E1
Interest calculations
Aims and overview
In this session students will apply algebraic and numerical skills in transforming formulae and solving equations, including exponential equations, in the context of simple and compound interest calculations.
Key concepts
– simple interest (flat or fixed rate), compound interest (reducing balance or effective rate), principal, amount, rate of interest, time interval, percentage, decimal equivalents, percentage of an amount, exponent (or index, logarithm, or power), base, nth root, laws of exponents.
Preparation
Students will need scientific calculators for use in Question 5.
Lesson
A: Introduction – Provide a mini-lesson on the variables and formulas for simple and compound interest calculations. In particular, review percentage equivalents, for example r = 5% = 0.05 for simple interest calculations and R = 1 + r = 105% = 1.05 for use in compound interest calculations. Also indicate how scientific calculators can be used to write a number as a power of 10. B: Work period – Monitor student work in Question 1, in particular, to see which students use step by step calculations rather than formulas. Promote discussion of their answers to part d. Encourage ‘same to both sides’ justification of each step in the algebraic work in Questions 2 to 4. – Make note of all the different ways in which scientific or simpler calculators can be used to do the numerical calculations of Questions 4c and 5. Support students in writing out the steps of the algebraic transformations involved in Questions 2, 3 and 4. C: Extension/revision – Discuss the alternative methods for finding solutions and their justifications in terms of the laws of exponents. Obtain agreement about the eleven formulas involved in the lesson. Ask students to provide the detail of the algebraic steps involved in obtaining one or more of these formulas.
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E1
Interest calculations
There are two main types of interest; simple interest (or flat or fixed rate interest) and compound interest (or reducing balance or effective rate interest). Simple interest is no longer used for consumer loans. Under Victorian legislation all consumer loans (housing, personal and credit card) must make available the compound interest rate, for which interest is paid on the amount still owing, not on the original amount borrowed. It is important to understand the two different types of interest, and why simple interest is less attractive for the borrower. The basic formulae involved are: Simple Interest I = P , where: rt I is the amount of interest P is the Principal, the amount borrowed r is the annual rate of interest, eg. 5% = 0.05 t is the time in years Compound Interest
I = P (1 + r ) n − P = P ( + r ) − 1 where: 1
I is the amount of interest P is the Principal, the amount borrowed r is the rate at which interest is calculated n is the number of time intervals for which interest is calculated
[
n
]
Exercises - Flat or reducing? 1. Sophie borrows $800 from her parents to help with the start-up costs (bond and furniture) for renting the house. She agrees to repay the loan plus interest charged at 6% per annum. Four years later she has enough saved to pay out the loan. a) Calculate the amount of interest if it is calculated as simple interest. b) Calculate the amount of interest if it is calculated as compound interest calculated at the end of each year. c) Calculate the amount of interest if it is calculated as compound interest calculated at 3% every six months rather than at 6% at the end of each year. 2. Do the algebra required to change the subject of the simple interest formula. a) Make P the subject of the formula. Make it P = etc. b) Make rthe subject of the formula. Make it r = etc. c) Make tthe subject of the formula. Make it t= etc. d) Tye borrows $800 from his parents. Five years later they put the pressure on him to repay an amount of $1000. Choose the appropriate formula to work out what this means as a simple interest calculation. 3. Some other formulae often used are ) – for simple interest A = P(1 + rt n – for compound interest, A = P (1 + r ) n – for compound interest, A = PR Compare these formulae with the formulas for the amount of interest and explain what is represented by the A and the R.
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continued
E1
Interest Calculations
4. Do the algebra required to change the subject of the formula A = PR n a) Make P the subject of the formula b) Make R the subject of the formula. c) Choose the appropriate formula to work out the annual rate of compound interest that Tye’s parents could have been charging (Details in Question 2d previously.) 5. To make n the subject of the formula you will have to use the laws of indices (also called logs). a) Explain why 1202.9 = 800 x 1 0 n can be written as 103.0802 = 102.9031 x 100.02531 x n .6 b) Solve the equation of part a. and interpret the question and answer as a problem involving compound interest. n c) Start with A = PR n and explain how this leads to 10 log A = 10 log P x 10 log R
(
)
and n =
log A − log P log R
d) What annual rate of compound interest would have been involved if Tye’s parents had asked him to pay them $1200 after four years?
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E1
Interest Calculations
Extension/revision activities Work with your partner to prepare your answers and explanations to the following questions. 1. In Question 1 ‘Flat or reducing?’ on Worksheet E1, you should have obtained three different answers. Explain why. 2. Choose one of the eleven formulae used in the lesson and make up a question that it could be used for. Give the question to your partner to solve. 3. Use mental estimation to give approximate answers to the following questions. a) What is the simple interest charged at 5% for 4 years on a principal of $1100? b) What is the compound interest charged at 5% for 4 years on a principal of $1100? (Assume that interest is calculated on an annual basis.) c) How many years would it take for the interest on a 5% per annum simple interest loan to be equal to the original principal? d) How many years would it take to repay the amount on a 5% per annum compound interest loan with double the original principal? To make your estimate you could use the ‘rule of 70’ that says that, for doubling the value of a compound interest investment, the annual rate times the number of years is approximately equal to 70, r x t ≈ 70 4. Now use a calculator to do the calculations for the problems in Question 3. 5. How close were your mental estimates to the correct values?
