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

Soo T. Tan

Chapter 4

Mathematics of Finance - all with Video Answers

Educators

Section 1

Compound Interest

Problem 1

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

Catt Huth

Catt Huth

Numerade Educator

Problem 1

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ yr.
$$
P=\$ 1000, r=7 \%, t=8, \text { compounded annually }
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 2

Find the simple interest on a $$\$ 1000$$ investment made for 3 yr 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 $9 \mathrm{mo}$ on an $$\$ 800$ $deposit in a bank paying simple interest at a rate of $6 \% /$ year.

Matt Just

Numerade Educator

Problem 4

Find the accumulated amount at the end of $8 \mathrm{mo}$ 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 $2 \mathrm{yr}$ and the simple rate of interest is $8 \% /$ year, then 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 mo. Find the principal.

Amy Monaghan

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 ordinary simple interest at the rate of $5 \% /$ 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 $5 \% /$ 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 mo. Find the interest
rate.

Amy Jiang

Numerade Educator

Problem 10

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

Kaylee Mcclellan

Kaylee Mcclellan

Numerade Educator

Problem 12

Find the accumulated amount $A$ if the principal $P$ is invested at the interest rate of $r /$ year for $t$ yr.
$$
P=\$ 1000, r=8 \frac{1}{2} \%, t=6, \text { compounded annually }
$$

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 $t$ yr.
$$
P=\$ 2500, r=7 \%, t=10, \text { compounded semiannually }
$$

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 $t$ yr.
$$
P=\$ 2500, r=9 \%, t=10 \frac{1}{2}, \text { compounded semiannually }
$$

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 $t$ yr.
$$
P=\$ 12,000, r=8 \%, t=10 \frac{1}{2}, \text { compounded quarterly }
$$

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 $t$ yr.
$$
P=\$ 42,000, r=7 \frac{3}{4} \%, t=8, \text { compounded quarterly }
$$

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 $t$ yr.
$$
P=\$ 150,000, r=14 \%, t=4, \text { compounded monthly }
$$

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 $t$ yr.
$$
P=\$ 180,000, r=9 \%, t=6 \frac{1}{4}, \text { compounded monthly }
$$

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 $t$ yr.
$$
P=\$ 150,000, r=12 \%, t=3, \text { compounded daily }
$$

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 $t$ yr.
$$
P=\$ 200,000, r=8 \%, t=4, \text { compounded daily }
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 21

Find the effective rate corresponding to the given nominal rate.
$10 \% /$ year compounded semiannually

Amy Jiang

Numerade Educator

Problem 22

Find the effective rate corresponding to the given nominal rate.
$9 \% /$ year compounded quarterly

Manisha Sarker

Numerade Educator

Problem 23

Find the effective rate corresponding to the given nominal rate.
$8 \% /$ year compounded monthly

Manisha Sarker

Numerade Educator

Problem 24

Find the effective rate corresponding to the given nominal rate.
$8 \% /$ year compounded daily

Manisha Sarker

Numerade Educator

Problem 25

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest.
$8 \% /$ year compounded semiannually

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 26

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest.
$8 \% /$ year compounded quarterly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 27

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest.
$7 \% /$ year compounded monthly

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 28

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest.
$9 \%$ year compounded daily

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 29

Find the accumulated amount after 4 yr if $$\$ 5000$$ is invested at $8 \% /$ year compounded continuously.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 30

Find the accumulated amount after 6 yr if $$\$ 6500$$ is invested at $7 \% /$ year compounded continuously.

Nick Johnson

Numerade Educator

Problem 31

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 $12 \%$ year compounded monthly?

Amy Jiang

Numerade Educator

Problem 32

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 $8 \% /$ year compounded monthly?

Amy Jiang

Numerade Educator

Problem 33

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 $9 \% /$ year compounded monthly?

Jillian Rae Villa

Jillian Rae Villa

Numerade Educator

Problem 34

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 $8 \% /$ year compounded daily?

