1 Some review of percentages

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1 1 Some review of percentages Recall that 5% =.05, 17% =.17, x% = x. When we say x% of y, we 100 mean the product x%)y). If a quantity A increases by 7%, then it s new value is }{{} P new value = }{{} A old value +.07A }{{} = 1.07A growth Likewise if a quantity A decreases by 7%, then it s new value is P }{{} new value 1. What is 14% of 400? Answer: What is 4% of 750? Answer: 30 = A }{{} old value.07a }{{} =.93A growth 3. If a stock sells for $98 per share at the beginning of trading today, and it increases in value by 3% by the end of the day, how much does the stock sell for at the end of the day. Answer: $ If your bar tab is $55 and you leave a 21% tip, what was your total bill? Answer: 55)1.21) = $ If a $400 suit is on sale for 35% off, what is its sale price? Answer: 400).65) = $260 2 Exponential change 6. Suppose you own an asset that is worth $100 now, and it increases in value by $10 every year. a) How much is the asset worth in one year? Answer: $110 1

2 b) How much is the asset worth in 2 years? Answer: $ 120 c) How much is the asset worth in 18 years? Answer: $ 280 d) If V is the value of the asset in t years, what is an equation relating V and t. Answer: V = t 7. Suppose you own an asset that is worth $100 now, and it increases in value by 5% every year. a) How much is the asset worth in one year? Answer: $105 b) How much is the asset worth in 2 years? Answer: $ c) How much is the asset worth in 3 years? Answer: $ d) How much is the asset worth in 18 years? Answer: $ e) If V is the value of the asset in t years, what is an equation relating V and t. Answer: V = t 8. The population of Salem now is about 41,000. Suppose that Salem is growing at a rate of 3% each year. a) If P is the population of Salem in t years, what is an equation relating P and t? Answer: P = t b) What does this model estimate the population of Salem will be in 12 years? Answer: 58, 456 2

3 In summary, an exponential function is a function of the form y = ab x If a quantity A grows by r% each time period, then after t time periods, the size of the quantity will be A1 + r) t 3 Some review of exponents 9. Evaluate the following without a calculator: a) 2 3) 4 Answer: 4096 b) 3 7) 1/7 Answer: Evaluate the following with a calculator: a) 2 19 b) Answer: 524, 288 Answer: c) Answer: d) 99 Answer: e) 2300 Answer: Solve each of the following equations use a calculator if necessary): a) 5b 2 = 500 Answer: b = 10 b) 6x 3 = 162 Answer: x = 3 c) 18t 9 = Answer: t =

4 4 Exponential models 12. What is a formula for an exponential function whose graph passes through 1, 30) and 3, 50)? Answer: y = x 13. The population of Boston in the year 2000 was 590,000, and it is 673,000 today. a) build an exponential model for the population of Boston as a function of time. Be clear about what the variables in your model mean. Answer: Letting t be the number of years since 2000 and P the population of Boston in thousands of people, P = t b) Use your model to predict the population of Boston in the year Answer: , so the model predicts about 723,400 people in Boston in At the end of 2004, the price of a share of Apple stock was about $5, and at the end of 2016, it was about $119. a) build an exponential model for the price of a share of Apple stock as a function of time. Be clear about what the variables in your model mean. Answer: Letting t be the number of years since 2004 and V the value of Apple stock in dollars, V = t b) Use your model to predict the price of a share of Apple stock in Answer: $ Compound interest 15. In problem 7, you modeled the value of an asset that grew by 5% each year. Suppose the asset grows at half that rate but twice as often. 4

5 a) How much is the asset worth in one year? Answer: $ b) How much is the asset worth in 2 years? Answer: $ c) How much is the asset worth in 3 years? Answer: $ d) How much is the asset worth in 18 years? Answer: $ e) If V is the value of the asset in t years, what is an equation relating V and t. Answer: V = r ) 2t Suppose the asset in problem 7 grew at one fourth the rate but four times as often. In this case, we say that the rate is compounded quarterly. a) How much is the asset worth in one year? Answer: $ b) How much is the asset worth in 2 years? Answer: $ c) How much is the asset worth in 3 years? Answer: $ d) How much is the asset worth in 18 years? Answer: $ e) If V is the value of the asset in t years, what is an equation relating V and t. Answer: V = r ) 4t Suppose the asset in problem 7 grew at n times as often, but at 1 n times the rate. In this case, we say that the rate is compounded n times per year. a) How much is the asset worth in one year as a function of n? Answer: r ) n n 5

