390 CHAPTER Mathematics of Finance (a) Decide which graph corresponds to which annuity (b) Use the graph to estimate (to the nearest 10 months) how long it will be before the larger annuity is $10,000 more than the smaller one Dollars $ time Months Find the future value of an annuity of $1300 paid at the end of each year for years, if interest is earned at a rate of 6%, compounded annually Find the future value of an annuity of $5000 paid at the end of each year for 10 years, if it earns 9%, compounded annually Find the future value of an ordinary annuity of $80 paid quarterly for years, if the interest rate is 8%, compounded quarterly 10 Find the future value of an ordinary annuity of $300 paid quarterly for years, if the interest rate is 12%, compounded quarterly 11 The Weidmans want to save $40,000 in years for a down payment on a house If they make monthly deposits in an account paying 12%, compounded monthly, what is the size of the payments that are required to meet their goal? 12 A sinking fund is established to discharge a debt of $80,000 in 10 years If deposits are made at the end of each 6-month period and interest is paid at the rate of 8%, compounded semiannually, what is the amount of each deposit? 13 If $2500 is deposited at the end of each quarter in an account that earns 5% compounded quarterly, after how many quarters will the account contain $80,000? 14 If $4000 is deposited at the end of each half year in an account that earns 6.2% compounded semiannually, how long will it be before the account contains $120,000? 15 In this section’s Application Preview, we considered the investment strategies of twins and found that starting early and stopping was a significantly better strategy than waiting, in terms of total contributions made as well as total value in the account at retirement Suppose now that twin invests $2000 at the end of each year for 10 years only (until age 33) in an account that earns 8%, compounded annually Suppose that twin waits until turning 40 to begin investing How much must twin put aside at the end of each year for the next 25 years in an account that earns 8% compounded annually in order to have the same amount as twin at the end of these 25 years (when they turn 65)? 16 (a) Patty Stacey deposits $2000 at the end of each of 5 years in an IRA If she leaves the money that has accumulated in the IRA account for 25 additional years, how much is in her account at the end of the 30-year period? Assume an interest rate of 9%, compounded annually (b) Suppose that Patty’s husband delays starting an IRA for the first 10 years he works but then makes $2000 deposits at the end of each of the next 15 years If the interest rate is 9%, compounded annually, and if he leaves the money in his account for additional years, how much will be in his account at the end of the 30-year period? (c) Does Patty or her husband have more IRA money? ANNUITIES DUE 17 Find the future value of an annuity due of $100 each quarter for 212 years at 12%, compounded quarterly 18 Find the future value of an annuity due of $1500 each month for years if the interest rate is 12%, compounded monthly 19 Find the future value of an annuity due of $200 paid at the beginning of each 6-month period for years if the interest rate is 6%, compounded semiannually 20 A house is rented for $3600 per quarter, with each quarter’s rent payable in advance If money is worth 8%, compounded quarterly, and the rent is deposited in an account, what is the future value of the rent for one year? 21 How much must be deposited at the beginning of each year in an account that pays 8%, compounded annually, so that the account will contain $24,000 at the end of years? 22 What is the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 7.8%, compounded semiannually, so that the account will have a future value of $120,000 at the end of 15 years? 23 A company wants to have $800,000 for office renovations If it can deposit $40,000 at the beginning of each quarter into an account that earns 5.2% compounded quarterly, how long before the company reaches its goal? 24 Seana wants to save $20,000 toward a new car purchase She can contribute $400 at the beginning of each month into an account that earns 4.8% compounded monthly How long until the account reaches her goal? MISCELLANEOUS PROBLEMS In Problems 25–42, (a) state whether the problem relates to an ordinary annuity or an annuity due, and then (b) solve the problem 25 Parents agree to invest $500 (at 10%, compounded semiannually) for their son on the December 31 or June 30 following each semester that he makes the dean’s list during his years in college If he makes the Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 6.3 26 27 28 29 30 31 32 33 34 35 36 37 38 39 dean’s list in each of the semesters, how much money will his parents have to give him when he graduates? Jake Werkheiser decides to invest $2000 in an IRA at the end of each year for the next 10 years If he makes these investments, and if the certificates pay 12%, compounded annually, how much will he have at the end of the 10 years? How much will have to be invested at the beginning of each year at 10%, compounded annually, to pay off a debt of $50,000 in years? If $1000 is