Financial mathematics Recall 1

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1 Worksheet Worksheet R. Worksheet R. R.3 Worksheet Worksheet R.4 R.5 Financial mathematics Recall Prepare for this chapter by attempting the following questions. If you have difficulty with a question, go to Pearson Places and download the Recall Worksheet from Pearson Reader. Convert the following percentages to decimals. (a) 7% (b) 6% (c) 9.% (d) 4.8% (e).0% (f) -- % 4 Calculate: (a) 5% of $0 000 (b) 6% of $0 000 (c) 9.5% of $ (d) 7 -- % of $ (a) Increase 40 by: (i) 00% (ii) 50% (iii) 5% (iv) 00% (b) Decrease 60 by: 4 Evaluate: (i) 50% (ii) 30% (iii) 0% (iv) 75% (a) 7 (b) 3 (c) 0 6 (d) (0.6) (e) -- (f) (.) Convert the following to a decimal number of years. Assume a year has 5 weeks or 365 days. (a) 6 months (b) 3 months (c) 8 years and 9 months (d) 6 weeks (e) 3 weeks (f) 46 days After completing this chapter you will be able to: use and extend the simple interest formula to develop the compound interest formula use the compound interest formula to solve problems involving different rates and different compounding periods use recursive techniques compare effective and nominal interest rates and solve related problems adapt the interest formulas to solve problems of appreciation, depreciation, and growth and decay. PEARSON mathematics 0 0A essentials edition

2 . Interest. Need to Know Simple interest formula: I = PRT where I = amount of simple interest P = principal R = rate of interest per annum (p.a.) as a decimal T = time in years Also: A = P + I where A = total amount repaid on the loan or collected from the investment P = principal I = simple interest Compound interest formula (compounded annually): A = P( + r) n where A = amount of the loan or investment after n years P = principal r = rate of interest (p.a.) as a decimal n = time in years Also: I = A P where I = compound interest owed or earned after a certain number of years A = amount of the loan or investment after the same number of years P = principal Increase in cost or value is called appreciation. The rate of increase in prices within the economy is called the inflation rate. The simple interest formula can be used to calculate a flat rate of appreciation. Worked Example $0 000 is borrowed for 5 years at an interest rate of 9.5% p.a. If the interest is calculated as simple interest, find: (a) the amount of interest charged over the period of the loan (b) the total amount to be repaid at the end of the loan. (a) I = PRT P = 0 000, R = 9.5% = 0.095, T = 5 I = = $9500 (b) A = P + I P = 0 000, I = 9500 A = = $9 500 Financial mathematics 3

3 . Worked Example Find the information that is missing in each of the following simple interest situations. (a) $00 interest is paid on an investment of $5000 after 6 months. (b) $350 interest is paid on an investment at 4% p.a. after 9 months. (c) $50 interest is paid on an investment of $3000 invested at 5% p.a. (a) I = $00 P = $5000 R =? 6 T = ---- = 0.5 I = PRT 00 = 5000 R R = = 0.08 The rate is 8% p.a. (b) I = $350 P =? R = T = ---- = 0.75 I = PRT 350 = P P = = The principal is $ (c) I = $50 P = $3000 R = 0.05 T =? I = PRT 50 = T 50 T = = The time is 8 -- years, 3 which is 8 years and 4 months. Worked Example 3 3 $ 500 is borrowed for 7 years at an interest rate of 5% p.a. If the interest is compounded annually, find: (a) the total value of the loan after 7 years (b) the amount of interest charged over the period of the loan. (a) A = P( + r) n P = 500, r = 5% = 0.5, n = 7 A = 500( + 0.5) 7 = = $ (rounded to the nearest cent) (b) I = A P A = , P = 500 I = = $ PEARSON mathematics 0 0A essentials edition