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E2
Buying a car
Aims and overview
In this session students will apply algebraic and numerical skills in transforming formulas, manipulating spreadsheets and solving equations, by studying loans with regular equal repayments.
Key concepts
– percentage rate, flat rate, effective rate, instalment, representing functions and solving equations, including solutions by numerical approximation.
Preparation
– All students will need access to technology, preferably spreadsheets. If graphic calculators are used instead of spreadsheets then adapt the lesson accordingly. – Prepare a spreadsheet or graphic calculator with projection capabilities to demonstrate the flexibility of loan calculations based on stored values of principal, rate and the instalment amount that is regularly repaid. Set initial amounts to those required for Question 1. – Assess the level of technology skills when grouping students for this lesson.
Lesson
A: Introduction – Discuss spreadsheet formulas and introduce the technology presentation by asking students to interpret the calculation. As the time interval is not specified, students can determine which time interval is the most realistic for 2% interest calculated on the same frequency as $185 instalments. Remind students of the relationships between flat and effective interest rates and that their spreadsheets or graphic calculator listings developed in this lesson need to be retained for the next lesson. B: Work period – Assist students in setting up the spreadsheet and/or graphic calculator lists. – Encourage students to use approximations and mental calculations as means of checking whether calculated answers are reasonable. Show how flat rate interest calculations can give approximate answers to effective rate calculations and that the calculations involved are much simpler. Students can access the Consumer Affairs Victoria website at www.consumer.vic.gov.au for further research. C: Extension/revision – Start the session with one student in charge of the spreadsheet, a second student in charge of a simple calculator and a third student armed with ‘pen and paper’ or board space and marker. – Complete the quantitative questions in the order given, seeking advice from the class at each stage on whether the spreadsheet or the calculator should be used and whether the approximate answer can be predicted through ‘pen and paper’ approximation and mental calculation.
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E2
Buying a car
One of the first major purchases you will make will be your first car. Do your homework to compare prices, interest rates and credit providers. If you do not understand the terms included in the contract of sale or the financial agreement DO NOT sign it until you have advice from a family member or Consumer Affairs Victoria. Exercises - The hidden costs 1. Nik takes out a bank loan of $2500 to pay the deposit for buying a second-hand car. Quarterly repayments of $400 must be made until the loan is completely paid off. Interest is charged at 8% per annum, calculated on the amount owing at the end of each quarter. a) Explain why the appropriate value of R is 1.02, rather than 1.08. b) Use a calculator to check the first four rows of calculations in the following spreadsheet. A 1 2 3 4 5 6 7 8 9 A 1 2 3 4 5 6 7 Principal, P = Rate, R = Repayment, Q = Time 1 = B1 = D7 = D8 Principal, P = Rate, R = Repayment, Q = Time 1 2 3 B $2,512.00 1.02 $185.00 Amount owing at the start Plus interest = B7 * B$2 = B8 * B$2 = B9 * B$2 Minus repayment = C7 - B$3 = C8 - B$3 = C9 - B$3 B $2,512.00 1.02 $185.00 Amount owing at the start $2,512.00 $2,377.24 $2,239.78 C Plus interest $2,562.24 $2,424.78 $2,284.58 D Minus repayment $2,377.24 $2,239.78 $2,099.58 C D
8 = A7 + 1 9 = A8 + 1
c) Use your calculator or a spreadsheet to find how many repayments have to be made for the loan to be completely repaid. Spreadsheet hints: Do not type the dollar signs and commas in cells B1 and B3. The dollar sign in cell D7 enables you to use Edit to Fill down the rest of column D without B3 changing to B4, B5 etc.