Amy Jiang

Numerade Educator

Problem 35

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

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 36

Use logarithms to solve each problem.
Find the interest rate needed for an investment of $$\$ 4000$$ to double in 5 yr if interest is compounded continuously.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 37

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 $7 \frac{1}{2} \%$ compounded continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 38

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 $8 \%$ compounded continuously?

Jillian Rae Villa

Jillian Rae Villa

Numerade Educator

Problem 39

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

Neel Faucher

Numerade Educator

Problem 40

Jennifer was awarded damages of $$\$ 150,000$$ in a successful lawsuit she brought against her employer 5 yr ago. Interest (simple) 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 41

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 mo. If the bank charges simple interest at the rate of $12 \% /$ year, how much will the Abdullahs owe the bank at the end of the term?

Wendi Zhao

Numerade Educator

Problem 42

David owns $$\$ 20,000$$ worth of 10 -yr bonds of Ace Corporation. These bonds pay interest every 6 mo at the rate of $7 \%$ /year (simple interest). How much income will David receive from this investment every $6 \mathrm{mo}$ ? How much interest will David receive over the life of the bonds?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 43

Maya paid $$\$ 10,000$$ for a 7 -yr 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 44

Write Equation (1b) in the slope-intercept form and interpret the meaning of the slope and the $A$ -intercept in terms of $r$ and $P$.
Hint: Refer to Figure 1 .

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 45

If the cost of a semiprivate room in a hospital was $$\$ 580 /$$ day 5 yr 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?

Whitney Dillinger

Whitney Dillinger

Numerade Educator

Problem 46

Today a typical family of four spends $$\$ 600 /$$ month for food. If inflation occurs at the rate of $5 \%$ lyear over the next $6 \mathrm{yr}$, how much should the typical family of four expect to spend for food 6 yr from now?

Carson Merrill

Numerade Educator

Problem 47

The Kwans are planning to buy a house 4 yr 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 $$\$ 210,000$$ ?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 48

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, due 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 49

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 $5.5 \%$ /year compounded semiannually over a period of 10 yr. At the end of this period, how much will the investment be worth?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 50

Five and a half years ago, Chris invested $$\$ 10,000$$ in a retirement fund that grew at the rate of $10.82 \% /$ year compounded quarterly. What is his account worth today?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 51

Jodie invested $$\$ 15,000$$ in a mutual fund 4 yr ago. If the fund grew at the rate of $9.8 \% /$ year compounded monthly, what would Jodie's account be worth today?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 52

A young man is the beneficiary of a trust fund established for him 21 yr 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 $8 \% /$ year compounded annually? Compounded quarterly? Compounded monthly?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 53

Find how much money should be deposited in a bank paying interest at the rate of $8.5 \% /$ year compounded quarterly so that, at the end of $5 \mathrm{yr}$, the accumulated amount will be $$\$ 40,000$$.

Willis James

Numerade Educator

Problem 54

An individual purchased a 4-yr, $$\$ 10,000$$ promissory note with an interest rate of $8.5 \%$ /year compounded semiannually. How much did the note cost?

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 55

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 $13 \mathrm{yr}$, how much should they set aside in the trust now if they can invest the money at $8 \frac{1}{2} \% /$ year compounded (a) annually, (b) semiannually, and (c) quarterly?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 56

Anthony invested a sum of money 5 yr ago in a savings account that has since paid interest at the rate of $8 \% /$ year compounded quarterly. His investment is now worth $$\$ 22,289.22$$. How much did he originally invest?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 57

In the last 5 yr, Bendix Mutual Fund grew at the rate of $10.4 \% /$ year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of $10.6 \% /$ year compounded semiannually. Which mutual fund has a better rate of return?

AG

Ankit Gupta

Numerade Educator

Problem 58

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 59

The proprietors of The Coachmen Inn secured two loans from Union Bank: one for $$\$ 8000$$ due in 3 yr and one for $$\$ 15,000$$ due in $6 \mathrm{yr}$, both at an interest rate of $10 \%$ /year compounded semiannually. The bank has agreed to allow the two loans to be consolidated into one loan payable in 5 yr at the same interest rate. What amount will the proprietors of the inn be required to pay the bank at the end of 5 yr? Hint: Find the present value of the first two loans.