6 b) How much is the asset worth in 2 years as a function of n? Answer: V = r ) 2n n c) How much is the asset worth in 3 years as a function of n? Answer: V = r ) 3n n d) How much is the asset worth in 18 years as a function of n? Answer: V = r ) 18n n Compound interest: If an investment of A dollars grows at a rate of r compounded n times per year, then the value of the investment after t years is V = A 1 + r ) nt n 18. Suppose you invest $100 at 8% interest. Fill in the values of the investment after 1 year under the different compounding schemes listed in the table below. compounding scheme value of $100 after 1 year semi-annually $ quarterly $ monthly $ daily $ The effective annual rate of a compounding scheme is the percentage by which the asset grows in 1 year. A straightforward way to determine this is to compute the value of $100 under the compounding scheme after one year. a) What is the effective annual rate of 4% compounded monthly? Answer: 4.07% b) What is the effective annual rate of 12% compounded daily? Answer: 12.75% 20. If you invest $500, determine how much your money will grow to under each of the following scenarios 6

7 a) The money grows at 6.2% compounded quarterly for 8 years. Answer: $ b) The money grows at 10.2% compounded monthly for 8 years. Answer: $1, c) The money grows at 2.8% compounded daily for 4 years. Answer: $ d) The money grows at 5.1% compounded daily for 14 years. Answer: $1, If you want to save $5000, how much must you invest to achieve this under each of the following scenarios. a) You have 7 years, and your money can earn 3.9% compounded daily. Answer: $3, b) You have 4 years, and your money can earn 5.3% compounded daily. Answer: $4, If you put $500 on your credit card, and the debt grows at 19% compounded daily, how much will you owe in three years? Make the simplifying, but hopefully unrealistic assumption that you make no payments.) Answer: $ If you invested $5 at 6% interest compounded daily 200 years ago, what would your investment be worth today? Answer: $812, Additional practice 24. If $10000 is invested at 12% interest, find the value of the investment at the end of 7 years if the interest is compounded a) annually Answer: $22,

8 b) monthly Answer: $23,067.2 c) daily Answer: $23, A bank pays interest at the nominal rate of 4.5% per year. What is the effective annual yield if compounding is: a) annual Answer: 4.5% b) monthly Answer: 4.59% c) daily Answer: 4.6% 26. If you need $40,000 four years from now, what is the minimum amount of money you need to deposit into a bank account that pays 7% annual interest, compounded a) annually Answer: $30,515.8 b) monthly Answer: $30, 256 c) daily Answer: $30, Al borrows $2270 from his uncle. Two years later, he borrows another $1370. If his uncle charges him 8% interest compounded annually, how much does Al owe 9 years after the first loan? Answer: $6, Continuous growth For large values of n, V = A 1 + r ) nt Ae rt n 8

9 where e is Euler s number, e An asset with initial value A grows at a continuously compounded rate r if it s value in t years is given by V = Ae rt 1) More generally, a quantity A grows exponentially with continuous growth rate r if its size in t units of time is given by 1) 28. Use a calculator to compute the following a) e b) e.02)14) c) 4000e.032)21) What will a $5000 asset be worth in 6 years if it grows at rate of 3% compounded a) daily? Answer: $5, b) continuously? Answer: $5, What would the value of an asset have to be now in order that it grow to a value of $10000 in 4 years if the asset grows at 6.2% compounded continuously? Answer: $7, In 2000, U.S. GDP was about 10.5 trillion dollars annually. Assuming that this rate grew at a continuous rate of 4%, what would GDP be today? In fact, U.S. GDP today is about trillion dollars annually.) Answer: million dollars annually. 32. Suppose that a 50 mg chunk of a radioactive substance decays at a continuous rate of 14% per year. How much of the substance will remain in 5 years? Answer: mg 9

10 7 The natural logarithm Definition 1 The natural logarithm is the function lnx) defined by That is, y = lnx) whenever x = e y e 2 = so ln7.389) = 2 e 1.36 = so ln3.896) = 1.36 You should be able to easily evaluate natural logarithms on your calculator. 33. Evaluate or simplify a) lne 4 ) = 4 b) lne 20 ) = 20 c) lne.02)10) =.2 d) lne x ) = x 34. Solve the following equations a) e x = 13 Answer: x = ln 13 b) 200e x = 800 Answer: x = ln 4 c) 5000e.04t = Answer: x = ln Suppose that a quantity is growing exponentially this means that we model it s growth using 1)). There were 400 units of the quantity initially and there are 500 units after 4 days. How many units will there be after 30 days? Answer: units 36. If a $100 asset grows at 4% compounded continuously, how long does it take the value of the asset to double? Answer: years 10