4 Answers. Interest page 48. If necessary, round the answers to the following questions to the nearest cent. Fluency For each of the following loans the interest is calculated as simple interest. Find: (i) the amount of interest charged over the period of the loan (ii) the total amount to be repaid at the end of the loan. (a) $ borrowed for year at 8% p.a. (b) $ borrowed for year at 9% p.a. (c) $ borrowed for 5 years at 7% p.a. (d) $ borrowed for years at 5% p.a. (e) $ borrowed for 5 years at 7.5% p.a. (f) $ borrowed for 30 years at 6.5% p.a. (g) $ borrowed for. years at 6.75% p.a. (h) $ borrowed for. years at 7.5% p.a. (i) (j) $3 000 borrowed for 5 years at 9 -- % p.a. $5 800 borrowed for 4 years at 0 -- % p.a. (k) $ borrowed for years at 5% p.a. (l) $ borrowed for 5 -- years at 7% p.a. (m) $0 000 borrowed for years and 3 months at 5% p.a. (n) $6 000 borrowed for 3 years and 9 months at 7% p.a. Find the information that is missing in each of the following simple interest situations. (a) $400 interest is paid on an investment at 0% p.a. after 3 years. (b) $700 interest is paid on an investment at 5% p.a. after 6 years. (c) $5 00 interest is paid on an investment of $0 000 after 9 years. (d) $ interest is paid on an investment of $ after years. (e) $ interest is paid on an investment of $ invested at 6.5% p.a. (f) $ interest is paid on an investment of $ invested at 7.5% p.a. 3 For each of the loans in Question the interest is now compounded annually. Find: (i) the total amount to be repaid at the end of the loan (ii) the amount of interest charged over the period of the loan. 4 Lily invested $0 000 for 5 years at an interest rate of 5.5% p.a. Jarrah invested $ for 7 years at an interest rate of 9.5% p.a. (a) Calculate the interest earned on each investment assuming: (i) annual compound interest (ii) simple interest. (b) Compare the compounded interest with the simple interest. 5 Calculate the total amount owing on a loan of $7000 after years, if the 6% interest p.a. is: (a) compounded annually (b) calculated as simple interest. 3 Financial mathematics 5

5 . 6 Hint Use A = P + I to find the simple interest first. Then use I = PRT to find R. Hint In some cases you may need to use guess and check to solve the equation. Hint Either choose a number for the amount Ahmed has to invest or let P be this amount. 6 $445 at 6.5% p.a. compound interest, compounded annually over 8 years, will amount to: A $445( ) B $445( ) 8 C $445( ) 8 D $445( ) 8 Understanding 7 At what simple percentage rate will an initial investment of $7500 grow to a total investment of $8400 in 0 years? 8 If the simple interest on an investment over 8 years at 4% is $93, what was the amount of the original investment? 9 Complete the following table assuming an annual compounding period. 0 For each of the following appreciation questions: PEARSON mathematics 0 0A essentials edition (i) state whether the problem should be modelled by the simple interest formula or by the compound interest formula (ii) solve the problem, rounding answers to the nearest whole number or dollar. (a) Martina currently gets $0 pocket money a week. She has been promised a $4 increase each year. What is the promised rate of increase? How much more pocket money will she get in 3 years than she is getting now? (b) An antique clock appreciates in value by % p.a. If it was valued 0 years ago at $350 how much is it worth now? (c) Yang paid $0 000 for a new motorbike. If the inflation rate is 6.5% p.a., what is the estimated cost of a similar new motorbike in 4 years time? (d) Rhea has been awarded an annual salary increase rate of 7% p.a. Her current salary is $4 000 p.a. Calculate how much she can expect to earn in 3 years time if the same rate continues. (e) Ethel was given a $500 ring 30 years ago. Each year it appreciated at a flat rate of 5%. What is it worth now? (f) In 993, a house was purchased for $ It was sold in 00 for $ What was the yearly average rate of appreciation? Costs in a business are growing at an annually compounding rate of 8% p.a. Currently they run at $780 per week but 7 years ago they were: $780 A B $ C $ D.08 7 Reasoning Amount (A) Principal (P) Interest rate p.a. (R) Time in years (T) $ % 3 Ahmed has some money to invest. He wants to at least quadruple his money. (a) How long will it take if he invests at a simple interest rate of 3% p.a.? (b) How long will it take if he invests at an annually compounded interest rate of 3%? Give your answers in whole years. $4 600.% 5 $ $ % $5.7 3% 0 $60.50 $50 $