Sreeraj P

Numerade Educator

Problem 60

Find the effective rate of interest corresponding to a nominal rate of $9 \% /$ year compounded annually, semiannually, quarterly, and monthly.

Gregory Higby

Numerade Educator

Problem 61

Juan is contemplating buying a zero coupon bond that matures in $10 \mathrm{yr}$ 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 62

Maxwell started a home theater business in 2005 . The revenue of his company for that year was $$\$ 240,000$$. The revenue grew by $20 \%$ in 2006 and by $30 \%$ in 2007 . Maxwell projected that the revenue growth for his company in the next 3 yr will be at least $25 \% /$ year. How much does Maxwell expect his minimum revenue to be for $2010 ?$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 63

Online retail sales stood at $$\$ 23.5$$ billion for the year 2000 . For the next 2 yr, they grew by $33.2 \%$ and $27.8 \%$ per year, respectively. For the next $6 \mathrm{yr}$, online retail sales were projected to grow at $30.5 \%, 19.9 \%$, $24.3 \%, 14.0 \%, 17.6 \%$, and $10.5 \%$ per year, respectively. What were the projected online sales for 2008 ?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 64

The inflation rates in the U.S. economy for 2003 through 2006 are $1.6 \%, 2.3 \%, 2.7 \%$, and $3.4 \%$, respectively. What was the purchasing power of a dollar at the beginning of 2007 compared to that at the beginning of $2003 ?$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 65

Investment A offers a $10 \%$ return compounded semiannually, and investment $\mathrm{B}$ offers a $9.75 \%$ return compounded continuously. Which investment has a higher rate of return over a 4-yr period?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 66

Leonard's current annual salary is $$\$ 45,000$$. Ten yr from now, how much will he need to earn in order 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 67

Having received a large inheritance, Jing-mei's parents wish to establish a trust for her college education. If $7 \mathrm{yr}$ from now they need an estimated $$\$ 70,000$$, how much should they set aside in trust now, if they invest the money at $10.5 \%$ compounded quarterly? Continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 68

Maria, who is now 50 yr 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 $15 \mathrm{yr}$ is $6 \% / \mathrm{year}$ compounded continuously? 8\%/year compounded continuously? $12 \% /$ year compounded continuously?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 69

REAL ESTATE INVESTMENTS 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
$$
V(t)=80,000 e^{\sqrt{V} / 2}
$$
where $V(t)$ is measured in dollars and $t$ is the time (in yr) from the present. If the rate of appreciation is expected to be $9 \%$ compounded continuously for the next $8 \mathrm{yr}$, find an expression for the present value $P(t)$ of the property's market price valid for the next 8 yr. What is $P(t)$ expected to be in 4 yr?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 70

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 $P r$ 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 71

Suppose an initial investment of $$\$ P$$ grows to an accumulated amount of $$\$ A$$ in $t$ yr. Show that the effective rate (annual effective yield) is
$$
r_{\text {eff }}=(A / P)^{1 / t}-1
$$
Use the formula given in Exercise 71 to solve Exercises $72-76 .$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 72

Martha invested $$\$ 40,000$$ in a boutique 5 yr ago. Her investment is worth $$\$ 70,000$$ today. What is the effective rate (annual effective yield) of her investment?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 73

Georgia purchased a house in January 2000 for $$\$ 200,000$$. In January 2006 she sold the house and made a net profit of $$\$ 56,000$$. Find the effective annual rate of return on her investment over the 6 -yr period.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 74

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

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 75

Nina purchased a zero coupon bond for $$\$ 6724.53 .$$ The bond matures in $7 \mathrm{yr}$ 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 accumulated
amount.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 76

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 mo ( 245 days) was $\$ 5170.42$. 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 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.
When simple interest is used, the accumulated amount is a linear function of $t$.

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.
If interest is compounded annually, then the accumulated amount after $t$ yr is the same as the accumulated amount under simple interest over $t$ yr.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 79

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 80

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-yr period. Therefore, Susan received annual increases of $5 \%$ over that period.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator