Yarra Valley Grammar Unit 3 Further Mathematics Module 4: Business related mathematics Name FMT Business Related Mathematics Page 1
FMT Business Related Mathematics Page 2
Further Mathematics: Study design Module 4: Business-related mathematics This module covers the application of numerical computations and graphical techniques to the formulation and solution of various problems in business and financial contexts, including the solution of related equations. Financial transactions and asset value, including: financial computations involving multiple transactions (such as bank account balances), percentage changes and charges (such as discounting, capital gains, stamp duty, GST); graphical and tabular representation and calculation of the value of money for a series of consumer price changes (inflation); the use and comparison of fl at rate, reducing balance and unit cost methods of calculating depreciation. Loans and investments, including: use and comparison of simple and compound interest in investment and loan applications without periodic payments; annuity investments involving a series of regular and equal deposits; consideration of the effects of initial and periodic deposit values, frequency of deposits, interest rate, and length of investment; the ordinary perpetuity as a series of regular payments from an investment that continues indefinitely; reducing balance loans as particular applications of annuities, with applications including housing loans, time payment plans (hire purchase) and credit/store cards; consideration of the effects of varying the repayment amount, the frequency of repayments, and the interest rate on the total repayment time and total interest paid; determination of effective interest rates from nominal (flat rate) interest rates. **** All answers should be given to 2 decimal places (since we are dealing with money) unless otherwise stated**** FMT Business Related Mathematics Page 3
Percentages For r% discount Discount = r 100 original price New price = original price discount = (100 r ) 100 original price Example: Find the markdown and the new price of the following a. $5.40 discounted by 20% Discount (markdown) = New price = b. $10.00 discounted by 12.5% Discount = New price = For r% increase Increase = r 100 original New price = original price + increase = (100 + r ) 100 original price Examples Find the mark up and the new price of the following. a. $40 is increased by 15% b. $100 increased by 2.5% Increase = Increase = New price = New price = FMT Business Related Mathematics Page 4
Percentage discount % discount = discount original 100 1 Percentage increase % increase = increase original 100 1 Examples a. Find the percentage discount when a $50.00 jumper is reduced to $45.00 b. Find the percentage increase if a Tshirt is increased from $40.00 to $48.00 Finding the original price For r% discount Original = new price x 100 (100 r) For r% increase Original = new price x 100 (100 + r) Examples a. Find the original price of a top that is discounted by 15% and is now $40 b. Find the original price of a toy car that is increased by 10% and now costs $24.00 Complete Ex 20A page 526-528 Q1-2 LHS, Q3-9, then even Q10-14 FMT Business Related Mathematics Page 5
Simple interest I = Prt 100 A = P + I = P + Prt 100 Examples 1. If $300 is invested at 6% pa simple interest for 5 years, find the interest earned. 2. $1000 is invested at 7.5% pa for 6 months. Find the interest earned. 3. $10000 is invested at 4.75% pa for 18 months. Find the final value of the investment. 4. If $20000 is invested at 11.25% pa simple interest for 50 weeks, find the final value of the investment. FMT Business Related Mathematics Page 6
r = 100I Pt 5. a. When $7000 was invested for 4 years, $426 interest was earned. What was the rate of interest? b. Determine the interest rate on a car loan of $15000 borrowed over 7 years if the interest charged is $15000 t = 100I Pr 6. a. $1500 earned $230 when the rate of interest was 5.1%. How long was the money invested? b. Determine the time taken to invest $10000 at 6% p.a simple interest to earn $2000 interest. Give your answer in both years (to 2 decimal places) and in years and months. P = 100I rt 7. a. An amount of money was invested for 8 years and earned $500 when the interest rate was 5.25%pa. How much was originally invested? P = A 1 + rt 100 FMT Business Related Mathematics Page 7
b. Calculate the amount to be invested over 3 years at 3.8% p.a simple interest if at the end of the investment period you want to have $10000 to buy a car. c. Steve invested $25 000 which pays a monthly interest at the simple interest rate of 12% p.a. (i) What interest rate is paid per month? (ii) How much interest does Steve earn each month? (iii) How much interest does Steve earn each year? (iv) How long does it take for the investment to pay out $7500 in interest? (v) How much interest does Steve earn in 10.5 years? Graph of simple interest Interest ($) Total ($) time (years) time (years) FMT Business Related Mathematics Page 8
Using the Classpad Example: determine the amount of time it would take for $50000 to earn $17000 if it is invested at 4.25% p.a simple interest Prt I = 100 = = 1. Using a table Open the sequence Select explicit Type in formula: Tap the table icon If needed, click on Application from the I = 2125n and to change input Main menu select Scroll down table until you reach $17000 2. Using a graph Define graph Select sequence Select analysis and grapher trace We can see that it takes 8 years to earn $17000 interest FMT Business Related Mathematics Page 9
Using e-activity Enter parameters and click solve Complete Ex 20B page 534-536 Q1-8, then even Q10-24 FMT Business Related Mathematics Page 10
Compound interest For only 1 compound period per year A = P 1 + r 100 t Graph of compound interest Interest ($) A = P 1 + r/n 100 nt A = P 1 + R 100 N Total($) A = final amount of the investment or loan P = initial investment/borrowing r = interest rate per annum n = number of compounds per annum t =number of years R=r/n N =n t time (years) time (years) Let s compare simple interest to compound interest, for a $5000 investment at 10% p.a. over 10 years SI = Compound interest Year 1: Year 6: Year 2: Year 7: Year 3: Year 8: Year 4: Year 9: Year 5: Year 10 FMT Business Related Mathematics Page 11
Year (n) Total Simple interest ($) Total compound interest ($) 1 2 3 4 5 6 7 8 9 10 Graph both the total simple interest and total compound interest on the grid below. 8000 Total interest ($) 7000 Examples 6000 1. 5000 4000 3000 2000 1000 2 4 6 8 10 Year FMT Business Related Mathematics Page 12
Complete the table below for the following multichoice questions to assist you in solving the problems. Question P r t n R N 1 2 3 4 5 6 Examples 1. The number of months that it will take for an investment to earn $700 when $3500 is invested at 5% per annum compound interest compounding monthly is (to the nearest month): A 3 B 4 C 21 D 43 E 44 2. Suppose that $20 000 is invested at 6.25% per annum compounding quarterly. The amount of interest the investment earns in the third year is closest to: A $1449 B $2641 C $4090 D $22 641 E $24 090 3. The compound interest, to the nearest dollar, on $2300 over a period of three years at an interest rate of 5.5% per annum compounded annually is: A $2701 B $380 C $2680 D $427 E $401 4. The amount of money which should be invested at 6.5% per annum compound interest, compounding monthly, if you require $30 000 in four years time is closest to: A $23 320 B $23 148 C $23 180 D $28 127 E $38 881 FMT Business Related Mathematics Page 13
