Credit and loans FINANCIAL MATHEMATICS

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1 !NNC Yr12 maths ch 03 Page 79 Wednesday, October 4, :42 AM Credit and loans FINANCIAL MATHEMATICS 3 At some stage in your life you will probably need to borrow money. This may be for a home, a car, a computer or a holiday. When taking out a loan you must consider how much money you need, your ability to repay the loan, how much interest you will pay as well as the terms and conditions of the loan. There are many institutions willing to lend money, but the borrower must be certain that they can pay off the loan. It is not always best to take up offers such as Buy now and pay nothing for 6 months or We will arrange finance for approved customers or Buy now and pay only $20 per week. Credit does not mean free. You will compare different loan and credit options in this chapter. In this chapter you will learn how to: calculate the principal, interest and repayments for a flat rate loan consider borrowing money on terms and deferred payment plans construct and calculate values in a table of loan repayments use published tables to determine monthly repayments on a reducing balance loan use technology to compare loans calculate credit card payments involving interest-free periods, interest rates, fees and charges. CREDIT AND LOANS 79

2 !NNC Yr12 maths ch 03 Page 80 Wednesday, October 4, :42 AM FLAT RATE LOANS A flat rate loan is one where flat (or simple) interest is charged on the amount borrowed (or principal) for the term of the loan. The term of the loan, usually from 1 to 5 years, is decided by the lender and is based on the borrower s capacity to meet the repayments. There may be a small charge to establish the loan. This is to cover administrative costs. This type of loan is used for purchasing goods such as a car, computer or stereo. Fees and charges often apply and can be included in the amount borrowed or paid up front. Some examples of lending institutions offering flat rate loans are: banks building societies (e.g. IMB) credit unions insurance companies (e.g. AMP, GIO, NRMA) finance companies (e.g. AVCO). Flat rate interest or simple interest is calculated using the formula I = Prn where P = principal r = rate per period expressed as a decimal n = number of periods. Example 1 Mark borrowed $2500 at a flat interest rate of 9% p.a. to buy a new computer. What is his monthly repayment (to the nearest cent) if he pays off the loan in 5 years? Solution P = $2500, r = 0.09, n = 5 Interest = Prn = $ = $1125 Amount to repay = principal + interest = $ $1125 = $3625 $3625 Monthly repayment = = $ Mark s monthly repayment is $ years = 5 12 months = 60 months Example 2 Donna wanted to buy a car stereo for $1500. Her bank offered her a loan at a flat interest rate of 12% p.a. to be repaid in fortnightly instalments over 2 years. The bank charges were stamp duty of $7.50, loan insurance of $8.50 and an establishment fee of $12. If Donna included the bank charges in the amount she borrowed, find: (a) the total amount to repay (b) the fortnightly repayment 80 NEW CENTURY MATHS GENERAL: HSC

3 !NNC Yr12 maths ch 03 Page 81 Wednesday, October 4, :42 AM Solution (a) Total amount borrowed = $ $ $ $12 = $1528 P = $1528, r = 0.12, n = 2 Interest = Prn = $ = $ Total amount to repay = amount borrowed + interest = $ $ = $ Interest is paid on the whole amount borrowed, including bank charges. (b) $ Fortnightly repayment = years = 52 fortnights = $ Each fortnightly repayment is $ Exercise 3-01: Flat rate loans 1. Guy borrowed $9000 to buy a second-hand car on a flat rate loan at 10% p.a. interest over 5 years. (a) What must he repay altogether? (b) What is his monthly repayment? 2. Heidi obtained a holiday loan of $3500 at 15% p.a. flat rate interest to be paid back in fortnightly instalments over 3 years. (a) How much will her holiday cost altogether? (b) How much is each instalment? 3. Brooke borrowed $9500 to help furnish her new home. She took a flat rate loan at 10.2% p.a. interest and paid off the loan in weekly instalments over 4 years. How much was each repayment? 4. Elton obtained a $1200 loan at a flat interest rate of 11.2% p.a. to help with the purchase of a new piano. He paid off the loan in monthly instalments over 3 years. (a) How much interest did he pay? (b) How much was each repayment? 5. Mabel bought a new washing machine and dryer and borrowed $1200 to help with the purchase. If the simple interest rate was 14.6% p.a. and the term of the loan was 2 years, how much was each fortnightly instalment? 6. Gaylene bought a new bed from Bob s Beds for $1850. She paid off the bed in weekly instalments over the next 18 months at a flat interest rate of 18.55% p.a. If she included $9 stamp duty, a dealer s commission of $18.50 and loan protection insurance of $45 in the amount borrowed, find: (a) the total amount borrowed (b) the amount of interest she paid (c) the weekly repayment (d) how much she would have saved if she had paid the charges up front 7. Katie borrowed $8500 over 3 years at a flat interest rate of 13.5% p.a. to help towards her wedding. She included the $7.50 stamp duty and $15 per year loan insurance in the amount borrowed and repaid the loan in monthly instalments. (a) How much did she borrow altogether? (b) How much interest did she pay? (c) What was her monthly repayment? CREDIT AND LOANS 81