6 AA_Pearson Maths 0 EE-0.fm Page 7 Friday, June 5, 0 4:38 PM. 3 A certain sum of money $P is deposited into a bank account at an annually compounding interest rate, R. After years it has grown to $48 95 and after 3 years it has grown to $ What is the interest rate? Open-ended 4 Find values for the amount of an investment and the annual interest rate, given that the investment compounds annually for years, to give a final value within $00 of $ Hint Write two equations with P and R in them. Then use these two equations to write in terms of P and R. 5 Calum s car was written off as a result of hail damage in a sudden and severe local storm. Problem Problem oblem solving The oldest cake problem pa What percentage of the cake has been promised to ge s His interstate cousin Jessica also had to replace her car after flood damage. Both needed to borrow the same amount of money. Calum was able to get a flat rate loan at 0% p.a. for 8 years. Jessica will end up paying the same amount of interest as Calum but her loan has a simple interest rate between 0% and 0% p.a. and will need to be repaid over a shorter period of time. Find possible values for the interest rate and period of Jessica s loan. Strategy options Elysia by her st birthday? How old will Elysia be when she finally gets Aunty Bee s present? Draw a diagram. Sa m pl e Solve a simpler problem. Financial mathematics 7

7 .. Compound interest the general formula Need to Know Answers page 48 4 Compound interest formula (general case): A = P( + r) n where A = amount owing or earned at the end of the complete process P = principal (original amount borrowed or loaned) r = interest rate for the period, expressed as a decimal n = number of compounding periods Worked Example 4 4 Clive borrowed $ 000 to buy a lathe for his factory. He agreed to pay interest at a rate of 5% p.a. and to repay the loan in full in 3 years. Calculate the amount to be repaid if the interest is compounded: (a) annually (b) half-yearly (c) quarterly. (a) A = P( + r) n. P = 000 r = 0.5 n = 3 = 3 A = 000 (.5) 3 = $ Fluency (b) A = P( + r) n P = r = = n = 3 = 6 A = 000 (.075) 6 = $ (c) A = P( + r) n Compound interest the general formula Calculate the amount to be repaid after years on a loan of $6 000 if the 4% interest p.a. is compounded: (a) annually (b) half-yearly (c) quarterly. P = r = = n = 3 4 = A = 000 (.0375) = $ PEARSON mathematics 0 0A essentials edition

8 . Calculate how much an investor would repay on a debt of $8000 after -- years if the % interest p.a. is compounded: (a) half-yearly (b) quarterly, i.e. every 3 months. 3 If a loan of $5800 is made at 6% p.a. compounded every half year, over 6 years the debt would grow to: A $5800( + 0.6) 6 B $5800( ) C $5800( + 0.6) D $5800( + 0.8) 4 If the loan in Question 3 were compounded quarterly, the interest accrued would be: A $5800( ) 6 B $5800( + 0.6) 6 $5800 C $5800( ) 4 $5800 D $5800( + 0.4) 4 $5800 Understanding 5 Calculate how much interest (to the nearest cent) is added over 0 years to an account paying 9% p.a. interest on an initial sum of $ if the interest is compounded: (a) half-yearly (b) quarterly (c) monthly. 6 How much more will an investor get on an investment of $3 000 over 4 years in an account offering 8.8% p.a. if the interest is compounded weekly rather than annually? (Assume 5 weeks in a year.) 7 Calculate the interest in Question 6 if it were to be compounded daily. 8 Martina won $5000 in the lottery and decided to invest it with a view to taking a trip to Paris sometime in the future. Martina was given two options when she invested the money. She could have an interest rate of 4.5% p.a. compounded quarterly or she could have 4.4% p.a. compounded daily. Which is the better deal, and by how much, if Martina invests the money for years? Reasoning 9 (a) $0 000 is to be invested for 3 years. Calculate the interest earned on each of the following to find which would give the best return. (i) 5% p.a. compounded quarterly (ii) 4.5% p.a. compounded every months (iii) 4% p.a. compounded monthly (b) Do you think the size of the investment would alter which option in part (a) would give the best return? (c) Prove your answer to part (b) algebraically by redoing part (a) with $P in place of $ Open-ended 0 The more frequent the compounding, the greater the interest. Explain what this statement means. Use a principal of $0 000 invested at 8% p.a. for 5 years and three different compounding periods (such as weekly, quarterly or monthly) to help you demonstrate your answer. Express the percentage increase correct to one decimal place. Tyson is having trouble understanding the difference between simple and compound interest. He can t see why compound interest is higher than simple interest after the first compounding period if the principal, interest rate and overall time period are the same. How would you explain the difference to Tyson? Use an example of $4000 invested for 3 years at 5% p.a. to help you explain your answer. Financial mathematics 9

9 .3.3 Compound interest further applications Worked Example 5 5 Aunt Eva leaves Heidi a legacy of some deposit stock that was invested for 0 years at 5.% p.a. compounded daily. The value of the cheque received was $ Calculate the initial deposit. Give your answer rounded to the nearest dollar. (Ignore the occurrence of leap years in the 0-year period.) A = P( + r) n A = r = n = = = P P = P = $ So, the principal was $ Worked Example 6 6 In the first year of his business Jack had takings of $ They grew to $ after years, increasing at the same rate every year. At what annual percentage rate has his business been growing? Round your answer to two decimal places. A = P( + r) n A = , P = 8 000, n = = = 8 000( + r) = ( + r) = ( + r) r = r = r = 7.% p.a. The business has been growing at a rate of 7.% p.a. 0 PEARSON mathematics 0 0A essentials edition

10 .3 Worked Example 7 7 How many years will it take a $45 stamp to be worth $000 if it increases in value by 7.5% p.a.? Assume that interest is compounding annually. Give your answer to the nearest whole number of years. A = P( + r) n A = 000, P = 45, r = = 45( ) n 000 = 45(.075) n = (.075) n = (.075) n.3 Fluency If n = 4, (.075) 4 = If n = 0, (.075) 0 = If n =, (.075) = If n =, (.075) = It will take years for the stamp to increase its worth to $000. Compound interest further applications In each of the following, use the compound interest formula to find the principal. (a) Calculate how much would need to be invested at 8% p.a. compounded each half year to accumulate to $9600 in 6 years. Round your answer to the nearest dollar. (b) How much would Li have to deposit in order to receive $0 000 in 7 years if she places her money in an account that pays 8.8% interest, compounded quarterly? Round your answer to the nearest dollar. (c) How much would Sylvester have to deposit to receive $8000 in 5 years time if he places money in an account that pays 7.8% interest, compounded half yearly? Round your answer to the nearest cent. (d) A deposit accumulates to $4500 in 9 months at % p.a. compounded quarterly. The initial deposit, in dollars, was: 4500 A B C D / 4 In each of the following, use the compound interest formula to find the annual interest rate. (a) Sales of $6 900 grow to $ in years. Calculate the percentage growth p.a. rounded to one decimal place. Assume the growth compounds annually. (b) Costs of $600 increase to $8000 in 8 years. Calculate the percentage increase p.a. rounded to two decimal places. Assume annual compounding. (c) Calculate the rate of interest p.a. which would allow $7900 to accumulate to $9000 in 5 years if interest is compounded each half year. Round your answer to one decimal place. Answers page Financial mathematics

11 .3 (d) $800 accumulates to $4500 in 3 years. The percentage rate of interest if it is compounded annually is: A B C 3 In each of the following, use the compound interest formula and the method of guess and check (or a CAS) to find the number of years. In each case, assume that interest is compounding annually. (a) How many years will it take for a $400 sapphire ring to be worth $8000 if it increases in value by 0.6% p.a.? Round your answer to the nearest whole number of years. (b) How many years will it take for a $ block of land to be worth $ if it increases in value by 8.5% p.a.? Round your answer to the nearest whole number. (c) How many years will it take for a $3400 porcelain dinner set to be worth $5000 if it increases in value by 9.7% p.a.? Round your answer to the nearest tenth of a year. Understanding 4 $5000 is invested for 4 years with the interest compounded quarterly. For the first years the interest rate is 4% p.a. and for the other years it is 4.8% p.a. (a) How much will the investment be worth at the end of the 4 years? (b) How much interest will be earned? (c) Would the investor be better off to invest for the whole 4-year period at 4.4% p.a.? How much interest would this investment earn? (d) How much interest would be earned if the first years were at 4.8% p.a. and the rest at 4% p.a.? (e) The conditions of investment change and interest is now compounded annually. What would the interest rate need to be to earn the same amount of interest as found in part (b)? (f) The interest rate is now 5% p.a. compounded annually. After how many years will the investment be at least as great as the result shown in part (a)? 5 Pene s savings are in a bank account that attracts 3.5% p.a. If her savings are to grow from $0 000 to over $0 000 in 0 years, what is the smallest number of compounding periods per year required? Reasoning A building society offered an interest rate of 7.5% p.a. on any investment over $3000. Mary deposited $3600 and received $5944 after 7 years. (a) How often was the interest compounded? (b) What simple interest rate would have given the same return in the same time? Round your answer to one decimal place. (c) Mary was a little disappointed because she d calculated a return of $597. Her calculations were based on an incorrect assumption. What was that assumption? 7 If savings in an account are to be at least tripled over 0 years with an interest rate of 5.55% p.a., how often should the interest be compounded? Give your answer in months. D PEARSON mathematics 0 0A essentials edition

