Finite Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 5

Mathematics of Finance - all with Video Answers

Educators

JT

Section 1

Compound Interest

Problem 1

Find the simple interest on a $\$ 500$ investment made for 2 years at an interest rate of $8 \%$ lyear. What is the accumulated amount?

Catt Huth

Catt Huth

Numerade Educator

Problem 2

Find the simple interest on a \$1000 investment made for 3 years at an interest rate of $5 \% /$ year. What is the accumulated amount?

Catt Huth

Numerade Educator

Problem 3

Find the accumulated amount at the end of 9 months on an $\$ 800$ deposit in a bank paying simple interest at a rate of $6 \%$ year.

Amy Jiang

Numerade Educator

Problem 4

Find the accumulated amount at the end of 8 months on a $\$ 1200$ bank deposit paying simple interest at a rate of $7 \% /$ year.

Amy Jiang

Numerade Educator

Problem 5

If the accumulated amount is $\$ 1160$ at the end of 2 years and the simple rate of interest is $8 \%$ /year, what is the principal?

Emily Frampton

Numerade Educator

Problem 6

A bank deposit paying simple interest at the rate of $5 \% /$ year grew to a sum of $\$ 3100$ in 10 months. Find the principal.

Trinity Steen

Numerade Educator

Problem 7

How many days will it take for a sum of $\$ 1000$ to earn $\$ 20$ interest if it is deposited in a bank paying simple interest at the rate of $2.5 \% /$ year? (Use a 365 -day year.)

Anas Venkitta

Numerade Educator

Problem 8

How many days will it take for a sum of $\$ 1500$ to earn $\$ 25$ interest if it is deposited in a bank paying simple interest at the rate of $5 \% /$ year? (Use a 365 -day year.)

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 9

A bank deposit paying simple interest grew from an initial sum of $\$ 1000$ to a sum of $\$ 1075$ in 9 months. Find
the interest rate.

Amy Jiang

Numerade Educator

Problem 10

Determine the simple interest rate at which $\$ 1200$ will grow to $\$ 1250$ in 8 months.

Catt Huth

Numerade Educator

Problem 11

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 1000, r=4 \%, t=8$, compounded annually

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 12

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 1000, r=5 \frac{1}{2} \%, t=6$, compounded annually

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 13

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 2500, r=4 \%, t=10$, compounded semiannually

Sandra Kudolo

Numerade Educator

Problem 14

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 2500, r=6 \%, t=10 \frac{1}{2}$, compounded semiannually

Sandra Kudolo

Numerade Educator

Problem 15

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 12,000, r=5 \%, t=10 \frac{1}{2}$, compounded quarterly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 16

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 42,000, r=4 \frac{3}{4} \%, t=8$, compounded quarterly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 17

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 150,000, r=4 \%, t=4$, compounded monthly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 18

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 180,000, r=6 \%, t=6 \frac{1}{4}$, compounded monthly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 19

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 150,000, r=9 \%, t=3$, compounded daily

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 20

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ years.
$P=\$ 200,000, r=8 \%, t=4$, compounded daily

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 21

Find the effective rate corresponding to the qiven nominal rate.
$6 \%$ year compounded semiannually

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 22

Find the effective rate corresponding to the qiven nominal rate.
$5 \% /$ year compounded quarterly

Claire Rochford

Claire Rochford

Numerade Educator

Problem 23

Find the effective rate corresponding to the qiven nominal rate.
$4 \% /$ year compounded monthly

Manisha Sarker

Numerade Educator

Problem 24

Find the effective rate corresponding to the qiven nominal rate.
$4 \% /$ year compounded daily

MS

Mike Stern

Numerade Educator

Problem 25

Find the present value of $\$ 40,000$ due in 4 years at the given rate of interest.
$4 \% /$ year compounded semiannually

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 26

Find the present value of $\$ 40,000$ due in 4 years at the given rate of interest.
$4 \% /$ year compounded quarterly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 27

Find the present value of $\$ 40,000$ due in 4 years at the given rate of interest.
$3 \% /$ year compounded monthly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 28

Find the present value of $\$ 40,000$ due in 4 years at the given rate of interest.
$5 \% /$ year compounded daily

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 29

Find the accumulated amount after 4 years if $\$ 5000$ is invested at $6 \%$ lyear compounded continuously.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 30

Find the accumulated amount after 6 years if $\$ 6500$ is invested at $5 \%$ /year compounded continuously.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 31

Mitchell has been given the option of either paying his $\$ 300$ bill now or settling it for $\$ 306$ after 1 month ( 30 days). If he chooses to pay after 1 month, find the simple interest rate at which he would be charged.