5. An investment account opens with $5000. Compound interest accrues at 5.7% per annum, credited quarterly. The least number of quarters needed for the value of this account to exceed $6000 is: A 3 B 4 C 12 D 13 E 16 6. Suppose that $215 000 is invested at 4.25% per annum compounding monthly. The amount of interest the investment earns in the fourth year is closest to: A $10 582 B $29182 C $39 764 D $244 182 E $254 764 Ex 20C Page 545-547 odd questions FMT Business Related Mathematics Page 14
Skill sheet Compound interest. (complete these problems in your workbook) 1 Calculate the compound interest on the following amounts, giving your answers to the nearest cent. a $12 000 invested at 5% per annum for 6 years b $100 000 invested at 4% per annum for 10 years c $1000 invested at 6% per annum for 3 years 2 Calculate the compound interest on the following amounts, giving your answers to the nearest cent. a $15 000 invested at 5% per annum compounded monthly for 5 years b $2500 invested at 4% per annum compounded quarterly for 10 years c $120 000 invested at 6% per annum compounded monthly for 7 years 3 Find the total amount owed on the following amounts, to the nearest cent. a $17 700 borrowed at 5.5% per annum compounded monthly for 5 years b $8500 borrowed at 6.7% per annum compounded quarterly for 8 years c $50 000 borrowed at 6% per annum compounded monthly for 3 years 4 How much should be invested reach the following amounts? Give your answers to the nearest cent. a $12 700 if the investment earns 5.5% per annum compounded monthly for 5 years b $8500 if the investment earns 6.7% per annum compounded quarterly for 8 years c $50 000 if the investment earns 3.6% per annum compounded monthly for 10 years 5 How long does it take to earn the following amounts of interest? a $3780, if $35 000 is invested at an interest rate of 5.5% per annum compounded monthly b $15 300, if $122 000 is invested at an interest rate of 4.75% per annum compounded quarterly c $29 580, if $235 000 is invested at an interest rate of 6.5% per annum compounded monthly 6 What interest rate is being paid on the following investments? Give your answers to one decimal place. a If $2000 as grown to $4000 in 12 months, with interest compounded monthly b If $12 000 as grown to $15 000 in 18 months, with interest compounded quarterly c If $132000 as grown to $254000 in 10 years, with interest compounded monthly FMT Business Related Mathematics Page 15
Reducing balance loans Reducing balance loans Pay more of the principal each time. Loan balance balance owed ($) 12000 10000 8000 6000 4000 2000 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Time (years) When the loan is fully paid off it is because the amount owing is. The time that over which the loan was paid is called the Examples: 1 Sam borrows $5000 to purchase a car. The interest rate charged is 12% p.a. compounded monthly. He makes payments of $400 per month. a. How much does Sam still owe after 4 months? b. How much of the principal will he have paid off after 4 months? Set up a table as shown below End of month Interest Repayment Balance of loan($) 0 - - 5000 1 2 3 4 Chapter 20D Page 548-549 Q1-7 Chapter 20 review P551-554 All questions FMT Business Related Mathematics Page 16
Applications of simple interest Income tax Capital gains tax Goods and services tax (GST) Stamp duty Bank balances Hire purchase Flat rate depreciation Income tax Tax is paid according to income levels, divided into subdivisions or tax brackets. Terms Gross salary: Net salary: Tax rates 2007-08 Taxable income $1 $6,000 Nil Tax on this income $6,001 $30,000 15c for each $1 over $6,000 $30,001 $75,000 $3,600 plus 30c for each $1 over $30,000 $75,001 $150,000 $17,100 plus 40c for each $1 over $75,000 $150,001 and over $47,100 plus 45c for each $1 over $150,000 Tax rates 2008-09 Taxable income $0 $6,000 Nil Tax on this income $6,001 $34,000 15c for each $1 over $6,000 $34,001 $80,000 $4,200 plus 30c for each $1 over $34,000 $80,001 $180,000 $18,000 plus 40c for each $1 over $80,000 $180,001 and over $58,000 plus 45c for each $1 over $180,000 Examples Questions 1-4 require the use of the following table Tax subdivision ($) (%) Tax payable(marginal rate) 1 0 6000 0 2 6 001 21 600 17 3 21 601 70 000 30 4 70 001 125 000 42 5 125 001 + 47 1 Richard earns $42 250 per year. What is his annual income tax? Give your answer correct to the nearest dollar. A $8847 B $6163 C $10 875 D $9867 E $6195 FMT Business Related Mathematics Page 17
2 Genevieve s gross salary is $62 000 per year. Assuming that there are exactly 52 weeks per year, how much does she pay in income tax per week, correct to the nearest dollar? A $284 B $358 C $203 D $304 E $233 3 Sally earns $49 540 per year. a. How much income tax does she pay? b. What is her take home (net) annual income? c. Sally receives a 5% pay increase (i) What is her new gross income? (ii) How much tax will she now pay? (iii) How much more money will she take home each year? Each week? 4 Calculate the income tax payable on an income of $83 000 FMT Business Related Mathematics Page 18
Capital gains tax (CGT) Tax paid on profit made from investments The profit made = capital gain Taxed according the marginal rate depending on your income 1 Mark has some money in an investment account at the bank, on which he is paid $14 885 interest in one financial year. If his other income for that financial year is $147 657, how much capital gains tax will he pay on the bank interest, correct to the nearest dollar? A $1510 B $2530 C $4466 D $6252 E $6996 2 John purchased a block of land for $190 000 and sold it 1 year later for $230 000 a. What is the profit he made on the sale? b. If he earned $70 000 during this financial year, determine the Capital Gains Tax he will need to pay. c. How much is John s actual profit on the land? 3 If Nathan makes a profit of $10 300 on his investments in one financial year, and his other income for that financial year is $72 500, how much capital gains tax will he pay on the profit he makes on his investments, correct to the nearest dollar? A $3090 B $1751 C $4841 D $4326 E $5150 FMT Business Related Mathematics Page 19
Income tax and capital gains tax problems Questions 3 and 4 require the use of the following table Tax subdivision ($) (%) Tax payable (marginal rate) 1 0 6000 0 2 6 001 21 600 17 3 21 601 70 000 30 4 70 001 125 000 42 5 125 001 + 47 3. Stacey earns $52 000 per year. What is her annual income tax to the nearest dollar? 4. Peter has some money in an investment account at the bank, on which he is paid $10 200 interest in one financial year. If his other income for that financial year is $98 000, how much capital gains tax will he pay on the bank interest, correct to the nearest dollar? FMT Business Related Mathematics Page 20