11 37. If a $1,000,000 asset grows at 4% compounded continuously, how long does it take the value of the asset to double? Answer: years 38. If $500 is invested at 7% compounded continuously, how long does it take for the investment to grow to be worth $800? Answer: years 39. If I want to have $5000 in five years to use as a downpayment on a car, how much must I invest now if I can make 4.5% interest compounded continuously? Answer: $3, The current U.S. inflation rate is about 1.94% per year. If this continues to be the case for the next decade, how much will a $300 television cost in ten years? assume that prices grow via a continuously compounded inflation rate) Answer: $ The population of the city of London was about 6.5 million in 1980 and is about 8.5 million people today in 2017). Build an exponential model of the form 1) that relates the population of London to time. Use it to predict the city s population in the year Answer: 9.34 million 42. Polonium-210 is a chemical that is radioactive, which means it disintegrates over time. A 10 pound quantity of this chemical will only weigh 6.05 pounds after 100 days. Use this information to build an exponential model of the form 1) that relates the weight of this chunk of Polonium-210 to time. Use it to predict how large the chunk will be in one year. Answer: pounds 8 Loans If I borrow A dollars, the lender charges me interest on the loan at some interest rate i. Interest on loans is typically compounded monthly, and this 11

12 is the only case we ll consider. So after one month, It would seem that I now owe the lender: ) i }{{} A + A = 1 + i ) A 2) principal }{{}}{{} interest new balance Notice that since i is an annual interest rate, we divide it by 12 to get the monthly interest rate. Since this rate is going to come up a lot, let s set r = i 12 and rewrite 2) as A }{{} principal + }{{} ra = 1 + r)a }{{} interest new balance Of course, 3) isn t quite right because I m also making monthly payments to reduce what I owe. If my monthly payment is P dollars, then after one month what I really owe the lender is A }{{} principal + ra }{{} interest P }{{} payment = 1 + r)a P }{{} new balance If the loan is structured well, the new balance will be less than the original principal, that is, my payment will be larger than the monthly interest charge, that way eventually I pay off the debt. Usually, the borrower and lender agree on a term for the loan, a number of months until the balance is paid off in full. The term of the loan dictates what the monthly payment will be. Deriving this relationship is beyond our scope, so we ll just state it, but we need two auxiliary quantities. With, n representing a number of months and r the interest rate, define two formulas as follows: 3) 4) s[n, r] = 1 + r)n 1 5) r s[n, r] a[n, r] = 6) 1 + r) n You don t have to memorize these, but you will have to be able to use them. The key relationship governing loans is A = a[n, r] P 7) For example, if I borrow $10,000 at 4% on a 60 month term, then r = =

13 and 7) becomes We compute the values of a and s: = a[60,.0033] P s[60,.0033] = = a[60,.0033] = = So P = = $ The monthly payment on this loan would be about $ Compute the following use a calculator) a) s[48,.06/12] b) a[36,.05/12] c) a[360,.0325/12] What is the monthly payment on a $12,000 loan at 6% with a 48 month term? Answer: $ If I can afford a $350 monthly payment for 48 months, how much can I borrow if the interest rate is 5%? Answer: $15, In the example I did above, I borrowed $10,000 at 4% on a 60-month term and found the monthly payment do be $184. How much do I actually pay the lender over the course of the loan? Answer: $ Suppose you take out a 30-year $250,000 mortgage at 4.2%. a) Compute the monthly payment. Answer: $1, b) How much interest you pay over the course of the mortgage? Answer: $190,114 13

14 c) How much do you pay total over the course of the mortgage? Answer: $440,114 d) If you invest $250,000 at 4.2% compounded monthly, how much will your investment be worth in 30 years? Answer: $879, Find the monthly payment on a $100,000, 25-year mortgage at 12% interest compounded monthly. Answer: $1, This problem concerns a $60,000 debt with a 10 year term and a 9% interest rate. a) What is the monthly payment? Answer: $ b) How much interest is paid in the first month? Answer: $450 c) How much of the first month s payment is applied to principal? Answer: $ d) What is the unpaid balance after one month? Answer: $59,689.9 e) What is the unpaid balance after 6 years? Answer: $30,542.8 f) How much interest has been paid during the first 6 years of the loan? Answer: $25,267.1 g) How much interest is paid over the entire term of the loan? Answer: $31,

1 Some review of percentages

1 Some review of percentages Recall that 5% =.05, 17% =.17, x% = x. When we say x% of y, we 100 mean the product (x%)(y). If a quantity A increases by 7%, then it s new value is }{{} P new value = }{{}