4 !NNC Yr12 maths ch 03 Page 82 Wednesday, October 4, :42 AM 8. The Education Credit Union published this table for flat rate loans. Giselle borrowed $8000 over 4 years. (a) How much does she repay per month? (b) What is the total amount to repay the loan? (c) What is the interest charged? (d) Calculate the flat interest rate per annum. Years to repay loan Monthly repayment (per $1000) 1 $ $ $ $ $ Ryan borrowed $2200 from the Education Credit Union to purchase new ski equipment. He agreed to repay the loan over 2 years. Use the table in question 8 to find: (a) how much he repays per month (b) the total amount to be repaid (c) the amount of interest charged (d) the simple interest rate per annum 10. Cameron borrowed $6300 from the Education Credit Union to consolidate his debts. He arranged to repay the loan over 5 years. Use the table in question 8 to find: (a) the monthly repayment (b) the total amount of interest 11. A bank advertised the following fixed rates per annum for secured personal loans. Loan security Amount of loan Greater than $ Less than $ New motor vehicle 9.4% 9.4% Used motor vehicle 9.9% 10.75% New or used motorbike, caravan or boat 10.49% 10.75% Three people had their loans approved: Alexandra (A) borrowed $ over 5 years for a new car. Basil (B) borrowed $9000 over 36 months for a second-hand motorbike. 1 Cecilia (C) borrowed $ over 4 -- years for a new boat. 2 For each person find: (a) the interest rate per annum charged by the bank (b) the amount of interest charged (c) the monthly repayment 12. The following fixed interest rates per annum, determined by the amount borrowed and the term of the loan, were published by a bank. Loan amount Term 1 2 years 3 5 years 6 7 years Greater than $ % 10.99% 11.99% Less than or equal to $ % 11.49% 12.49% 1 Note: 30 months (2 -- years) loans will receive the 3 5 years interest rate; 66 months 1 2 (5 -- years) will receive the 6 7 years interest rate NEW CENTURY MATHS GENERAL: HSC

5 !NNC Yr12 maths ch 03 Page 83 Wednesday, October 4, :42 AM Consider these cases: Adrian (A) gets a loan of $7500 for a second-hand car to be paid back over 2 years. Beryl (B) obtains a loan of $ over 4 years for a new bathroom. Christian (C) takes out a $9800 loan for an overseas holiday, to be repaid in 30 months. For each person find: (a) the interest rate on their loan (b) the amount of interest they will pay (c) their monthly repayment 13. The OzExpress Credit Union advertised the following personal loans. Loan type Terms and conditions Interest rate (p.a.) New Car Loan Loan availability up to $ Maximum term 10 years Car Loan Loan availability up to $ Maximum term 8 years One-Stop Travel Loan Loan availability up to $ Minimum loan $2000 Maximum term normally 5 years New Computer Loan Loan availability up to $ Minimum loan $2000 Maximum term normally 4 years 9.25% 9.90% 9.50% 9.50% Calculate the repayment for the following OzExpress Credit Union personal loans. (a) $2000 for a new computer to be repaid monthly over 3 years (b) $ for a new car to be repaid fortnightly over 8 years (c) $4500 for a Fiji holiday to be repaid weekly over 3 years (d) $ for a used car to be repaid monthly over 6 years Investigation: Current interest rates 1. Collect current interest rates for flat rate loans (often called personal loans) from as many different lending institutions as you can. These can be found by approaching a bank (or other lending institution) and asking for pamphlets, by consulting the money section of a newspaper or by accessing Internet websites. Here are seven websites that may be of help: Investigate the different types of flat rate loans and their current interest rates. 3. What terms and conditions for flat rate loans does each lending institution impose on its borrowers? CREDIT AND LOANS 83

6 !NNC Yr12 maths ch 03 Page 84 Wednesday, October 4, :42 AM Spreadsheet activity: Calculating flat rate loan repayments 1. Create a spreadsheet to calculate monthly, fortnightly or weekly repayments for a flat rate loan by entering amounts in cells B1, B2, B3 and B4 and appropriate formulas in cells B6, B7 and B8. A B C D 1 Principal (P) $ Interest rate (% p.a.) 9 3 Term (years) 8 4 Periods per year Total interest $ Total repaid $ Repayment per period $ Show that the fortnightly repayment required to pay off a loan of $ in 8 years at 9% p.a. interest is $ What weekly repayment is needed to pay off a loan of $7600 at 8% p.a. interest in 4 years? Just for the record AUSTRALIA S FIRST BANK The first bank in Australia, the Bank of New South Wales, was opened in Macquarie Place, Sydney on 8 April 1817 in what was the British colony of NSW. In the 1970s it became known as The Wales Bank. In 1982 it expanded to the Western Pacific region and changed its name as it was no longer restricted to New South Wales. What is its new name? BUYING ON TERMS Many shops and businesses sell goods to customers on terms. Another name for this is time payment as the customer signs an agreement to pay for the goods over a certain period of time. It is also called hire purchase because the customer actually hires (or borrows) the goods until they are paid off. If you buy on terms, you do not actually own the goods until they are fully paid for, and goods can be repossessed (taken back) if you fail to meet the repayments. Example 3 A refrigerator is advertised for $1200 cash or $200 deposit and $57 per month for 2 years. (a) How much more will the refrigerator cost if bought on terms? (b) What is the flat rate of interest charged? 84 NEW CENTURY MATHS GENERAL: HSC