12 Open-ended 8 Jacob and Hafida are planning an overseas trip that will cost $ They have decided to pool their money and invest it at 6.5% p.a. compounding quarterly for year. They did the following calculations:.3 How much do we need to invest? P =?, A = 0 000, n = 4, r = Checking: A =?, P = , n = 4, r = = P = P(.0) = P.08 0 P = P = A = A = 858.5( + 0.0) 4 A = When they collect their money a year later, they are almost $50 below their target. Where did they go wrong? 9 Give different rates and compounding periods that will result in savings being increased by at least 50% over 4 years. Financial mathematics 3

13 Half-time. Round answers to the nearest cent, unless told otherwise. If a loan of $700 is made at 7% p.a. compounded quarterly over 5 years, the debt would grow to: A 700( + 0.7) 5 B C D 700( + 0.7) Calculate how much would need to be invested at 6% p.a. compounded monthly to accumulate to $4 000 in 8 years. 3 Calculate the total amount owing on a loan of $6000 after 3 years if the 4% interest p.a. is compounded annually. 4 $ grows to $ in year. The rate of growth is compounding every 6 months. Find the value of the annual interest rate. 5 How much interest is added over 8 years to an account paying 8.5% p.a. interest on the initial sum of $ if the interest is compounded: (a) half-yearly (b) monthly (c) daily? 6 An item originally valued at $ increased in value by a total of 55.33% over a number of years. Each year it appreciated at a rate of 5% of its value the previous year. Find the number of years, rounded to the nearest whole number. 7 Catherine and Ken invested $0 000 for 6 years at an interest rate of 7.5% p.a. (a) Calculate the value of the interest earned on the investment if it was compounded annually. (b) Calculate the value of the interest earned on the investment if simple interest was paid instead. (c) Compare the compounded interest with the simple interest. Round your answers to the nearest dollar. 8 A painting was purchased years ago for $ It has just been sold for $ Calculate the flat rate of appreciation p.a. Round your answer to one decimal place. 4 PEARSON mathematics 0 0A essentials edition

14 Chapter review D.I.Y. Summary Key Words adjusted value exponential growth reducing balance depreciation appreciation flat rate residual value compound interest inflation rate salvage value cumulative depreciation initial cost scrap value depreciation interest simple interest depreciation value nominal interest rate straight-line depreciation effective interest rate per annum total possible depreciation expected life prime cost unit cost depreciation exponential decay principal written-down value Copy and complete the following using the words and phrases from the list, where appropriate, to write a summary for this chapter. A word or phrase may be used more than once. The original amount of money that is borrowed or loaned is called the. Interest rates are normally expressed with p.a. after them. This stands for and means for every year. 3 is always higher than after the first compounding period. 4 Increase in cost or value is called and decrease in cost or value is called. 5 The stated interest rate is known as the. The adjusted rate that takes into account the effect of any compounding in the first year is called the. 6 The amount of depreciation on a certain item is $0 each year for 7 years. This is an example of. 7 Depreciation that has a decreasing value from one year to the next is called. 8 If the depreciation is $5 in the first year and $4 in the second year, the over the first two years is $9. 9 The value of an item after depreciation is known as its or. 0 The formula V T = P D T has three, V T, P and D T. In this formula, P stands for the or. Any value that an item still has at the end of its useful life is known as, or. The initial purchase price less any scrap value is the. 3 This value divided by the expected life gives the. 4 The growth of bacteria is an example of. The opposite process is called. Financial mathematics 9