MS

Mike Stern

Numerade Educator

Problem 32

Jennifer was awarded damages of $\$ 150,000$ in a successful lawsuit she brought against her employer 5 years ago. Simple interest on the judgment accrues at the rate of $12 \% /$ year from the date of filing. If the case were settled today, how much would Jennifer receive in the final judgment?

Sirat Shah

Numerade Educator

Problem 33

To help finance the purchase of a new house, the Abdullahs have decided to apply for a shortterm loan (a bridge loan) in the amount of $\$ 120,000$ for a term of 3 months. If the bank charges simple interest at the rate of $10 \% /$ year, how much will the Abdullahs owe the bank at the end of the term?

Willis James

Numerade Educator

Problem 34

David owns $\$ 20,000$ worth of 10 -year bonds of Ace Corporation. These bonds pay interest every 6 months at the rate of $3 \%$ year (simple interest). How much income will David receive from this investment every 6 months? How much interest will David receive over the life of the bonds?

Charles Carter

Numerade Educator

Problem 35

Maya paid $\$ 10,000$ for a 7-year bond issued by a city. She received interest amounting to $\$ 3500$ over the life of the bonds. What rate of (simple) interest did the bond pay?

Majid Borumand

Numerade Educator

Problem 36

Isabella purchased $\$ 20,000$ worth of 13-week T-Bills for $\$ 19,875$. What will be the rate of return on her investment?

Narayan Hari

Numerade Educator

Problem 37

Maxwell purchased $\$ 15,000$ worth of 52-week T-Bills for $\$ 14,650$. What will be the rate of return on his investment?

Narayan Hari

Numerade Educator

Problem 38

The value of Maria's investments increased by $20 \%$ in the first year and by a further $10 \%$ in the second year. The value of Laura's investments grew $10 \%$ in the first year, followed by a gain of $20 \%$ in the second year. Both Maria and Laura started out with a $\$ 10,000$ investment. Whose investment increased more in the 2-year period? Explain.

Sandra Kudolo

Numerade Educator

Problem 39

The value of Alan's stock portfolio grew by $20 \%$ in the first year, followed by a growth of $10 \%$ in the second year. It dropped $10 \%$ and $20 \%$ in the third and fourth years, respectively. Is the value of Alan's stock portfolio after 4 years the same as that when he started out? Explain.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 40

The value of Jack's investment portfolio fell by $20 \%$ in the first year but rebounded by $20 \%$ in the second year. Did Jack regain all of the money that he lost in the first year at the end of the second year? Explain.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 41

The value of Arabella's stock portfolio dropped $20 \%$ in the first year. Find the annual rate of growth, compounded yearly, that she must achieve in the next 2 years to bring the value of her stock portfolio back to its initial value (its value at the beginning of the first year).

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 42

Write Formula (1b) in the slope-intercept form, and interpret the meaning of the slope and the $A$ -intercept in terms of $r$ and $P$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 43

If the cost of a semiprivate room in a hospital was $\$ 680 /$ day 5 years ago and hospital costs have risen at the rate of $8 \% /$ year since that time, what rate would you expect to pay for a semiprivate room today?

Monica Miller

Numerade Educator

Problem 44

Today, a typical family of four spends $\$ 880 /$ month for food. If inflation occurs at the rate of $3 \% /$ year over the next 6 years, how much should the typical family of four expect to spend for food 6 years from now?

Carson Merrill

Numerade Educator

Problem 45

The Kwans are planning to buy a house 4 years from now. Housing experts in their area have estimated that the cost of a home will increase at a rate of $5 \% /$ year during that period. If this economic prediction holds true, how much can the Kwans expect to pay for a house that currently costs $\$ 260,000$ ?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 46

A utility company in a western city of the United States expects the consumption of electricity to increase by $8 \% /$ year during the next decade, owing mainly to the expected increase in population. If consumption does increase at this rate, find the amount by which the utility company will have to increase its generating capacity in order to meet the needs of the area at the end of the decade.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 47

The managers of a pension fund have invested $\$ 1.5$ million in U.S. government certificates of deposit that pay interest at the rate of $2.5 \% /$ year compounded semiannually over a period of 10 years. At the end of this period, how much will the investment be worth?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 48

Five and a half years ago, Chris invested $\$ 10,000$ in a retirement fund that grew at the rate of $6.82 \% /$ year compounded quarterly. What is his account worth today?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 49

Jodie invested $\$ 15,000$ in a mutual fund 4 years ago. If the fund grew at the rate of $7.8 \% /$ year compounded monthly, what would Jodie's account be worth today?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 50

A young man is the beneficiary of a trust fund established for him 21 years ago at his birth. If the original amount placed in trust was $\$ 10,000$, how much will he receive if the money has earned interest at the rate of $6 \% /$ year compounded annually? Compounded quarterly? Compounded monthly?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 51

Find how much money should be deposited in a bank paying interest at the rate of $3.5 \%$ /year compounded quarterly so that at the end of 5 years, the accumulated amount will be $\$ 40,000$.