7 !NNC Yr12 maths ch 03 Page 85 Wednesday, October 4, :42 AM Solution (a) Deposit = $200 Total repayments = $57 24 = $1368 Total cost on terms = $ $200 = $1568 Difference between cash price and terms price = $1568 $1200 = $368 By buying on terms the refrigerator will cost $368 more than the cash price. (b) The buyer borrows $1000 since $200 of the $1200 are paid in cash. P = $1000, I = $368, n = 2 I = Prn 368 = 1000 r 2 = 2000 r 368 r = = The flat rate of interest charged is 18.4% p.a. Deferred payment plan The deposit is sometimes called a down payment. The difference is the interest on the loan. No deposit Nothing to pay until June Buy now and pay nothing for 6 months! Some stores advertise deferred payment schemes to entice customers to purchase goods, in the expectation that they are receiving something for nothing. If the customer pays for the goods before the end of the interest-free period, this type of plan may be beneficial. A customer who pays for the goods after the interest-free period will be charged interest from the date of purchase. Example 4 Michelle wants to purchase a home gym for $900. She cannot afford to pay cash and must choose one of the following options: The Loan Arranger: Pay 10% deposit and 12 monthly repayments of $80. Flashy Finance: No deposit, nothing to pay for 6 months, then 12 monthly repayments of $100. (a) For each option find: (i) the total cost (ii) the amount of interest (iii) the flat interest rate per annum (correct to 1 decimal place) (b) Which of the two loan options should Michelle choose? Why? Solution (a) The Loan Arranger (i) Deposit = 10% $900 Total repayments = $80 12 = $90 = $960 Total cost = deposit + total repayments = $90 + $960 = $1050 CREDIT AND LOANS 85

8 !NNC Yr12 maths ch 03 Page 86 Wednesday, October 4, :42 AM (ii) Interest = $1050 $900 = $150 (iii) I = $150, P = $810, n = 1 I = Prn 150 = 810 r r = = The flat interest rate is 18.5% p.a. Flashy Finance (i) Total cost = $ = $1200 (ii) Interest = $1200 $900 = $300 (iii) I = $300, P = $900, n = 1.5 I = Prn 300 = r = 1350 r 300 r = Amount borrowed = $900 $90 = $810 With this option, interest is charged for 18 months; that is, n = 1.5 years = The flat interest rate is 22.2% p.a. (b) Michelle should choose The Loan Arranger. By paying a deposit and then paying the rest over 12 months she pays less interest. Exercise 3-02: Buying on terms 1. Darby and Joan retired and bought a second-hand motorhome for $ to travel around Australia. They paid $1500 deposit and 36 monthly instalments of $685. (a) How much did they pay altogether for the motorhome? (b) How much interest did they pay? (c) What was the flat rate of interest charged? 2. Budget Bob sells used cars on terms. For sale he has the following: BUDGET BOB S BEAUT DEALS Toyota Celica $ cash or $500 deposit and $95 per week over 5 years Holden Ute $ cash or $300 deposit and $78 per week over 4 years Kombi Van $6200 cash or $180 deposit and $70 per week over 2 years Price includes registration, stamp duty and delivery for approved customers (a) How much would you pay for the Kombi Van if you took Bob s terms? (b) How much would you save on the Celica by paying cash rather than terms? (c) What is the total amount a buyer would pay for the Ute on terms? (d) What is the flat rate of interest per annum, correct to 1 decimal place, charged on each vehicle? (e) Can you give a reason why Bob charges different rates of interest for different cars? 86 NEW CENTURY MATHS GENERAL: HSC

9 !NNC Yr12 maths ch 03 Page 87 Wednesday, October 4, :42 AM 3. Del and Barry bought an old cottage for $ and had it transported to their farm in the country. They paid the vendor 10% deposit and agreed to pay him the rest at $85 per week over 5 years. (a) How much did they actually pay for the house? (b) What was the vendor s flat interest rate, correct to 3 significant figures? 4. The Ripoff Finance company had the following hire purchase agreement with Mr B. Spender: Cash price of yacht $ Deposit $450 Trade-in allowance on old yacht $1200 Stamp duty $ Registration $ monthly repayments of $680 (a) What was the total amount paid for the yacht if the deposit, stamp duty and registration were paid by Mr Spender at the time of purchase? (b) What was the flat interest rate charged p.a. by the finance company? Answer correct to 1 decimal place. (Note: Principal borrowed = cash price deposit trade-in) 5. Jayden bought a new bedroom suite from Dad s Discounts. The cash price was $2850 but Jayden chose to take Dad s terms and paid 10% deposit and $42.50 per week for 18 months. (a) How much did she actually pay for the bedroom suite? (b) How much would she have saved by paying cash? (c) What was the flat rate of interest charged? 6. Billy Joe bought a new stereo TV, cash price $999, and chose the deferred payment plan offered at Main Man discounts. He paid $20 deposit and his first payment of $60 was 3 months later. If he made 24 monthly repayments, find: (a) the total cost of the TV (b) the flat rate of interest charged (correct to 1 decimal place) 7. Steven purchased an engagement ring and paid $100 down, no payments for 6 months and then $60 a fortnight for 3 years. (a) How much more than the advertised price of $3000 did he pay? (b) What was the flat rate of interest charged? 8. Brian wants to buy a new computer desk that retails for $1150. He chooses the deferred payment plan of no deposit, nothing to pay for 6 months and then 12 monthly instalments of $130. (a) Find the total cost of the desk using the deferred payment plan. (b) How much extra does the plan cost? (c) What is the flat interest rate charged? 9. Write down one advantage and one disadvantage of: (a) paying cash for goods (b) buying on terms (c) using a deferred payment plan CREDIT AND LOANS 87