15 Fluency (a) The amount an investment of $800 amounts to after years, if 9.5% p.a. interest is compounded annually, is closest to: A $558 B 63 C $7790 D $983 (b) The amount an investment of $4000 amounts to after years, if 3.5% p.a. interest is compounded annually, is closest to: A $85 B $49 C $480 D $485 Michael borrowed $4 000 to buy a new piano. He agreed to pay interest at a rate of 6% p.a. and to repay the loan in full after 4 years. Calculate the amount to be repaid if the interest is compounded: (a) half-yearly (b) quarterly (c) monthly. 3 Helga deposited some money into a savings account. The account guaranteed a return of 9.5% p.a. compounded daily. Calculate the initial deposit if the savings had grown to $ after 5 years. Give your answer rounded to the nearest dollar. 4 Find the effective interest rate for each of the following: (a) $ invested at a simple rate of 8.7% p.a. (b) $ invested at an annually compounding rate of 9.5% p.a. (c) the interest owed on a loan of $700 after year was $8 (d) 7.4% p.a. compounded every months 5 The initial price 5 years ago was $ The rate of depreciation is 7% p.a. (a) Assuming straight-line depreciation, find the current written-down value. (b) Assuming reducing balance depreciation, find the current written-down value. Round all answers to the nearest cent. 6 The population of a small rural community has been declining at a rate of % a year. If there were 46 residents in 006, estimate the population in 06. Understanding 7 Simple interest is paid on an investment of $ 500 and yields $ in 5 years. What is the rate of interest being paid? 8 Nina s salary has increased by 6% p.a. each year over the past 3 years to $ p.a. Calculate her salary 3 years ago, to the nearest dollar. 9 Assume an inflation rate of 5.6% p.a. In 4 years, an ice-cream, currently $.0, will cost: A $.0(.04) 4 B $.0(.056) 4 C $.0(.04) 6 D $.0(.56) 4 0 Calculate the rate of interest p.a. (rounded to one decimal place) that would allow $6700 to accumulate to $0 000 in 5 years if interest is compounded each quarter. A dining suite was purchased 3 -- years ago for $4600. It depreciated at a fixed percentage of its reduced value each year and has just sold for $800. The rate of depreciation is closest to: A 5.5% p.a. B 3.% p.a. C 3% p.a. D.8% p.a PEARSON mathematics 0 0A essentials edition

16 Copy and complete the following. (a) If the prime cost of a photocopier is $0 800 and it has a scrap value of $500, then the total possible depreciation is $. If it has an expected life of pages, then the unit cost depreciation is $ per page. After it copies 5 000, the depreciation is $ and the written-down value is $. (b) If the prime cost of an item is $5 500 and it has a scrap value of $750 then the total possible depreciation is $. If it has an expected life of 5000 days, then the unit cost depreciation is $ per day. After it does a year s work, the depreciation is $ and the written-down value is $. 3 The population of Metown increases by.% each year. What was the population 6 years ago if its current population is ? 4 The temperature inside a room decreases by 3% every 5 minutes. Find the temperature at 7 p.m. if it is 6 C at 5 p.m. Round your answer to one decimal place. Reasoning 5 How many years will it take for an investment of $0 000 to increase by at least 50% if the interest rate of 0% p.a. is compounded half-yearly? Give your answer to the nearest tenth of a year. 6 The following represent loan options. In each case, choose the better option without calculating the effective rate and justify your choice. (a) % p.a. compounded quarterly (b) 0% p.a. compounded daily (c) 7% p.a. compounded daily (d) % p.a. flat rate for years 4% p.a. compounded monthly 0% p.a. compounded monthly 7.5% p.a. compounded daily % p.a. compounded annually for years 7 One pair of breeding rats can become a colony of 000 in one year. Estimate the population rate of growth per month. Give your answer rounded to one decimal place. 8 A new city restaurant had only customers for lunch on its opening day. Business improved and the number of lunchtime customers followed an exponential growth curve. After years the restaurant had reached its capacity of 0. What was the growth rate per month over these first years? Round your answer to the nearest whole number. Financial mathematics 3