Goods and services tax (GST) GST is a flat rate of 10% on most items. It therefore operates like simple interest GST = 10% of pre GST price Selling price = 110% of pre GST price (i.e the pre GST price plus the GST) Examples 1 Geoff s telephone bill one month is $58.50, including GST, how much would the phone bill be without GST? A $64.35 B $52.65 C $65.00 D $53.18 E $50.87 Selling prices = 1.1 pre GST price GST = selling price 11 2 A plumber quoted $300, excluding GST (Goods and Services Tax), to complete a job. A GST of 10% is added to the price. The full price for the job will be: A $3 B $30 C $303 D $310 E $330 [VCAA FMT 2008] 3 The price of a book is $22. It includes a Goods and Services tax (GST) of 10%. The price before this GST is added is: A $19.80 B $20.00 C $24.00 D $24.20 E $24.40 [VCAA FMT 2006] 4 Determine the GST for a dress worth $120 before the GST has been added to give the final price FMT Business Related Mathematics Page 21
5 Find the GST payable on an electricity bill of $240 (GST inclusive) 6 A flat screen TV is advertised for $1050, without GST. How much would it cost with GST? 7 Given a GST rate of 10%, find the total due on the following invoices after GST. a Chemist shop supplies goods worth $165. b Bookseller supplies books worth $380. c Plumber supplies goods and services to a total of $2870. 8 Find the missing amounts in this invoice supplied by an electrical company. Shaes Electrical Contractor Company Tax invoice date: 12/8/06 Materials supplied (wiring, switches, fittings) $870 Labour supplied 4 hours @$150/hour Total GST Total due (plus GST) payable within 14 days FMT Business Related Mathematics Page 22
Stamp duty Tax (or duty) charged by the government Commonly charged for land and property and vehicles Purchase price of property Stamp duty payable 1 0 20 000 1.4% 2 20 001 115 000 $280 plus 2.4% of the value in excess of $20 000 3 115 001 870 000 $2560 plus 6% of the value in excess of $115 000 4 More than 870 000 5.5% of the value Examples 1 The value of the stamp duty payable when a house is purchased for $665 500 is closest to: A $30 378 B $33 030 C $35 590 D $13 212 E $7707 2 The price of a property purchased in 2006 was $200 000. Stamp duty was paid on this purchase according to the schedule in the table above. The amount of stamp duty paid was: A $2560 B $2800 C $5100 D $7660 E $9460 3 Calculate the stamp duty payable on a property purchased for $890 000 [VCAA FMT 2007] 4 Calculate the stamp duty payable on a property purchased for $480 000 Ex 21A p558 560 Q1-16 FMT Business Related Mathematics Page 23
Bank balances Debit: out of account (withdrawal) Credit: into account (deposit) Two methods Minimum monthly balance (mmb): lowest balance throughout the month : often interest is only paid on mmb Balance used for calculating interest on minimum daily balances If money credited (deposited): balance before the deposit If money debited (withdrawn): balance after withdrawal Examples 1. The bank statement opposite shows transactions over a one month period for a savings account. a. Complete the balance column for the statement for the month of May. Date Transaction Debit Credit Balance 1 May 250.54 6 May Purchase 109.00 19 May Salary 457.23 1 June Interest b. If interest is calculated on a minimum monthly balance at a rate of 3%p.a, how much is earned this month? c. Complete the rest of the table. d. If the interest is instead calculated on the minimum daily balance (still at 3% p.a), calculate the interest due and the new balance. FMT Business Related Mathematics Page 24
2. The bank statement opposite shows transactions over a 1month period for a savings account that earns simple interest at a rate of 3.95% per annum calculated daily and paid monthly. Date Transaction Debit Credit Balance 1 April 260.25 5 April Cash 150.00 410.25 21 April Cash 150.00 560.25 1 May Interest a. How much interest is earned in April? Complete the entry for interest and balance for 1 May. b. If interest is calculated on a minimum monthly balance instead, how much interest is earned in April? 3 A savings account was opened on 20 th May and the bank statement shows the transactions on the bank account over two months. a. Complete the balance column for the statement. Date Transaction Debit Credit Balance 20 May Opening 450.00 deposit 28 May Withdrawal 80.00 31 May Fee 0.27 3 June Pay 364.00 10 June Withdrawal 220.00 17 June Pay 362.00 30 June Withdrawal 300.00 b. If interest is paid at 1.5% p.a. on the minimum monthly balance, find the interest paid on this account for the month of June. c. If instead the interest is paid on the daily balance, what now would be the interest paid? FMT Business Related Mathematics Page 25
Using CAS to calculate the number of days Ex 21B p561 563 Q1-8 FMT Business Related Mathematics Page 26
Bank balances extra questions and past exam questions 1 Interest on a savings account is calculated at 0.35% per month based on the minimum daily balance. The following statement lists all the transactions in an account for April Jun one year. Date Transaction detail Debit Credit Balance 01 April balance brought forward 2360.10 07 April withdrawal 200.00 2160.10 03 May deposit 100.00 2260.10 15 Jun deposit 50.00 2310.10 The interest, in dollars, for the month of May is: A $8.03 B $7.56 C $7.91 D $79.10 E $8.26 2 Interest on a savings account is calculated at 0.3% per month based on the minimum monthly balance. The following statement lists all the transactions on an account for January March one year. Date Transaction detail Debit Credit Balance 01 Jan balance brought 275.00 forward 07 Jan withdrawal 50.00 225.00 03 Feb deposit 846.10 1071.10 04 Mar withdrawal 250.00 821.10 The interest, in dollars, for the month of February is: A $0.68 B $0.83 C $3.20 D $2.46 E $2.54 FMT Business Related Mathematics Page 27
3. A bank statement for the month of October is shown below. Date Transaction detail Debit Credit Balance 01 October Opening balance 853.92 01 October Withdrawal internet banking 380.00 473.92 16 October Deposit - Cheque 518.15 992.07 18 October Credit card payment 125.56 866.51 23 October Withdrawal Internet bank 250.00 616.51 31 October Closing balance 616.51 Interest on this account is calculated at a rate of 0.15% per month on the minimum monthly balance. The interest payment for the month of October will be: A $0.19 B $0.57 C $0.71 D $0.92 E $1.28 [VCAA 2006] 4 Tamara s bank statement for September has been damaged by spilt ink as shown below. Date Transaction detail Debit Credit Balance 01 Sept Opening balance 2143.50 01 Sept Interest 2.45 13 Sept Payment - Telstra 616.40 30 Sept Pay/Salary 30 Sept Totals at the end of the month 616.40 1971.75 Tamara s salary was deposited on 30 September. What is the value of this deposit? A $441.95 B $442.20 C $444.65 D $785.70 E $788.15 [VCAA 2005] FMT Business Related Mathematics Page 28
5. The following is an extract from a bank account showing all transactions for the period 1 January to 30 June, 2003. Date Transaction detail Debit Credit Balance 01 Jan 2003 Balance brought forward 4320.00 15 Mar 2003 Deposit 2100.00 6420.00 31 Mar 2003 Interest 32.40 6452.40 22 May 2003 Withdrawal 460.00 5992.40 30 June 2003 Interest Interest on this account is calculated at a rate of 0.25% per month on the minimum monthly balance and paid into the account quarterly. Interest for the June period (April to June) is paid on 30 June. The balance in the account after interest is paid on 30 June 2003 is A $6039.64 B $6038.49 C $6024.76 D $6023.51 E $6022.36 6. The bank statement below shows the transactions on Michelle s account for the month of July. Date Transaction detail Debit Credit Balance 01 July Opening balance 6250.67 11 July Deposit - Cash 6870.67 14 July Withdrawal - cheque 749.81 6120.86 19 July Deposit internet transfer 838.23 6959.09 31 July Closing balance 6959.09 a. What amount, in dollars, was deposited in cash on 11 July? [2003] Interest for this account is calculated on the minimum monthly balance at a rate of 3% per annum. b. Calculate the interest for July, correct to the nearest cent. [VCAA 2008] FMT Business Related Mathematics Page 29