10 !NNC Yr12 maths ch 03 Page 88 Wednesday, October 4, :42 AM REDUCING BALANCE LOANS Loans taken over long periods, such as home loans, are usually calculated using reducible interest. This means that interest is calculated on the balance still owing, not on the total principal borrowed as with flat rate interest. The balance of the loan reduces after each repayment and continues in this way until the loan is fully repaid that is, until the balance owing is zero. Example 5 Shelley borrowed $ to buy a new car at 8% p.a. reducible interest. She made monthly payments of $180. (a) Draw up a table showing the progress of the loan in the first 6 months. (b) How much had Shelley paid off the principal after her 6th payment? (c) How much interest did she pay in the first 6 months? (d) How much interest did she save compared with a flat interest rate loan at 8% p.a.? Solution (a) For the first month: Amount owing (principal) at the start of the month $ Interest on this principal for 1 month $ = $ Amount owing before repayment $ $100 = $ Balance owing after repayment $ $180 = $ Principal for the second month $ Continue in this way until you have calculated the values for the first 6 months. Loan for Shelley s new car Amount borrowed $ Interest rate p.a. 8% Monthly repayment (R) $180 No. of months (n) Principal ($P) Reducing balance loan table Interest ($I) Amount owing before repayment $(P + I) Balance $(P + I R) (b) At the end of the first 6 months: Amount owing = $ Amount paid off principal = $ $ = $ (c) Interest paid in first 6 months = $ Find the sum of the interest column. 88 NEW CENTURY MATHS GENERAL: HSC

11 !NNC Yr12 maths ch 03 Page 89 Wednesday, October 4, :42 AM (d) For a flat rate of interest for 6 months: P = $15 000, r = 0.08, n = 0.5 n = 6 months = 0.5 years I = Prn = = $600 Interest saved for first 6 months = $600 $ = $8.06 Exercise 3-03: Reducing balance loans 1. (a) Copy the table in Example 5 and add rows to show the progress of the loan for the next 6 months. (b) How much had Shelley paid off the principal after 12 months? (c) How much interest did she pay in the first year? (d) Calculate her saving in interest for the first year compared to the same loan at a flat rate interest of 8% p.a. 2. Isabelle borrowed $ to purchase her first studio apartment. The bank offered to lend her the money at 10% p.a. reducible interest and fortnightly repayments of $540. No. of fortnights (n) Principal ($P) Interest ($I) Amount owing $(P + I) Balance $(P + I R) (a) Copy and complete the table above showing the progress of her loan for the first 6 fortnights. (b) How much had she paid off the principal after 6 repayments? (c) How much interest did she pay: (i) in the first 3 fortnights? (ii) in the next 3 fortnights? (d) What is happening to the amount of interest as the number of repayments increases? 3. Greg borrowed $2000 to buy new golf clubs. The interest rate is 10% p.a. and he makes quarterly repayments of $500. (a) Draw up a loan repayment table and calculate values until the balance is zero. The last payment may be less than $500. (b) How much will his last payment be? (c) How much interest will he pay? (d) How many years will it take to pay off the loan? (e) How much more interest would he pay if the loan had a flat rate interest of 10% p.a.? 4. Gillian borrowed $2500 at 7% p.a. interest for a South Pacific Cruise. She repaid the loan in regular fortnightly repayments of $300. (a) How many weeks did it take to pay off the loan completely? (b) What was the amount of the last payment? (c) How much interest was charged on this loan? (d) How much interest would she have paid if this loan had had a flat interest rate of 7% p.a.? CREDIT AND LOANS 89

12 !NNC Yr12 maths ch 03 Page 90 Wednesday, October 4, :42 AM USING PUBLISHED LOAN REPAYMENT TABLES Because the calculations for large loans are complicated, financial institutions publish tables related to loans. For example, the table below gives the monthly repayments for a $1000 reducing balance loan for various terms and interest rates. Monthly repayments reducing balance loan of $1000 Interest rate (%) Term (years) Example 6 (a) Mr and Mrs Pitt obtain a premium home loan of $ at 7% p.a. reducible interest for a term of 20 years. Find: (i) the monthly repayment (ii) the total amount repaid (iii) the total interest paid (b) How much more interest would they pay by choosing a 25-year term rather than a 20-year term? Solution (a) Term = 20 years, interest rate = 7% p.a. From the table above: (i) Monthly repayment for $1000 = $7.75 Monthly repayment for $ = 370 $7.75 = $ (ii) Total amount repaid = $ = $ (iii) Interest paid = total amount repaid amount borrowed = $ $ = $ (b) Term = 25 years, interest rate = 7% p.a. From the table above: Monthly repayment for $1000 = $7.07 Monthly repayment for $ = 370 $7.07 = $ Total amount repaid = $ = $ Interest paid = $ $ = $ Difference in interest paid = $ $ = $ They would pay $ more interest by choosing a 25-year term. 90 NEW CENTURY MATHS GENERAL: HSC

13 !NNC Yr12 maths ch 03 Page 91 Wednesday, October 4, :42 AM Study tips FINDING THE KEY When you study a novel, your aim is to find the underlying theme of the book. The same is true when you study a mathematics topic. Look for the key, the main idea that threads through the topic: the reason for studying it. Once you understand and appreciate the point of a topic, a lot of its concepts and ideas will fall into place. For example, what are the main themes of topics in this chapter? To help you find the key: Analyse the content of the chapter for meaning and relevance. Why did you learn the topics? How and where will you use the mathematics? Use your own words to write your understanding of the topics in the chapter so that you take ownership of your mathematics. Use the chapter introduction and chapter review to help you. Ask yourself and your teacher questions about the topic. Have an opinion. Sound informed. Finding the key will unlock the door to mathematical understanding! Exercise 3-04: Using published loan repayment tables 1. Use the table of loan repayments on page 90 to determine monthly repayments for the following loans. (a) $ at 12% over 15 years (b) $ at 9% over 25 years (c) $ at 6% over 20 years (d) $ at 11% over 10 years 2. Use the table of loan repayments on page 90 to determine, for each loan below: (i) the monthly repayment (ii) the total amount repaid (iii) the total interest paid (a) $ at 7% over 30 years (b) $ at 8% over 25 years (c) $ at 10% over 30 years (d) $ at 9% over 20 years 3. This table gives the monthly repayments required to pay off a loan in 10 years. Monthly loan repayments 10-year term Principal borrowed Interest rate 6% p.a. 7% p.a. 8% p.a. 10% p.a. $5000 $ $ $ $55.52 $ $ $ $58.06 $ $ $ $60.67 $ $ $ $66.08 $ $ $ For each of the following 10-year loans find: (i) the total repayments (ii) the amount of interest paid over the term of the loan (a) $ at 8% p.a. (b) $ at 6% p.a. (c) $5000 at 7% p.a. (d) $ at 10% p.a. (e) $ at 8% p.a. (f) $ at 7% p.a. CREDIT AND LOANS 91