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E2
Buying a car
d) Calculate the total amount of interest paid and do the calculations to show that these terms are equivalent to a flat rate of interest of approximately 4.46% per annum. e) Change the numbers at the top of your spreadsheet until you find the effective rate of interest for a loan of $2500 that is repaid in four equal instalments of $755 over a period of four years. f) Do the calculations to show that the flat rate of interest for the same loan is 5.2%. 2. Approximate formulas often used for matching flat rates (F) to effective rates (E) are
E=
2nF and F = E ( n + 1) , where n is the number of instalments paid. n +1 2n
Fill in the following table to test how accurate these formulas are in comparison with the results from your spreadsheet calculations. Actual E For Questions 1c, 1d 8% For Questions 1e, 1f Actual F 4.46% 5.20% E by formula 8% 5.20% F by formula
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continued
E2
Buying a car
Extension/revision activities Work with your partner to prepare your answers and explanations to the following questions. 1. A slightly simpler approximate formula is that the effective rate of interest is approximately twice the flat rate of interest. a) Under what circumstances does this formula give a reasonably accurate answer for someone who wants to buy a car on terms? b) Under what circumstances does this formula give an answer that is accurate enough for a seller who must quote both flat and effective rates of interest? 2. Nik wants to buy the car on terms, but his current budget only allows for monthly repayments of $250. The second hand car dealer has a policy that terms should be at 8% flat rate and should never extend beyond four years. Part of the sales-spiel is that 8% is about the same rate as that currently quoted by the major banks. a) What size loans are within Nik’s budget? b) Nik believes the bank might be prepared to extend the personal loan up to $4000? What financial advice would you give him? c) What other legal and technical details should Nik check before he signs any deal? 3. The following extract was taken from the Consumer Affairs Victoria website. If you buy goods, services or land now but agree to pay later and are charged extra for this, you are being provided with credit. From 1 July 2003, banks and other credit providers are required to give consumers a better idea of how much they’re really paying for their credit. The new rate, called a comparison rate, includes both the interest rate and the fees and charges, reduced to a single percentage figure. How could knowing about this requirement have assisted Nik in deciding which deal to accept? 4. Sophie has been offered $2000 for her old car in a private sale and $2500 as a trade in on a new car with a retail price of $20,500, including all on-road costs. a) Terms for paying for the new car would be 36 monthly instalments of $620. How much would she end up paying if the trade in value was used as deposit? b) Alternatively she could sell the old car privately for $2000 and pay the additional $18,500 using a bank loan guaranteed by her sister Nina. The bank would charge 9% per annum interest, calculated monthly, and she could repay $620 per month until the loan was paid off. Would this be a better deal? Include calculations using either: – the approximate formulas, or – a spreadsheet, or – the Savings calculator on the web site www.choice.com.au
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E3
Home loan
Aims and overview
This lesson involves the application of algebraic and numerical skills in transforming formulas, manipulating spreadsheets and solving equations, using a study of long term loan calculations.
Key concepts
The same concepts apply as those for E2 Buying a car, with the addition of the relationship between loans (for borrowing) and annuities (for investing).
Preparation
Students will need access to the spreadsheet formulated in the previous lesson, the website www.choice.com.au and computer projection facilities. A small library of Year 12 Further Mathematics or other textbooks that contain the annuities formula will also be useful.
Lesson
A: Introduction to the lesson Introduce the problem and discuss the method of approximation for finding the monthly repayment. The major points are: – It is much easier to work out monthly repayments for flat rate interest than for effective rate interest. – What flat rate of interest would involve approximately the same monthly payment as for a 6% effective rate for a long term loan? (Answer – a flat rate of 3% approximately). – Organize the students into groups of three as they must explain to each other the three different methods used in Question 2. B: Work period – Monitor group work on Question 1 to ensure their understanding of the answer before they do the tasks required in Question 2. – Provide hints for Question 2 leaving students to do the investigations. They will soon find that the answer, which needs to be justified, is $1159.74 – Remind students that they will need to explain their method to other group members. C: Extension/revision Have a student explain the detail of the solution to Question 1. Further questions to ask are: – Is this an underestimate or an overestimate of the more accurate answer? Explain why. – Would using V of 6% = 3% have been just as good? (as an approximation, yes.) – Seek explanations of the Question 2 solution by the annuities formula and then, using the computer, obtain student explanations of solutions by the other two methods. – Share answers to the extension questions.
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E3
Home loan
Sophie’s older sister Nina is thinking of buying a unit with a purchase price of $200 000. – Stamp duty and legal fees add an extra 10% to the price. A deposit of 10% of the purchase price would need to be paid immediately – and the rest within three months. – Nina thinks that she has saved enough to pay the immediate costs of the deposit, the Stamp Duty and the legal fees. – Nina is fairly sure that she could get a 25 year bank loan with interest calculated monthly at an initial rate of 6% per annum. Exercise - Paying the mortgage Nina needs to know what the monthly repayment amount would be. Calculate it for her, using any of the following: a) an approximate method, based on the formula F =
2nE n +1
b) the spreadsheet developed in the Buying a Car lesson c) the Savings Calculator on the ‘Money and Rights’ page at www.choice.com.au d) an annuities formula. How close to the correct value is the answer obtained by the approximate method?
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E3
Home loan
Extension/revision activities in groups of three prepare your answers and explanations to the following questions. 1. What advice or assistance could Sophie offer Nina if she decides that the monthly repayment would be beyond her present budget? 2. Under what circumstances does the formula F = give a reasonably n +1 accurate answer for someone who wants to buy a car on terms? 3. Some adults who can afford to purchase a home, still choose to rent rather than buy. Discuss with an adult the financial aspects of this choice. 4. Using the Venn diagram provided below, compare all aspects of renting versus purchasing a home. Some aspects may be similar so these can be listed in the area where the circles overlap. Renting Similarities Purchasing
2nE
5. Write a short report on your findings from questions 3 and 4 above.
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