6 Find the balance in this savings account at the end of February. Date Debit Credit Balance 1 Feb 0.36 2856.23 8 Feb 483.48 14 Feb 20.00 14 Feb 140.00 22 Feb 299.95 28 Feb 1020.78 7 Find the balance in this savings account at the end of December. Date Debit Credit Balance 1 Jan 2856.23 3 April 856.05 8 June 550.00 23 Sept 450.72 10 Oct 225.00 28 Dec 420.00 8 Find the interest payable for a savings account (i) For September which had a minimum monthly balance of $858.25 and interest was credited at 1.5% p.a. (ii) In January on a minimum monthly balance of %5026.38 where the interest was paid at 3.6% p.a. 9 Interest on Terry s bank account is paid yearly on 30 th June and is calculated on the minimum monthly balance. The interest rate is 6% per annum. For the year 2003-04, the complete statement for Terry s account, before adding interest is shown. Date Credit Debit Balance 30 June 2003 13 April 2004 Interest 37.00 1025.00 2137.00 1112.00 Assuming no other deposits or further withdrawals were made after 13 th April 2004, the total interest in dollars to be credited to this account on 30 th June 2004 is given by the expression A 0.005 x (2137 x 9 + 1112 x 3) B 0.005 x (2137 x 10 + 1112 x 2) C 0.005 x (2000 + 2137 x 8 + 1112 x 3) D 0.06 x (2137 x 9 + 1112 x 3) E 0.06 x (2137 x 10 + 1112 x 2) FMT Business Related Mathematics Page 30
10 Interest on a bank account is added on 1 July annually at a rate of 8% p.a. calculated monthly on the minimum monthly balance. A sum of $1620 was used to open an account on 15 th May 2005. Assuming there were no withdrawals or deposits, the interest payable on 1 July 2005 was A $8.00 B $10.80 C $16.20 D $21.60 E $32.40 FMT Business Related Mathematics Page 31
Hire purchase skillsheet 1 Calculate, to one decimal place, the flat interest rate per annum on each of the following loans. a $2500, to be repaid in 30 monthly instalments of $90 b $1500, to be repaid in 52 weekly instalments of $35 c $10 000, to be repaid in 24 monthly instalments of $450 d $27 500, to be repaid in fifteen quarterly instalments of $2200 e $6250, to be repaid in 18 monthly instalments of $400 FMT Business Related Mathematics Page 32
2 Evaluate the total interest paid, and hence the repayments (to the nearest cent), on the following contracts. a $3000 at a flat interest rate of 17.5% per annum over one year, with monthly repayments. b $5000 at a flat interest rate of 12% per annum over 18 months, with quarterly repayments. c $22 500 at a flat interest rate of 14% per annum over 72 months, with quarterly repayments d $1250 at a flat interest rate of 9.75% per annum over six months, with monthly repayments e $42 000 at a flat interest rate of 11.25% per annum over 5 years, with fortnightly repayments FMT Business Related Mathematics Page 33
3 Calculate, to one decimal place, the effective interest rate per annum on each of the following loans. a $5000, to be repaid in 60 monthly instalments of $100 b $15 000, to be repaid in 5 quarterly instalments of $3400 c $12 000, to be repaid in 24 monthly instalments of $650 d $750, to be repaid in six quarterly instalments of $135 e $7250, to be repaid in 26 weekly instalments of $300 4 The local electrical store advertises a television set for $4500 or $100 deposit and $35 per week for three years. a How much does the television set end up costing under the hire-purchase scheme? FMT Business Related Mathematics Page 34
b What is the flat rate of interest per annum? Give your answer to two decimal places. c What is the effective rate of interest per annum? Give your answer to one decimal place. 5 A car costing $18 000 is bought for $5000 deposit and repayments of $360 per month over a period of five years. a How much does the car end up costing under the hire-purchase scheme? b What is the flat rate of interest per annum? Give your answer to two decimal places. c What is the effective rate of interest per annum? Give your answer to one decimal place. 6 A boat costing $32 500 is purchased for $10 000 deposit and repayments of $785 per month for three years. a How much does the boat end up costing under the hire-purchase scheme? FMT Business Related Mathematics Page 35
b What is the flat rate of interest per annum? Give your answer to two decimal places. c What is the effective rate of interest per annum? Give your answer to one decimal place. Answers 1 a 3.2% b 21.3% c 4.0% d 5.3% e 10.1% 2 a $525, $293.75 b $900, $983.33 c $18 900, $1725 d $60.94, $218.49 e $23 625, $504.81 3 a 7.9% b 17.8% c 28.8% d 9.1% e 29.2% 4 a $5560 b 8.03% c 16.0% 5 a $26 600 b 13.23% c 26.0% 6 a $38 260 b 8.53% c 16.6% FMT Business Related Mathematics Page 36
Hire purchase People buy on hire-purchase when they cannot afford to buy the goods for cash Where, instead of buying item outright, purchaser hires item from vendor and makes periodic payments at an agreed rate of interest If purchaser fails to make a payment, item is returned to vendor Hire purchase agreements usually require a deposit and then regular payments. First determine the total payments (deposit plus payments number of payments), so that you can then calculate the interest. The retailer arranges a contract with a financial institution and the purchaser pays regular instalments including interest at a flat rate to the financial institution. A flat rate is the same as simple interest rate. The interest charged is added onto the balance owing and then divided into the equal instalments. Advantages of this form of buying are: 1. the purchaser has the use of the goods while paying them off 2. the cost of the goods is spread over a long term in small amounts. The disadvantages are more complex: 1. the purchaser pays more for the goods in the long run 2. the goods are legally owned by the finance company until they are fully paid off 3. any forfeit on making the regular payments entitles the finance company to repossess the goods as well as retain all past payments made. The main stages of hire-purchase interest and total price calculations are: Step 1. Check the price of the goods. Step 2. Pay any deposit. Step 3. Set up the balance as a loan. Loan amount, P = price of goods deposit paid Step 4. Calculate the interest on the loan using the simple interest formula. I = Prt 100 Step 5. The total amount to be repaid on the loan is the sum of the balance and the interest. Total paid on loan = P+I Step 6. Establish regular payments/instalments. P = r = t = installment amount = total amount number of installments Step 7. Determine the total cost of the goods Total cost of goods = deposit + loan + interest = deposit + repayment amount x number of repayments FMT Business Related Mathematics Page 37