14 !NNC Yr12 maths ch 03 Page 92 Wednesday, October 4, :42 AM 4. The table below shows the monthly repayment required to pay off a reducing balance loan ranging from $ to $ at an interest rate of 8.50% p.a. over terms from 15 to 30 years. For each loan below find: (i) the monthly repayment (ii) the amount of interest paid over the term of the loan (a) $ for 19 years (b) $ for 25 years (c) $ for 30 years (d) $ for 17 years (e) $ for 18 years (f) $ for 15 years (g) $ for 25 years (h) $ for 15 years Monthly loan repayments (interest 8.50% p.a.) Principal ($) Term (years) Use the table of monthly loan repayments on page 90 to: (a) find the interest rate per annum on a: (i) 5-year loan if the monthly repayment is $21.74 per $1000 (ii) 20-year loan if the monthly repayment is $8.36 per $1000 (b) find the term of the loan at: (i) 7% p.a. interest if the monthly repayment is $7.75 (ii) 11% p.a. interest if the monthly repayment is $9.80 Think: Fortnightly vs monthly repayments If you pay half the monthly repayment each fortnight you actually pay 13 monthly repayments per year instead of 12. This means the balance of the loan reduces more rapidly, you are charged less interest and your loan is paid off in less time. The graph shows the progress of a $ loan at 6.7% p.a. reducible interest over 30 years where repayments are made monthly and fortnightly. By making fortnightly repayments, the loan is paid off 6 years earlier. Write two or three sentences describing this graph. Balance ($) Reducing balance loan (interest 6.7% p.a.) Fortnightly repayments Monthly repayments Years 92 NEW CENTURY MATHS GENERAL: HSC

15 !NNC Yr12 maths ch 03 Page 93 Wednesday, October 4, :42 AM USING TECHNOLOGY TO COMPARE HOME LOANS For this section you will need access to the Internet, a graphics calculator and a spreadsheet package. Using the Internet Some good sources of information on current loan rates, terms, conditions, fees and charges are the websites of banks and credit unions (page 83). Here are three more websites to investigate: (has a loan repayment calculator with graph) (general information on all lenders) (calculators and glossary of mortgage terms) Use a loan calculator to: estimate what repayments might be at a higher or lower interest rate estimate the impact of changing from monthly to fortnightly repayments change the term of the loan and see the effect on repayments. The financial mode of a graphics calculator The financial mode (or TVM time, value, money) of a graphics calculator can be used to simulate a reducing balance loan by using the compound interest formula. On a Casio graphics calculator, access TVM mode then Compound Interest. By entering five out of these six quantities, you can find the unknown quantity: n = number of periods (and hence payments) in the term of the loan I% = interest rate per annum PV = present value or loan amount borrowed PMT = payment for each period FV = future value or unpaid balance (this is zero when the loan is paid off) P/Y = number of periods per year Example 7 Use a graphics calculator to find the number of payments needed to pay off a loan of $ at 8% p.a. if monthly repayments of $1500 are made. Solution n = Leave n blank. I% = 8 PV = PMT = 1500 FV = 0 P/Y = 12 The payment is entered as a negative because this is being subtracted from the loan balance. The number of payments (n) is , which would be 330 payments of $1500 and one payment of $1020 to finish the loan. CREDIT AND LOANS 93

16 !NNC Yr12 maths ch 03 Page 94 Wednesday, October 4, :42 AM Constructing a spreadsheet The spreadsheet below shows the progress of a home loan of $ with an interest rate of 8% p.a. and a monthly repayment of $1500. The screen has been split to show the beginning and end of the loan. Construct your spreadsheet as follows. 1. Enter the information in rows 1 4 and row 7 (cells B1 to B4 may need to be formatted). 2. Enter the number 1 in cell A8 then the formulas as shown in this table into cells B8 to F8 and A9 and B9. Copy the formulas down in each column. A B C D E F 1 Amount borrowed (principal P) $200,000 2 Interest rate per annum 8.00% 3 Repayment per period ($) No. periods per year 12 7 Period no. Principal (P) Interest (I) Payment (R) P+I P+I-R 8 1 =B1 =B8*$B$2/$B$4 =$B$3 =B8+C8 =E8-D8 9 =A8+1 =F8 =B9*$B$2/$B$4 =$B$3 =B9+C9 =E9-D9 10 =A9+1 =F9 =B10*$B$2/$B$4 =$B$3 =B10+C10 =E10-D10 3. Your spreadsheet will look like this. Note: Including a payment (R) column means you can increase, decrease or miss a repayment at different times during the loan and see the effect on the term of the loan. 4. Use the Chart option to draw a line graph showing the progress of the loan. Amount owing Number of payments 5. When is the loan balance $ ? 6. What is the balance of the loan at the end of 10 years? 94 NEW CENTURY MATHS GENERAL: HSC

