-hand car is advertised for $5575 cash or $600 deposit and 24 monthly intalments of $ Calculate the flat rate of interest per annum. 5 Exercise gym equipment, which normally costs $750, can be bought through hire-purchase with a $200 deposit and $26.40 a month for 30 months. Calculate: a the amount of interest being charged b the flat rate of interest per annum. 6 A microwave oven is advertised with a marked priced of $576 and the opportunity to buy it on hire-purchase, with no deposit and an interest rate of 10% repayable over a year with four equal instalments. Calculate: a the amount of interest c the amount of each instalment. b the total amount to be repaid 7 A student buys a tent for bushwalking, priced at $200. He pays a deposit of $50 and agrees to repay the balance of $150 plus interest at 14% over the period of 1 year, in two half-yearly instalments. Calculate: a the amount of each half-yearly instalment b the total price of the tent.

38 Chapter 6 Financial arithmetic A customer bought a new car for $ on hire-purchase. A deposit of $4000 was paid and the loan plus interest is to be repaid over 18 months in six quarterly repayments of $ Calculate: a the total amount to be repaid b the flat interest rate per annum. 9 A student bought a new computer costing $2225 on hire-purchase. She traded in an old computer for $200 and paid a deposit of $150. The balance was paid by monthly instalments of $ over 2 years. Calculate: a the total interest paid b the flat rate of annual interest.

39 262 Essential Standard General Mathematics Review Key ideas and chapter summary Percentage increase or decrease Simple interest Amount of the investment or loan (simple interest) Minimum monthly balance Compound interest Amount of the investment or loan (compound interest) Depreciation Percentage incresase or decrease is the amount of the increase or decrease expressed as a percentage of the original value. amount of change Percentage change = original value 1 Simple interest is paid on an investment or loan on the basis of the original amount invested or borrowed, called the principal (P). The amount of simple interest is constant from year to year, and thus is linearly related to the term of the investment. The amount of interest earned ($I )when a principal ($P) isinvested at r% per annum for t years is given by: I = Prt The total amount owed or invested ($A) after simple interest has been added to a principal ($P) for t years at r% per annum is given by: A = P + I = P + Prt where $I is the amount of interest earned. The lowest amount an account contains in each calendar month is its minimum monthly balance. Under compound interest, the interest paid on a loan or investment is credited or debited to the account at the end of each period, and the interest for the next period is based on the sum of the principal and previous interest. The amount of the compound interest increases each year, and thus there is a non-linear relationship between compound interest and the term of the investment. The amount of interest earned ($I )when a principal ($P) isinvested at r% per annum for t years, compounded n times a year, is given by: ( I = A P = P 1 + r/n ) nt P where $A is the amount of the investment after t years. The total amount owed or invested ($A) after compound interest has been added to a principal ($P) for t years at r% per annum, compounded n times a year, given by: ( A = P 1 + r/n ) nt Depreciation is the amount by which the value of an item decreases over time.

40 Chapter 6 Financial arithmetic 263 Book value Scrap value Flat rate depreciation Book value (flat rate depreciation) Reducing balance depreciation Book value (reducing balance depreciation) Hire-purchase Flat rate of interest The book value of an item is its depreciated value. The scrap value of an item is the value at which it is no longer of value to the business, so it is replaced. Flat rate depreciation is when the value of an item is reduced by the same percentage of the purchase price, or amount, for each year the item is in use. It is equivalent, but opposite, to simple interest. The depreciation ($D) ofanitem that has a purchase price of $P and is depreciating at a flat rate of r% per annum for t years is given by: D = Prt The book value ($V )ofanitem that has a purchase price of $P and is depreciating at a flat rate of r% per annum for t years is given by: V = P D = P Prt where $D is the amount of depreciation. Reducing balance depreciation is when the value of an item is reduced byaconstant percentage for each year it is in use. It is the equivalent, but opposite, situation to compound interest. The depreciation ($D) ofanitem that has a purchase price of $P and is depreciating at a rate of r% per annum, compounded annually for t years, is given by: ( D = P V = P P 1 r ) t where $V is the book value of an item after t years. The book value ($V )ofanitem that has a purchase price of $P and is depreciating at a rate of r% per annum, compounded annually for t years, is given by: ( V = P 1 r ) t Under a hire-purchase agreement, the purchaser hires an item from the vendor and makes periodic payments at an agreed rate of interest. At the end of the period of the agreement, the item is owned by the purchaser. Flat rate of interest is when the total interest paid is given as a percentage of the original amount owed, and annualised. Flat rate of interest per annum, r f = I Pt where $I is the total interest paid, $P is the principal owing after the deposit has been deducted and t is the number of years. Review

41 264 Essential Standard General Mathematics Review Skills check Having completed this chapter you should be able to: calculate the amount of the discount and the new price when r% discount is applied calculate the amount of the increase and the new price when r% increase is applied calculate the percentage discount or increase that has been applied, given the old and new prices calculate the original price, given the new price and the percentage discount or increase that has been applied use the formula for simple interest to find the value of any one of the variables I, P, r or t when the values of the other three are known determine the interest payable on a bank account, paid on the minimum monthly balance, over a period when up to three transactions have been made calculate the amount of an investment after simple interest has been added plot the value of simple interest (I) against time (t) toshow a linear relationship use the formula for compound interest to find the amount of an investment after compound interest has been added calculate the amount of compound interest payable on an investment or loan plot the value of compound interest (I) against time (t) toshow a non-linear relationship calculate flat rate depreciation, book value and the length of time for which an asset will be in use under flat rate depreciation calculate the amount of depreciation of an asset under reducing balance rate depreciation determine the flat rate of interest per annum for a hire-purchase agreement. Multiple-choice questions 1 The amount saved if a 10% discount is offered on an item marked $120 is: A $20 B $12 C $1.20 D $10.90 E $10 2 If a 20% discount is offered on an item marked $30, the new discounted price of the item is: A $10 B $24 C $6 D $25 E $28 3 If a 15% increase is applied to an item marked $60, the new price of the item is: A $69 B $9 C $75 D $67 E $70 4 How much interest is earned if $2000 is invested for 1 year at a simple interest rate of 4% per annum? A $2080 B $160 C $8 D $800 E $80

42 Chapter 6 Financial arithmetic The total value of an investment of $0 after 3 years if simple interest is paid at the rate of 5.5% per annum is: A $55 B $1055 C $1165 D $ What is the interest rate, per annum, if a deposit of $1500 earns interest of $50 over a period of 6 months? A 0.56% B 5.45% C 5.55% D 6.45% E 6.67% 7 $2400 is invested at a rate of 4.25% compound interest, paid annually. The value of this investment after 6 years is: A $ B $ C $3012 D $ E $ How much interest is earned if $5000 is invested for 3 years at an interest rate of 4%, if the interest is compounded quarterly? A $600 B $ C $ D $ E $ Machinery costing $ depreciates by 10% in its first year, then by 7.5% in each subsequent year. The book value of the machinery after 3 years is approximately: A $ B $ C $ D $ E $ The value of $8500 compounded annually for 5 years at 6% per annum is closest to: A $2550 B $ C $ D $ E $ An investment account is opened with a deposit of $5000. Compound interest is paid on the investment at a rate of 12% per annum, credited monthly. The amount in the account after 18 months, if no withdrawals have been made, is closest to: A $5981 B $ C $5926 D $5900 E $ Anew computer costs $3600. If depreciation is calculated at 15% per annum (reducing balance), the computer s value at the end of 4 years will be closest to (in dollars): A 3600( ) B 3600( ) C 3600/( ) D E ( ) 4 13 Office equipment is purchased for $ It is anticipated that it will last for 10 years and have a scrap value of $1250. The amount of depreciation that would be allowed per year, assuming a flat rate of depreciation, would be: A $1200 B $1075 C $1325 D $1250 E $1500 The following information relates to Questions 14 and 15 To buy a car costing $23 000, Janet pays $5000 deposit and then payments of $440 per month for the next 5 years. 14 How much interest does Janet pay under this scheme? A $3400 B $8400 C $3120 D $1600 E $ What flat rate of interest per annum does this amount to? A 46.7% B 9.3% C 36.5% D 7.3% E 14.6% Review

43 266 Essential Standard General Mathematics Review Short-answer questions 1 Rabbit Easter Eggs were reduced in price from $2.99 to $2.37 as they were not selling quickly. Bilby Easter Eggs were discounted to $3.83 from $4.79. a Which type of Easter Egg had the larger percentage reduction? b Calculate the difference in the percentage rates. 2 After Christmas, all stock in JD s was discounted by 20%. The sale price of a pair of cross-trainers was $110. Calculate the original marked price. 3 How much additional interest is earned if $8000 is invested for 7 years at 6.5% when interest is compounded annually, as compared with simple interest paid at the same rate? 4 A person wishes to have $ available for a world trip in 10 years time. Calculate the principal to be invested over 10 years at 6% per annum for this to be achieved, if the interest is compounded monthly. 5 A computer system costing $4400 depreciates at a reducing balance rate of 20% per annum. a What is the value of the system after 4 years? b By what constant yearly amount would the system be reduced to give the same value at the end of 4 years? 6 A television set, which normally costs $880, can be bought through hire-purchase with a $200 deposit and a payment of $30 a month for 30 months. Calculate: a the amount of interest being charged b the flat rate of interest per annum. Extended-response questions 1 a The wholesale price of a digital camera is $350. The maximum profit that a retailer is allowed to make when selling this particular camera is 75% of the wholesale price. Calculate the maximum retail price of the camera. b Suppose that the wholesale price of the camera increases at 5% per annum simple interest for the next 5 years. i By how much will the wholesale price have increased at the end of 5 years? ii What is the new wholesale price of the camera? iii What is the new retail price of the camera (with 75% profit)? iv What percentage increase is this in the retail price determined in part a? 2 Peter has $ that he wishes to invest for 5 years. a Aussie Bank offers him 6% per annum simple interest. How much will he have at the end of the 5 years under this plan? b Bonza Bank offers him 5.5% per annum compound interest, compounding monthly. How much will he have at the end of the 5 years under this plan? c Find, correct to 1 decimal place, the simple interest rate that Aussie Bank should offer if the two investments are to be equal after 5 years.