deposited at the beginning of each quarter into an account that earns 8%, compounded quarterly, how long until the account contains $31,000? A family wants to have a $200,000 college fund for their children at the end of 20 years What contribution must be made at the end of each quarter if their investment pays 7.6%, compounded quarterly? If $2000 is deposited at the end of each quarter into an account that earns 6% compounded quarterly, how long until the account reaches $50,000? A couple has determined that they need $1.5 million to establish an annuity when they retire in 25 years How much money should they deposit at the end of each month in an investment plan that pays 10%, compounded monthly, so they will have the $1.5 million in 25 years? Sam deposits $500 at the end of every months in an account that pays 8%, compounded semiannually How much will he have at the end of years? A company deposits $12,000 at the beginning of each quarter into an account that earns 7.2% compounded quarterly How much will be in this account after 5 years? Sofia can deposit $350 at the beginning of each month into an account that earns 4.2% compounded monthly How long before the account contains $17,000? Mr Gordon plans to invest $300 at the end of each month in an account that pays 9%, compounded monthly After how many months will the account be worth $50,000? For years, $400 is placed in a savings account at the beginning of each 6-month period If the account pays interest at 10%, compounded semiannually, how much will be in the account at the end of the years? Grandparents plan to open an account on their grandchild’s birthday and contribute each month until she goes to college How much must they contribute at the beginning of each month in an investment that pays 12%, compounded monthly, if they want the balance to be $180,000 at the end of 18 years? How much money should a couple deposit at the end of each month in an investment plan that pays 7.5%, compounded monthly, so they will have $800,000 in 30 years? Jane Adele deposits $500 in an account at the beginning of each 3-month period for years If the account Future Values of Annuities 391 pays interest at the rate of 8%, compounded quarterly, how much will she have in her account after years? 40 A company establishes a sinking fund to discharge a debt of $750,000 due in years by making equal semiannual deposits, the first due in months If the investment pays 12%, compounded semiannually, what is the size of the deposits? 41 A sinking fund is established by a working couple so that they will have $60,000 to pay for part of their daughter’s education when she enters college If they make deposits at the end of each 3-month period for 10 years, and if interest is paid at 12%, compounded quarterly, what size deposits must they make? 42 A property owner has several rental units and wants to build more How much of each month’s rental income should be deposited at the beginning of each month in an account that earns 6.6%, compounded monthly, if the goal is to have $100,000 at the end of years? COMBINED APPLICATIONS Problems 43–46 are complex financial problems that require several skills, perhaps some from previous sections 43 Suppose a recent college graduate’s first job allows her to deposit $100 at the end of each month in a savings plan that earns 9%, compounded monthly This savings plan continues for years before new obligations make it impossible to continue If the accrued amount remains in the plan for the next 15 years without deposits or withdrawals, how much money will be in the account 23 years after the plan began? 44 Suppose a young couple deposits $1000 at the end of each quarter in an account that earns 7.6%, compounded quarterly, for a period of years After the years, they start a family and find they can contribute only $200 per quarter If they leave the money from the first years in the account and continue to contribute $200 at the end of each quarter for the next 1812 years, how much will they have in the account (to help with their child’s college expenses)? 45 A small business owner contributes $3000 at the end of each quarter to a retirement account that earns 8% compounded quarterly (a) How long will it be until the account is worth $150,000? (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $5000 at the end of each quarter What will the total value of the account be after 15 more years? 46 A young executive deposits $300 at the end of each month for years and then increases the deposits If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of $400,000 after 25 years? Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... account be worth $50,000? For years, $400 is placed in a savings account at the beginning of each 6-month period If the account pays interest at 10%, compounded semiannually, how much will be in... will have $800,000 in 30 years? Jane Adele deposits $500 in an account at the beginning of each 3-month period for years If the account Future Values of Annuities 391 pays interest at the rate... of their daughter’s education when she enters college If they make deposits at the end of each 3-month period for 10 years, and if interest is paid at 12%, compounded quarterly, what size deposits