17 !NNC Yr12 maths ch 03 Page 95 Wednesday, October 4, :42 AM Example 8 Use your spreadsheet to find: (a) the term of a $ loan with an interest rate of 8% p.a. and a monthly repayment of $1500 (b) the final repayment required to reduce the balance to zero (c) how long it would take to repay the loan if the monthly repayment was doubled to $3000 per month (d) the monthly repayment needed to repay the loan in 10 years (e) the final repayment required to reduce the balance to zero in 10 years Solution (a) 331 monthly repayments are made, so the term of the loan is 331 months or 27 years and 7 months. (b) The final repayment required to reduce the balance to zero is $ (c) By changing cell B3 to $3000, it can be seen that the loan would be repaid in 89 months or 7 years and 5 months. (d) Split the screen so you can view the rows for periods 1 to 12 and 118 to 120. Keep trying different values in cell B3 until the period 120 is the last period for the loan. 10 years = 120 months The monthly repayment required to repay the loan in 10 years is $2430. (e) The final repayment required to reduce the balance to zero in 10 years is $ Exercise 3-05: Using technology to compare home loans Equipment: A graphics calculator and your home loan spreadsheet. Use a graphics calculator for questions 1 to Find the monthly repayment for each of the following loans. (a) a $ home loan over 30 years at 6.4% p.a. (b) a $ reducing balance loan over 9 years at 9.3% p.a. 2. Find the fortnightly repayment for each of these loans. (a) a $ reducing balance loan with an interest rate of 7.35% p.a. over 6 years (b) a $ home loan over 30 years at 8.35% p.a. interest 3. Find the principal that can be borrowed for: (a) 20 years at 7% p.a. if a monthly repayment of $5000 is made (b) 10 years at 6.5% p.a. if a weekly repayment of $260 is made 4. Find the interest rate per annum on a reducing balance loan of: (a) $ repaid at $1270 per month over 13 years (b) $ if fortnightly repayments of $740 are made over 15 years Use your home loan spreadsheet for questions 5 to Find the terms of these loans and the final repayments. (a) $ with an interest rate of 8% p.a. and a monthly repayment of $2400 (b) $ with an interest rate of 7.3% p.a. and a monthly repayment of $ Find the monthly repayment and final repayment needed to pay off each of these loans. (a) $ with an interest rate of 8% p.a. in 10 years (b) $ with an interest rate of 8.2% p.a. in 15 years CREDIT AND LOANS 95

18 !NNC Yr12 maths ch 03 Page 96 Wednesday, October 4, :42 AM Use the most appropriate technology for questions 7 to (a) What is the monthly repayment on a loan of $ at 7.5% p.a. interest over a term of 20 years? (b) What effect does missing one repayment have on the term of the loan? (c) What happens if the monthly repayment is reduced to: (i) $1500? (ii) $1000? (d) What is the least amount (to the nearest dollar) that must be repaid per month to have a reducing balance? 8. Consider a loan of $ over 5 years at 6% p.a. interest. (a) What is the monthly repayment? (b) What is the effect on the monthly repayment if: (i) the amount borrowed is halved? (ii) the amount borrowed is doubled? (iii) the term of the loan is halved? (iv) the term of the loan is doubled? (c) What is the effect on the term of the loan if: (i) the monthly repayment is increased by $50? (ii) the monthly repayment is doubled? (iii) fortnightly repayments are made? (iv) the 10th payment is missed? Study tips DEVELOPING AN EXAM TECHNIQUE Getting used to completing exams and knowing what to expect in them minimises exam anxiety. Getting into an exam routine saves time and worry and allows you to stay focused on answering questions and using your time wisely rather than panicking. Use the reading time to plan your exam. How much can you do here without actually writing? Browse through the exam paper to see the work that is ahead of you. Calculate the average amount of time you should spend on each question/section. Take note of the marks allocated per question/section. Try to stick to your schedule. Easier questions are usually the first ones. Do an easy question first to boost your confidence. It will also save time. Allocate more time to harder questions. Leave them if you get stuck and come back to them later. Attempt every question. It is better to do a bit of every question and get some marks than to leave questions entirely blank and receive no marks for them. Show all working. Even if you get the wrong answer, you will be awarded some marks for correct working. Draw diagrams if necessary. 96 NEW CENTURY MATHS GENERAL: HSC

19 !NNC Yr12 maths ch 03 Page 97 Wednesday, October 4, :42 AM CREDIT CARD PAYMENTS Many people prefer to shop without cash by using credit cards, sometimes referred to as plastic money. Credit cards act as short term loans and are a good way to purchase goods if managed properly. Unfortunately, credit cards encourage some people to spend beyond their means. Monthly statements are issued to consumers listing the purchases for the previous month. Fees, charges and interest are added. There are two main types of credit cards: no interest-free period and no annual fee an interest-free period and an annual fee. A period is only classed as interest-free if the account is paid in full before this period ends; otherwise, interest is charged from the date of purchase. Cash advances do not have an interest-free period and are charged interest from the date of the advance. The due date on a statement is the day when the interest-free period ends. Example 9 Manuel has a credit card with no interest-free period and an interest rate of 14% p.a. He makes the following purchases for the period 1 August to 31 August: 2 August Dinner set $ August Pair trousers $ August Haircut $ August Dinner $ August White shirt $32.00 (a) What is the total amount of his purchases? (b) Manuel pays his account in full on 3 September. How much does he pay? Solution (a) Total purchases = $ $ $ $ $32.00 = $ (b) Interest is charged on each purchase from the date of purchase until the date payment is received. For example, the dinner set is bought on 2 August and paid for on 3 September. Number of days = = 32 Interest rate per annum = 14% = Interest rate per day = Purchase amount No. of days interest Interest to 3 September ($) $65.50 $85.00 $24.00 $36.80 $ = = = = = Total interest = $ $1.67 Manuel s total payment = $ $1.67 = $ Don t round off interest until the end. CREDIT AND LOANS 97