Examples 1. A ring with a marked price of $1800 is offered to the purchaser on the following terms: $200 deposit and the balance to be paid over 24 equal monthly instalments with interest charged at 11.5% p.a. flat rate. Find: a. the amount borrowed on hire purchase b. the total interest paid c. the monthly repayments c. the total cost of the ring 2. Debbie and Peter purchased a lounge suite on hire-purchase. The cash price was $2500. Peter and Debbie paid $250 deposit and signed an agreement to pay the balance in 36 equal monthly instalments. If the hire-purchase company charges 14% p.a. simple interest, find: a. the amount borrowed though hire purchase b. the total interest paid c. the monthly repayments d. the total cost of the lounge suite. FMT Business Related Mathematics Page 38
3. When buying new appliances for a recently renovated kitchen, Cheryl bought, from the same supplier, a refrigerator worth $490, a stove worth $350 and a dishwasher worth $890. If she paid $450 deposit and paid the balance over 48 months in equal monthly instalments at 12% p.a. simple interest, find: a. the total cost of the goods purchased. b. the amount borrowed through hire purchase. c. the interest charged. d. The cost of the goods over the 48 months. e. Cheryl s monthly instalments f. the total amount Cheryl paid for the goods. 4. While on holidays in Noosa, Jan saw a bracelet she could not live without. The marked price was $2000. The jewellery shop owner offered her a discount of 15% if she paid a deposit of $250. Jan paid the deposit and signed a hire-purchase agreement that she would pay the balance of the bracelet s cost at 15% p.a. flat rate with 24 equal monthly instalments. a What was the price of the bracelet after the 15% discount? b Calculate the balance Jan was to pay back through hire purchase. c Calculate the interest Jan paid. d Calculate Jan s monthly instalment. e How much did Jan pay altogether for the bracelet? FMT Business Related Mathematics Page 39
5. An electric guitar is bought on hire-purchase for a $250 deposit and monthly instalments of $78.50 for 3 years. The cash price for this guitar is $2500. The interest rate is closest to: A 9.5% B 7% C 8.5% D 8% E 7.5% 6. Carpeting the home is not cheap, Rob stated. Hire-purchase is the answer, replied Tom. The cost of the carpet for the house is $9500. Rob and Tom place a deposit of $1500 and plan to pay it back weekly over 4 years at 13% interest per year. The weekly instalment is: A $253.37 B $62.20 C $46.20 D $58.46 E $462.00 7. A salesman told a couple that if they bought a television at $890 today, he would allow a deposit of $100 plus $8.65 weekly for 2 years. The interest rate charged is: A 10% B 7% C 6.5% D 9½% E 7.5% 8. A company advertised a dining room suite for $2500. You could pay: a cash and receive a 10% discount, or b $200 deposit and 5% p.a. interest on the remainder for 3 years, or c $300 deposit and 4.5% p.a. on the remainder for 3½ years. What is the total paid on each deal? a. b. c. FMT Business Related Mathematics Page 40
Weekly installment advertising Many retailers use the option of hire-purchase to attract new sales. They also choose to advertise the instalment amount as it can seem to be very manageable. Buyers should investigate the entire arrangement offered and find answers to questions such as: What is the interest rate? How does it compare to bank rates? What is the total cost of the item? Example 1. The following advertisement for a laptop computer was found in a newspaper. If there is a total of 104 weekly instalments and a third deposit, find: a the interest charged b the interest rate (correct to 1 decimal place) c the total cost of the computer. Flat rate of interest r f = 100I Pt r f = r e n + 1 2n Flat rate does not consider that periodic repayments have been made, but is still calculating interest on the full amount borrowed effective rate of interest will be higher. Effective rate of interest r e = r f 2n n + 1 r e = 100I 2n Pt n + 1 If n is large we can estimate effective interest rate r e r f 2 FMT Business Related Mathematics Page 41
Examples 1 Repayments on a car loan with flat rate interest of 15% p.a. If the loan is paid off after 7 years, a. calculate the effective rate of interest. b. calculate an approximate value of the effective interest rate using r e r f 2 2 The local department store advertises a television set for $1085 or $100 deposit and $12.80 per week for three years. a. How much does the television end up costing under the hire purchase scheme? b. What is the flat rate of interest per annum? c. What is the effective rate of interest per annum? 3 A personal loan of $8000 over a period of 18 months costs $590 per month. a. What is the flat rate of interest per annum? b. What is the effective rate of interest per annum? FMT Business Related Mathematics Page 42
4 Sandi borrows $3000 and makes repayments of $220 per month over a period of 16 months. a. What is the flat rate of interest per annum? b. What is the effective rate of interest per annum? 5 Jim purchases a computer costing $3000 on hire purchase. He agrees to pay $500 deposit and then make 12 equal monthly payments of $230 for the rest of the year. a. What is the total amount Jim pays for the computer? b. How much interest does he pay? c. What is the flat rate of interest paid? FMT Business Related Mathematics Page 43
Past exam questions 1 VCAA 2006 A $2000 lounge suite was sold under a hire-purchase agreement. A deposit of $200 was paid. The balance was to be paid in 36 equal monthly installments of $68. The annual flat rate of interest applied to this agreement is A 10.0% B 11.4% C 12.0% D 22.4% E 36.0% 2 VCAA 2007 Brad investigated the cost of buying a $720 washing machine under a hire purchase agreement. A deposit of $180 is required and the balance will be paid in 24 monthly repayments. A flat rate of interest of 12% per annum applies to the balance. Brad correctly calculated the monthly repayment to be: A $22.50 B $25.20 C $26.10 D $27.90 E $29.70 3 VCAA 2009 Jamie bought a $500 games console on a hire-purchase plan.he paid $50 deposit and monthly installments of $25 for two years.the flat interest rate charged per annum is closest to A 15.0% B 16.7% C 30.0% D 33.3% E 66.7% FMT Business Related Mathematics Page 44
The following information relates to Question 4 and 5 Sandra has purchased a $4200 plasma television under a hire purchase agreement. She paid $600 deposit and will pay the balance in equal monthly installments over one year. A flat rate of 6% is charged. 4 The amount of each monthly installment is: A $300 B $303 C 318 D $350 E $371 5 The annual effective interest rate that Sandra pays under this agreement is closest to: A 10% B 11% C 12% D 13% E 14% Ex 21C Q1a,c,e, Q2a,c,e, Q3a, c, e, Q4a, c, e, Q5, 7, 9, 11, 13 FMT Business Related Mathematics Page 45
Inflation The growth of prices with time Continuing, upward movement in general level of prices of goods and services Results in a loss in the spending power of money Expressed as a percentage rate per year indicates annual increase in the price of a fixed set of goods and services e.g. gas, water, electricity. The effect of inflation (price increases) is that money loses its value and therefore its purchasing power (how much we can buy with it). Examples 1. Determine the growth factor for the following inflation rates a. 3% b. 8% c. 10% d. 4.5% 2. Consider an average inflation rate of 8% p.a. and a current price of $1.65 for a carton of milk. What will be the price in 2 years time? Growth factor = Growth factor = 1 + inflation% 100 Price in 1 year s time= Price in 2 year s time = So, inflation rate works in much the same way as compound interest (and therefore is an example of exponential growth). In fact we can use the compound interest formula as follows: A = P 1 + or r 100 t A = A = PR t P = R = (Growth factor) t = Examples 1 Suppose that inflation is recorded as 4.1% in 2006, 3.2% in 2007 and 3.8% in 2008, and that the rent on a certain property is $230 per week at the end of 2005. If the real estate agent increases the rent in line with inflation: a. what is the weekly rent at the end of 2008? Give your answer to the nearest cent. A = P = r = t = FMT Business Related Mathematics Page 46