Katelyn Chen

Numerade Educator

Problem 52

An individual purchased a 4 -year, $\$ 10,000$ promissory note with an interest rate of $5.5 \% /$ year compounded semiannually. How much did the note cost?

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 53

The parents of a child have just come into a large inheritance and wish to establish a trust fund for her college education. If they estimate that they will need $\$ 100,000$ in 13 years, how much should they set aside in the trust now if they can invest the money at $4 \frac{1}{2} \% /$ year compounded (a) annually, (b) semiannually, and (c) quarterly?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 54

Anthony invested a sum of money 5 years ago in a savings account that has since paid interest at the rate of $4 \%$ lyear compounded quarterly. His investment is now worth $\$ 22,289.22$. How much did he originally invest?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 55

In the last 5 years, Bendix Mutual Fund grew at the rate of $6.4 \%$ /year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of $6.5 \% /$ year compounded semiannually. Which mutual fund has a better rate of return?

Marc Lauzon

Numerade Educator

Problem 56

Fleet Street Savings Bank pays interest at the rate of $4.25 \%$ /year compounded weekly in a savings account, whereas Washington Bank pays interest at the rate of $4.125 \% /$ year compounded daily (assume a 365 -day year). Which bank offers a better rate of interest?

JT

Julie Tarr

Numerade Educator

Problem 57

The proprietors of The Coachmen Inn secured two loans from Union Bank: one for $\$ 8000$ due in 3 years and one for $\$ 15,000$ due in 6 years, both at an interest rate of $8 \% /$ year compounded semiannually. The bank has agreed to allow the two loans to be consolidated into one loan payable in 5 years at the same interest rate. What amount will the proprietors of the inn be required to pay the bank at the end of 5 years?

Alanya Huang

Numerade Educator

Problem 58

Find the effective rate of interest corresponding to a nominal rate of $5.5 \% /$ year compounded annually, semiannually, quarterly, and monthly.

Claire Rochford

Claire Rochford

Numerade Educator

Problem 59

Juan is contemplating buying a zero coupon bond that matures in 10 years and has a face value of $\$ 10,000$. If the bond yields a return of $5.25 \% /$ year, how much should Juan pay for the bond?

MS

Mike Stern

Numerade Educator

Problem 60

Maxwell started a home theater business in 2011 . The revenue of his company for that year was $\$ 240,000$. The revenue grew by $20 \%$ in 2012 and by $30 \%$ in 2013. Maxwell projected that the revenue growth for his company in the next 3 years will be at least $25 \% /$ year. How much does Maxwell expect his minimum revenue to be for 2016 ?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 61

Online retail sales stood at $\$ 141.4$ billion for the year 2004 . For the next 2 years, they grew by $24.3 \%$ and $14.0 \%$ per year, respectively. For the next 3 years, online retail sales were projected to grow at $30.5 \%$, $17.6 \%$, and $10.5 \%$ per year, respectively. What were the projected online sales for 2009 ?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 62

The inflation rates in the U.S. economy for 2009 through 2012 are $2.7 \%, 1.5 \%, 3.0 \%$, and $1.7 \%$, respectively. What was the purchasing power of a dollar at the beginning of 2013 compared to that at the beginning of $2009 ?$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 63

Investment $A$ offers an $8 \% /$ year return compounded semiannually, and Investment $B$ offers a $7.8 \%$ /year return compounded continuously. Which investment has a higher rate of return over a 4-year period?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 64

Leonard's current annual salary is $\$ 65,000$. Ten years from now, how much will he need to earn to retain his present purchasing power if the rate of inflation over that period is $3 \% /$ year compounded continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 65

Having received a large inheritance, Jing-mei's parents wish to establish a trust for her college education. Seven years from now they need an estimated $\$ 120,000$. How much should they set aside in trust now if they invest the money at $6.6 \% /$ year compounded quarterly? Continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 66

Maria, who is now 50 years old, is employed by a firm that guarantees her a pension of $\$ 40,000 /$ year at age $65 .$ What is the present value of her first year's pension if the inflation rate over the next 15 years is $3 \% /$ year compounded continuously? $4 \% /$ year compounded continuously? $6 \%$ lyear compounded continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 67

An investor purchased a piece of waterfront property. Because of the development of a marina in the vicinity, the market value of the property is expected to increase according to the rule
$$
V(t)=80,000 e^{\sqrt{t / 2}}
$$
where $V(t)$ is measured in dollars and $t$ is the time (in years) from the present. If the rate of appreciation is expected to be $5 \%$ /year compounded continuously for the next 8 years, find an expression for the present value $P(t)$ of the property's market price valid for the next 8 years. What is $P(t)$ expected to be in 4 years?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 68

The simple interest formula $A=P(1+r t)$ [Formula (1b)] can be written in the form $A=$ Prt $+P$, which is the slope-intercept form of a straight line with slope $\operatorname{Pr}$ and $A$ -intercept $P$.
a. Describe the family of straight lines obtained by keeping the value of $r$ fixed and allowing the value of $P$ to vary. Interpret your results.
b. Describe the family of straight lines obtained by keeping the value of $P$ fixed and allowing the value of $r$ to vary. Interpret your results.