20 !NNC Yr12 maths ch 03 Page 98 Wednesday, October 4, :42 AM Example 10 Consider this statement for a credit card account with an interest-free period of 55 days. Your Account Summary Mr B Spender Balance from previous statement $ Fast Lane Payment and other credits $ CR Decimal Point 2178 Purchases, cash advances $ Interest and other charges $24.65 Closing balance $ Statement period: 14 December 2003 to 13 January 2004 Account no: Credit limit: $2000 Available credit: $ Your Transaction Record Minimum payment required $80.00 Due date 7 February 2004 Date Reference Details Amount 15 Dec Payment thank you $ CR 15 Dec 2003 Interest $ Dec Elio s Restaurant $ Dec ATM withdrawal $ Dec Cash advance fee ATM $ Dec Sydney Opera House $ Jan 2004 Annual fee $ Jan Drew s Menswear $ Jan Shelley s Scissor Magic $58.40 Annual percentage rate Daily percentage rate 15.95% % (a) What is Mr Spender s credit limit? (b) For this period, what was the total amount of: (i) purchases? (ii) withdrawals (cash advances)? (iii) interest and other charges? (c) How much was paid by the cardholder and on what date? (d) On what dates does the interest-free period start and end? (e) What interest rate is charged (i) per annum (ii) per day if the account is not paid in full by the due date? (f) What interest will be charged if this account is paid in full on 6 February. Why? 98 NEW CENTURY MATHS GENERAL: HSC

21 !NNC Yr12 maths ch 03 Page 99 Wednesday, October 4, :42 AM Solution (a) $2000 (b) (i) $ $ $ $58.40 = $ (ii) $ (ATM withdrawal) (iii) $ $ $20.00 = $24.65 (or from account summary) (c) $500 on 15 December 2003 (d) Interest-free period starts on 14 December 2003 and ends on 7 February (e) (i) 15.95% (ii) % (f) Unlike the other items, cash advances (ATM withdrawals) do not have an interest-free period, so interest will be charged for the 47 days from 21 December to 6 February. Interest on cash advance = $ % 47 = $ $2.05 Just for the record FIRST CREDIT CARD The first international credit card was the Diners Club card in The American inventor of the Diners Club card, Frank McNamara, came up with the idea when he was entertaining friends at a restaurant and was short of cash. However, it was not until 1974 that Australia had its first credit card: Bankcard. Exercise 3-06: Credit card payments 1. Ian has a credit card with no interest-free period and an interest rate of 14% p.a. He makes the following purchases for the period 1 October to 31 October: 4 October Swim trunks $ October Fins and mask $ October Swimming lessons $ October Wetsuit $ October Towel $32.00 (a) What is the total amount of his purchases? (b) Ian pays his account in full on 5 November. How much does he pay? 2. Mrs Hoggett has a credit card with no interest-free period and an interest rate of 15% p.a. She makes the following purchases for the period 1 May to 31 May: 2 May Dress $ May Shoes $ May Hat $ May Pig $ May Hay $ May Mower $ (a) What is the total amount of her purchases? (b) Mrs Hoggett pays her account on 10 June. How much does she pay in interest for her May purchases? CREDIT AND LOANS 99

22 !NNC Yr12 maths ch 03 Page 100 Wednesday, October 4, :42 AM 3. Andre has a credit card with an interest-free period of 40 days and an interest rate of 18% p.a. He makes the following purchases for the period 1 November to 30 November: 4 November Tennis racquet $ November Shoes $ November Clothes $ November Tennis lessons $ November Haircut $26.00 (a) What is the total amount of his purchases? (b) Andre pays his account in full on 24 December. How much interest does he pay? 4. Tiger has a credit card with an interest-free period of 55 days and an interest rate of 17% p.a. He makes the following purchases for the period 1 February to 28 February: 2 February Golf club $ February Golf shoes $ February Green fees $ February Golf lessons $ February Tees $6.00 (a) What is the total amount of his purchases? (b) Tiger does not pay the due amount until 1 May. How much does he pay altogether? 5. Consider the monthly statement for Mr Spender in Example 10 (page 98). (a) Draw up his next monthly account for the period 14 January to 13 February, given that: he pays his last account in full on the due date he makes the following purchases: 18 January Elio Restaurant $ February Big M Groceries $ February Robert s Roses $55.00 he withdraws $100 from an ATM on 28 January (incurring a fee of $1). (b) How much interest is due on 13 February? (c) The minimum payment (5% of account or $5, whichever is greater) is due 55 days from the first day of the account period. How much is the minimum payment (to the nearest dollar) and when is it due? Investigation: Plastic credit 1. Collect monthly statements for as many credit, charge, store and debit cards as you can and use these to determine the advantages and disadvantages of each type. Look at interest rates, terms and conditions as well as fees and charges. Also consider any rewards or benefits to the consumer. 2. Collect newspaper articles on the use and misuse of credit in the community. 100 NEW CENTURY MATHS GENERAL: HSC