b. what is the total percentage rise in the rent over the three year period? Give your answer to one decimal place. 2 If a litre of milk cost $2.00 at the start of 2012 and the inflation rate in 2011 was 3.6% and 2.7% in 2009, find the cost of a litre of milk at the start of 2009. 3 a. If inflation is 2.7% in 2006 and 3.5% in 2007, determine the cost of a loaf of bread at the end of 2007 if it costs $1.50 at the start of 2006 assuming that the bread price rises with inflation. What happens over a longer period? b. What will the price of the bread be (from the previous example) in 20 years if the average inflation rate is 2.3% 4 Suppose that a particular house is sold at auction for $700 000. If the price of houses increases with the inflation rate, what will be the price of the house in 10 years time (to the nearest one hundred dollars): a. if the average inflation rate over the 10-year period is 2.5%? b. if the average inflation rate over the 10-year period is 6.5%? FMT Business Related Mathematics Page 47
Investigating purchasing power (remember: purchasing power is the amount we can buy with the money it will decrease over time if not invested) 1. If savings of $10 000 are hidden in a mattress at the start of 2000, what will be the purchasing power of this money in January 2010 if the average inflation rate over this period is 2.9%. Give your answer to the nearest dollar. 2. If Elise hides $5 000 in her garage in 1995, what will be the purchasing power of this money 15 years later if the average inflation rate over this period is 3.7%. Give your answer to the nearest dollar. Problems In each of the following cases: i complete a table of values to show the cost of each item for each of the next 4 years using the given inflation rate ii display the cost of the item for the next 8 years graphically. a The cost of a magazine is currently $4.50 and the average inflation rate is 5% p.a. Year 0 1 2 3 4 5 6 7 8 Value ($) 4.50 FMT Business Related Mathematics Page 48
b The cost of a loaf of bread is currently $3.45 and the average inflation rate is 4% p.a. Year 0 1 2 3 4 5 6 7 8 Value ($) 3.45 Ex21D p572-573 Q1-11 FMT Business Related Mathematics Page 49
Depreciation Most equipment used by companies will reduce ( ) in value over time or with use. Terms Depreciation: the amount an item reduces in value, may be Book value: value of item after depreciation (cannot be less than zero) Scrap value: the price the item is expected to get when it can no longer be used profitably by the company. Unit cost depreciation: Cost per unit of production. (Where the value of the item is based on the amount of work it has done e.g 2 cars that have completed different numbers of km will have different unit cost productive output) Unit cost Depreciation based on how many units (items) have been produced or used by the machine being depreciated Unit cost = purchase price scrap value total productive output Unit depreciation and book value Depreciation (D) = cost price scrap value Depreciation (D) = unit cost n where n is the number of units produced Book value (V) = purchase price unit cost n = purchase price - depreciation Predicting useful life Length of time item will be in use (t) = purchase price scrap value. average production per year unit cost FMT Business Related Mathematics Page 50
Unit cost depreciation examples: 1. A machine, originally costing $50,000 is expected to produce 100 000 items. The output of the machine in its first 3 years is 15 000, 14 000 and 14 000 units respectively. If the anticipated scrap value of the machine is $8000, find: a. the book value at the end of the three years b. the number of years the machine would be expected to last if the average production per year for the remainder of its effective life is 14 000 items. Steps: Part a finding the book value 1. Determine the unit cost Unit cost = (purchase price scrap value) production 2. Calculate the book value at the end of the first year (purchase price cost for units produced in first year ) V 1 = Purchase price no. of units in 1 st year unit cost 3. Calculate the book value at the end of the second year (book value at start of second year cost for units produced in second year) V 2 = V 1 no. of units in 2nd year unit cost 4. Calculate the book value at the end of the third year (book value at start of third year cost for units produced in third year) V 3 = V 2 no. of units in third year unit cost Steps: Part b finding the number of years of its effective life 1. Calculate the average depreciation per year Average depreciation = average production for remainder of life unit cost 2. Calculate how much more the item can be depreciated before it will be scrapped Further depreciation = book value at end of period scrap value 3. Calculate how many years this will be at the average rate per year Remaining time (years) = further depreciation average depreciation 4. Find total life of the machine Total life of item = number of years depreciated + remaining years FMT Business Related Mathematics Page 51
2. Below are depreciation details for 3 vehicles. In each case find: i the annual depreciation ii the useful life (km). Purchase price ($) Scrap value ($) Average rate of depreciation (cents/km) Distance travelled in first year (km) a 25 000 10 000 26 12 600 b 21 400 8 000 21.6 13 700 c 29 600 12000 28.5 14 000 a. b. c. 2 A company buys a $32 000 car which depreciates at a rate of 23 cents per km driven. It covers 15 340 km in the first year and has a scrap value of $9500. a. What is the annual depreciation? b. What is the item s useful life? 3 A new taxi is worth $29 500 and it depreciates at 27.2 cents per km travelled. In its first year of use it travelled 28 461 km. Its scrap value was $8200. a. What is the annual depreciation? b. What is the item s useful life? FMT Business Related Mathematics Page 52
4. A photocopier is bought for $8600 and it depreciates at a rate of 22 cents for every 100 copies made. In its first year of use, 400 000 copies are made and in its second year, 480 000 copies are made. Find a the depreciation for each year b the book value at the end of the second year 5 A photocopier purchased for $7200 depreciates at a rate of $1.50 per 1000 copies made. In its first year of use, 620 000 copies were made and in its second year, 540 000 were made. Find a the depreciation for each year b the book value at the end of the second year 6 A printing machine was purchased for $38 000 and depreciated at a rate of $1.50 per million pages printed. In its first year 385 million pages were printed and 496 million in its second year. Find a the depreciation for each year b the book value at the end of the second year FMT Business Related Mathematics Page 53