Catt Huth

Numerade Educator

Problem 69

Suppose an initial investment of $\$ P$ grows to an accumulated amount of $\$ A$ in $t$ years. Show that the effective rate (annual effective yield) is
$$
r_{\mathrm{eff}}=\left(\frac{A}{P}\right)^{1 / t}-1
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 70

Use the formula given in Exercise 69 .
Martha invested $\$ 40,000$ in a boutique 5 years ago. Her investment is worth $\$ 60,000$ today. What is the effective rate (annual effective yield) of her investment?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 71

Use the formula given in Exercise 69 .
Georgia purchased a house in September 2008 for $\$ 300,000$. In September 2014 , she sold the house and made a net profit of $\$ 66,000$. Find the effective annual rate of return on her investment over the 6-year period.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 72

Use the formula given in Exercise 69 .
Steven purchased 1000 shares of a certain stock for $\$ 25,250$ (including commissions). He sold the shares 2 years later and received $\$ 32,100$ after deducting commissions. Find the effective annual rate of return on his investment over the 2 -year period.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 73

Use the formula given in Exercise 69 .
Nina purchased a zero coupon bond for $\$ 6724.53$. The bond matures in 7 years and has a face value of $\$ 10,000$. Find the effective annual rate of interest for the bond. Hint: Assume that the purchase price of the bond is the initial investment and that the face value of the bond is the accumu-
lated amount.

Kaylee Mcclellan

Kaylee Mcclellan

Numerade Educator

Problem 74

Use the formula given in Exercise 69 .
Carlos invested $\$ 5000$ in a money market mutual fund that pays interest on a daily basis. The balance in his account at the end of 8 months ( 245 days) was $\$ 5070.42$. Find the effective rate at which Carlos's account earned interest over this period (assume a 365 -day year).

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 75

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
When simple interest is used, the accumulated amount is a linear function of $t$.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 76

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If interest is compounded annually, then the accumulated amount after $t$ years is the same as the accumulated amount under simple interest over $t$ years.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 77

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If interest is compounded annually, then the effective rate is the same as the nominal rate.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 78

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
Susan's salary increased from $\$ 50,000 /$ year to $\$ 60,000$ / year over a 4-year period. Therefore, Susan received annual increases of $5 \%$ over that period.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 79

Use logarithms to solve each problem.
How long will it take $\$ 5000$ to grow to $\$ 6500$ if the investment earns interest at the rate of $6 \% /$ year compounded monthly?

Amy Jiang

Numerade Educator

Problem 80

Use logarithms to solve each problem.
How long will it take $\$ 12,000$ to grow to $\$ 15,000$ if the investment earns interest at the rate of $4 \% /$ year compounded monthly?

Amy Jiang

Numerade Educator

Problem 81

Use logarithms to solve each problem.
How long will it take an investment of $\$ 2000$ to double if the investment earns interest at the rate of $5 \% /$ year compounded monthly?

Victor Salazar

Numerade Educator

Problem 82

Use logarithms to solve each problem.
How long will it take an investment of $\$ 5000$ to triple if the investment earns interest at the rate of $5 \% /$ year compounded daily?

Victor Salazar

Numerade Educator

Problem 83

Use logarithms to solve each problem.
Find the interest rate needed for an investment of $\$ 5000$ to grow to an amount of $\$ 6000$ in 3 years if interest is compounded continuously.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 84

Use logarithms to solve each problem.
Find the interest rate needed for an investment of $\$ 4000$ to double in 8 years if interest is compounded continuously.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 85

Use logarithms to solve each problem.
How long will it take an investment of $\$ 6000$ to grow to $\$ 7000$ if the investment earns interest at the rate of $5 \frac{1}{2} \% /$ year compounded continuously?

Amy Jiang

Numerade Educator

Problem 86

Use logarithms to solve each problem.
How long will it take an investment of $\$ 8000$ to double if the investment earns interest at the rate of $8 \%$ /year compounded continuously?

Jillian Rae Villa

Jillian Rae Villa

Numerade Educator