23 !NNC Yr12 maths ch 03 Page 101 Wednesday, October 4, :42 AM Chapter review Credit and loans 1. Flat rate loans 2. Buying on terms 3. Reducing balance loans 4. Using published loan repayment tables 5. Using technology to compare home loans 6. Credit card payments Topic summary In the Preliminary Course, you learnt about saving and investing money. This chapter, Credit and loans, examined the financial mathematics of using credit and borrowing money. You should be competent in making interest calculations involving flat rate and reducing balance loans as well as credit cards. You have constructed spreadsheets, used the financial function on a graphics calculator and investigated loans, credit cards and associated fees and charges. You should be able to make an informed decision on different loan and credit options. Make a summary of this topic. Use the chapter outline above as a guide. An incomplete mind map has also been started below. Use your own words, symbols, diagrams, boxes and reminders. Use the questions in Your say below to think about your understanding of the topic. Gain a whole picture view of the topic and identify any weak areas. Spreadsheets Loans reducing balance flat rate Credit and loans Who has the best loan? Loan repayment tables Credit cards Buying on terms Your say: Reflecting about the topic Have you satisfied the outcomes listed at the front of this chapter? What was the most important thing that you learned? How did you feel about the topic? Did you enjoy it? What was new? What are your weaknesses? What will you need to study more? How will you revise and summarise this topic? CREDIT AND LOANS 101

24 !NNC Yr12 maths ch 03 Page 102 Wednesday, October 4, :42 AM Chapter assignment 1. Jack borrows $3000 to buy a second-hand car on a flat rate loan at 12% p.a. interest over 4 years. (a) How much will he repay altogether? (b) What is his monthly repayment? 2. Huynh bought a motorcycle for $ The flat rate loan was at 15% p.a. interest over 4 years, to be repaid in equal monthly instalments. If there were additional charges of $100 delivery, $35 stamp duty, $250 registration and third-party insurance, and $80 per year loan insurance, find: (a) the total amount borrowed (b) the amount of interest charged (c) the monthly instalment 3. Juliet bought a new bedroom suite from Romeo s Discounts. The cash price was $999 but Juliet chose to buy on terms and paid 10% deposit and $20 per week for 15 months. (a) How much did she actually pay for the bedroom suite? (b) What was the flat rate of interest charged per annum? 4. Liz and Phil retired and wanted to go on a QE2 cruise. The price of the cruise was $ each. They paid a $6000 deposit each and paid off the remainder in 24 monthly instalments of $1120. (a) How much did they pay altogether for the cruise? (b) How much interest did they pay? (c) What was the flat rate of interest charged per annum? 5. Xi borrowed $ for a new bathroom at 10% p.a. reducible interest. He made monthly payments of $120. (a) Draw up a table showing the progress of the loan in the first 5 months. (b) How much had Xi paid off the principal after his 5th payment? (c) How much interest did he pay in the first 5 months? (d) How much interest did he save in the first 5 months compared with a flat interest rate of 10% p.a.? 6. Tina has a credit card with an interest-free period of 45 days and an interest rate of 16% p.a. She makes the following purchases for the period 1 November to 30 November: 7 November Clothes $ November Shoes $ November Clothes $ November Singing lessons $ November Makeup $ November Beauty treatment $ (a) What is the total amount of her purchases? (b) Tina pays her account in full on 15 December. How much interest does she pay? (c) How much interest would she pay if she settled her account on 31 December? 102 NEW CENTURY MATHS GENERAL: HSC

25 !NNC Yr12 maths ch 03 Page 103 Wednesday, October 4, :42 AM 7. Monthly loan repayments 25-year term Principal ($) Interest rate (% p.a.) Use the loan repayment table over a 25-year term to find the following. (a) the monthly repayment for a loan of $ at 7% p.a. (b) the amount borrowed at 8.5% p.a. if the monthly repayment is $725 (c) the interest rate per annum for a loan of $ if the monthly repayment is $671 (d) the monthly repayment for a loan of $ if the interest rate is 7% p.a. (e) the amount borrowed at 7.5% p.a. interest if the monthly repayment is $665 (f) the interest rate per annum for a principal of $ if the monthly repayment is $ Monthly loan repayments (interest 14% p.a.) Amount of loan ($) Period of loan (months) The table above gives the monthly repayments on a reducing balance loan with an interest rate of 14% p.a. Calculate for each loan below: (i) the monthly repayment (ii) the total amount repaid (iii) the total interest paid (iv) the interest charged if the loan was at a flat interest rate of 14% p.a. (a) $ over 36 months (b) $ over 18 months (c) $ over 5 years (d) $ over 2 years CREDIT AND LOANS 103

26 !NNC Yr12 maths ch 03 Page 104 Wednesday, October 4, :42 AM 9. Edita has a gross income of $ p.a. and wants to purchase a studio apartment. The bank will allow her to repay up to 20% of her gross income per annum. The current loan rate is 6.84% p.a. and fees and charges amount to $4000. She wants to borrow the maximum possible and make monthly repayments. (a) How much can she repay: (i) per annum? (ii) per month? (b) Would the bank lend her $ over 10 years? Justify your answer. (c) If no, what amount will the bank lend Edita over 10 years? 10. Reducing balance loan monthly repayments Amount owing ($) No. of repayments The graph shows the progress of a reducing balance loan of $ with monthly repayments of $3000. From the graph, estimate: (a) the number of monthly repayments to pay off the loan (b) the balance owing on the loan after 10 years (c) the time taken to reduce the balance to $ The graph compares the progress of a $ loan when repayments are made monthly and fortnightly. (a) What are two benefits of paying Reducing balance loan fortnightly instead of monthly? (b) Estimate the amount owing on the loan after 10 years if repayments are made: (i) monthly (ii) fortnightly Monthly repayments (c) Estimate the number of years it Fortnightly repayments takes to reduce the balance to $ if repayments are made: (i) monthly 0 (ii) fortnightly Loan term (years) Loan amount ($) 104 NEW CENTURY MATHS GENERAL: HSC