Flat rate depreciation Where the value is depreciated by the same percentage of the purchase price each year. If an item depreciates by the flat rate method then its value decreases by a fixed amount each unit time interval, generally each year. This depreciation value may be expressed in dollars or as a percentage of the cost price. This method of depreciation may also be referred to as prime cost depreciation. Since the depreciation is the same for each unit time interval, the flat rate method is an example of straight line (linear) decay. Depreciation D = purchase price interest rate (per annum) length of time (years) 100 = Prt 100 Book value V = purchase price depreciation = P Prt 100 Graph is linear book value ($) V = book value after t years P = purchase price of item r = flat rate of depreciation, p.a t = time (years) time (years) Can use CAS for table or graph refer back to page 8 Reducing balance depreciation Where the value of the item is reduced by a percentage of its value of the preceding year V = P 1 r t 100 D = P - V V = book value after t years P = purchase price of item r = compound of depreciation, p.a, compounded annually t = time (years) Graph is non-linear book value ($) time (years) FMT Business Related Mathematics Page 54
Depreciation tables flat rate Example A computer purchased for $5000 depreciates at 15% p.a flat rate a) By how much does the computer depreciate per year? b) Complete the depreciation table below c) What is the book value after 3 years? d) How many years does it take for the book value to become zero? Age of computer (years) Yearly depreciation ($) Book Value ($) 0 0 5000 Depreciation tables reducing rate Example: a. Compare the same computer if it is depreciated using the reducing balance method at 18% p.a. for the first ten years by completing the table below b. When do they have the same book value? Age of computer (years) Yearly depreciation ($) Book Value ($) 0 0 5000 FMT Business Related Mathematics Page 55
Graph the book value over time for both the flat rate and reducing balance depreciations book value ($) 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 1 2 3 4 5 6 7 8 9 10 time (y ears ) Examples 1 A machine costs $17 000 new and depreciates at a flat rate of 12.5% per annum. If its scrap value is $4250 find: a the book value of the machine after 3 years b the number of years until the machine can be written off 2 The value of their computer system is considered by a company to decrease at a rate of 12% of the previous year s value each year. If the cost of purchase of the system is $128 000, find: a the book value after 7 years FMT Business Related Mathematics Page 56
b the total depreciation after 7 years c the number of years until the value of the computer system is below the scrap value of $40 000 3 A car costing $32500 depreciates at a rate of 14.5% per year, on a reducing balance basis. a What is the book value of the car at the end of 5 years? b What is the total amount of depreciation after 5 years? c What would be the equivalent rate of flat rate depreciation over the same time period? Ex 21E page 582 583 Q1-13 FMT Business Related Mathematics Page 57
Applications of finance Annuities formula: (for compound interest with repayments) A = PR n Q (Rn 1) R 1 A = amount owing or invested after n payments n=x t where t = no. of years x = number of compounds per year P = amount borrowed or invested R = 1 + r/x 100 r= annual interest rate Q = the amount paid each period Reducing balance loans: loan taken out under compound interest and periodic payments are made (PV and PMT will have opposite signs) ****PV is positive**** ***PMT is negative*** Annuities: opposite of reducing balance loan. Money invested and periodic withdrawals made (PV and PMT will have opposite signs) ****PV is negative**** ***PMT is positive**** Annuity investment (Adding to an investment): PV and PMT will have same sign ****PV is negative*** ***PMT is negative*** Comparing loans: ie comparing simple interest with compound interest Interest only loans: where only the interest is paid back FV remains the same throughout the life of the loan ****PV is positive**** ****FV is negative*** Perpetuities: an investment that pays out an equal amount forever (Use annuities formula but we want A=P) ***PV is negative*** ****FV=-PV**** A = PR n Q (Rn 1) R 1 P = 100Q r Q = Pr 100 A = amount invested after n periods P = A R = 1+ r/x 100 x = number of compounds per year r = annual interest rate Q = payment received per year n = number of periods (=nt) where t = number of years FMT Business Related Mathematics Page 58
RULES: Bank gives you money: negative POSITIVE NEGATIVE You give the bank money: LOAN ANNUITY INVESTMENT THAT IS ADDED ON A REGULAR BASIS PV positive PMT negative FV 0 or negative PV negative PMT positive FV 0 or positive PV negative PMT negative FV positive Bank lends you money You pay off the loan by making regular payments to the bank At the end of the loan, it is either fully paid out (FV=0) or you still have to pay money to the bank You give money to the bank The bank makes regular payments to you (from the investment) At the end of the annuity, there is either no money left (FV=0) or the bank pays you back what is left You give money to the bank You increase the amount of the investment by making regular payments to the bank At the end of the investment, the bank pays you the total final value of your investment FMT Business Related Mathematics Page 59
Application of finance solvers Reducing balance loan Annuities Adding to an investment Comparing loans Interest only loans Perpetuities Amortise fully pay off Reducing balance loan Examples Compound interest with regular repayments 1. John and Joan have purchased a property for $550,000. They have a deposit of $150,000 and will take out a mortgage at 7.5% p.a reducing balance with monthly repayments of $3200 for the remainder. a. How long will it take to pay off the loan? Give your answer in both months, to the nearest month and years, to 1 decimal place. b. What will be their final payment? 2. a. Find the quarterly payment necessary to amortise a loan of $25000 over 5 years given a reducing balance interest rate of 12.5% p.a compounding quarterly. b. Find the interest paid on the loan c. the amount owing after 3 years FMT Business Related Mathematics Page 60
3. a. Find the fortnightly payment if $25,000 is borrowed at 12% p.a over 6 years. (Assume the interest is compounded over the same time period as the repayments) b. How much has been paid off the principal after 3 years? c. Over the term of the loan, what is the total interest paid? 4 Find the term of the loan if $40 000 is borrowed at 9% p.a compounded monthly with a monthly repayment of $380 5 Ozzie Home Loans offers its customers loans at a compound interest rate of 4.85% per annum, adjusted monthly. If Cathy and Tom borrow $220 000 to buy their new home, then the monthly repayment necessary to pay off the loan in 20 years is closest to: A $1434 B $11473 C $954 D $10670 E $1833 FMT Business Related Mathematics Page 61
Annuities Opposite to reducing balance loan Money is invested and periodic withdrawals made until no money is left. Often want to find the size of the withdrawal we can make over a period of time or else how long the investment will last. Examples 1. Tiffany purchases a $40 000 annuity investment at 3.5% p.a. compound interest, compounded monthly. a. She wants to receive monthly payments for 10 years. How much will she receive? Use TVM N = I% = PV = PMT = FV = P/Y = C/Y = b. (i) If she receives monthly payments of $1000, how long will the annuity last? Give answer to the nearest month Use TVM N = I% = PV = PMT = FV = P/Y = C/Y = (ii) State the value of the last payment FMT Business Related Mathematics Page 62
Adding to an investment Also called an annuity investment Money invested and periodic payments are made Examples 1. Ginny saves $100 per month, deposited directly into her bank account on payday (the last day of each month). The account earns 12% p.a., compounding monthly. How much will she earn at the end of 10 years, assuming the bank continues to pay the same interest rate? Use TVM N = I% = PV = PMT = FV = P/Y = C/Y = 2. Indira starts an investment with a deposit of $4000. At the end of every successive month, she makes an additional deposit of $300. The investment has a fixed interest rate of 10% p.a., compounding monthly. a. What would be the value of Indira s investment, to the nearest dollar, after: i 5 years? ii 20 years? N = N = I% = PV = PMT = FV = P/Y = C/Y = b. Indira plans to use her accumulated investment to put a deposit on a house and land package worth $500 000. She will need 15% of this amount as the deposit. To the nearest month, find how long it will take for the investment to realise Indira s goal. N = I% = PV = PMT = FV = P/Y = C/Y = I% = PV = PMT = FV = P/Y = C/Y = FMT Business Related Mathematics Page 63
Comparing loans Examples 1. James is considering taking a loan of $6000 for tertiary fees to be repaid over 5 years and is offered two different sets of terms and conditions: Plan A: Flat rate of interest of 7% p.a. Plan B: Interest of 8.5% p.a., adjusted monthly on the reducing balance. Determine which plan is best and by how much. 2. An amount of $6000 is borrowed over 3 years at 8% p.a. a Find the monthly repayment if the terms are: i interest is calculated at a flat rate. ii interest is calculated on the quarterly reducing balance. b How much is saved using the reducing balance method, to the nearest dollar? c What would the interest rate need to be for the flat rate loan to be equivalent to the reducing balance loan? FMT Business Related Mathematics Page 64
Interest only loans examples Where the value of the purchase (say an investment property) will increase, and will be sold to pay off the original loan Payments are made to pay the interest only. 1. Marcia borrows $150 000 to buy an investment property. If the interest on the loan is 7.35% p.a, compounding monthly, what will her monthly repayment be on an interest only loan? Use TVM N = I% = PV = PMT = FV = P/Y = C/Y = 2. Alex borrows $280 000 to purchase a property. If the interest is charged at 7.01% p.a. compounding monthly, what will be the monthly payments on an interest only loan? Use TVM N = I% = PV = PMT = FV = P/Y = C/Y = FMT Business Related Mathematics Page 65
Perpetuities An investment that pays an equal amount forever Use TVM where possible but sometimes need Examples P = 100Q r or Q = Pr 100 Q = annual payment Since these investments go on forever, when using TVM we can use ANY value for N 1. Sue invests $100000 in a perpetuity with an interest rate of 2.95% p.a. compounded quarterly. What quarterly payment will she receive? 2. Joanne decides to invest in a perpetuity to support disadvantage students. She wants to give scholarships of $1200 each year for ever. The perpetuity pays 3.5% p.a.. compounded annually. How much does Joanne need to invest? 3. Instead, Joanne decides she would prefer to give the recipient payment per month. She finds a different perpetuity offering 4% p.a. compounded monthly and invests $40000. What monthly payment will the scholarship recipients receive? FMT Business Related Mathematics Page 66
Using the annuities formula questions 1. Gemma borrowed $220 000 at 7.2% per annum compounding monthly. The repayments are $1 800 per month. The balance of the loan $A at the end of five years can be found using the rule A = P R n Q ( R n 1) R 1 where A. P = 220 000, R = 1.006, Q = 1 800 and n = 5 B. P = 220 000, R = 1.072, Q = 1 800 and n = 5 C. P = 220 000, R = 1.006, Q = 1 800 and n = 60 D. P = 220 000, R = 1.006, Q = 21 600 and n = 5 E. P = 220 000, R = 1.072, Q = 21 600 and n = 60 2. Jane has calculated that she needs $2 200 per month for living expenses. The amount, to the nearest dollar, that she will need to invest at 6.75% per annum interest to provide for her monthly living expenses for an indefinite period is A. $32 593 B. $289 104 C. $137 398 D. $391 111 E. $365 652 3. Tom is about to buy his first home which costs $460 000. To secure the purchase of the house, he has to pay a deposit of 15%. (a) Calculate the amount of deposit that Tom must pay. Tom intends to finance the purchase of the house with a bank loan of $250 000. The interest on the loan is charged at 7.8% per annum, compounded monthly and calculated on the reducing balance. Tom plans to make monthly repayments over the loan period of 25 years. (b) The monthly repayments can be calculated using the annuities formula A = PR n Q ( R n 1 ) R 1 State the values of A, P, R and n that need to be substituted into this formula to determine Tom s monthly repayments. A = P = R = n = FMT Business Related Mathematics Page 67
(c) Tom s monthly repayments work out to $1 896.54. Find the total amount that he will end up paying for his loan. (d) Calculate the amount of interest, in dollars, that is charged on the loan over the 25-year period. (e) What will be the amount that Tom still owes on his loan after 12 years? Give your answer to the nearest dollar. 4. Wendy sees an oven for sale at Hot Ovens Discount Store, also for $17 280. The terms of the sale there require no deposit and monthly repayments over three years at an interest rate of 6.4% per annum, calculated monthly on a reducing balance. The monthly repayments can be determined using the annuities formula: The loan is paid out in three years. a. What are the values of n, P and A? A = PR n Q ( R n 1 ) R 1 b. What is the monthly repayment for this loan? Write your answer in dollars, correct to two decimal places. c. What is the total cost of the oven from Hot Ovens Discount store on these terms? Write your answer correct to the nearest dollar. FMT Business Related Mathematics Page 68
Using the Amortization function on the CAS TVM Calculations Amortization Select End of period or Beginning of period in format Use the help screens to assist If you have already entered data in compound interest screen this will be copied across PM1 is the 1 st payment you are interested in PM2 is the final payment you are interested in Particularly useful in problems where you are required to find the balance owed after time or the total paid off the principal during any period. Example Daniella has been paying off a loan of $42 000 reducing monthly at an interest rate of 10.3% p.a., with repayments of $644 per month. Find the: a length of the loan. TVM Calculations Compound interest enter all of the initial constraints find N b balance owed after 1 year or using amortization FMT Business Related Mathematics Page 69
c d the principal portion of the first payment the interest portion of the first payment e f total amount of principal repaid during the first year. total amount of interest paid during the first year g amount of principal paid in the final month h the small amount payable in the final payment FMT Business Related Mathematics Page 70