Section 4.5.H: Homework
- Page ID
- 216984
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Section 4.5: Annuities - Homework Exercises
The Gonzalez family plans a big family vacation in five years. They save and are going to deposit $600 semiannually in an account that earns 6.2% interest, compounded semiannually. What is the future value?
- Answer
-
A = $6910.09
Naoko wants to redesign her living room in three years. She deposits $130 per month, at the end of each month, into an account earning 4.9% interest, compounded monthly.
- What will her savings grow to in 3 years?
- How much interest will she earn?
- Answer
-
- A = $5,030.44
- I = $350.44
Sonia plans to enroll in a professional certification program in two years. She saves $140 per month, depositing at the end of each month, in an account earning 5.1% interest, compounded monthly.
- What is the future value of her savings?
- How much interest did she earn?
- Answer
-
- A = $3,529.45
- I = $169.45
Andy plans to landscape his backyard in four years. He deposits $500 at the end of each quarter into an investment account earning 6.4% interest, compounded quarterly.
- What is the future value after four years?
- How much interest did he earn?
- Answer
-
- A = $9,035.55
- I = $1,035.55
Bobby wants to start a college fund for his nephew. He deposits $2,000 at the end of each year into an account earning 6.7% interest, compounded annually, for 15 years.
- What is the future value of the college fund?
- How much interest did it earn?
- Answer
-
- A = $49,111.86
- I = $19,111.86
Debbie wants to supplement her pension with additional savings. She begins at age 35 and invests $250 each month, deposited at the end of the month, into a fund earning 6.8% annual interest, compounded monthly. She plans to save for 30 years.
- Calculate the future value of Debbie’s savings after 30 years.
- Determine how much interest she accumulated.
- Compare the amount earned from interest to the amount she deposited.
- Answer
-
- A = $293,225.82
- I = $203,225.82
- Deposited = $90,000, so a little over double was made in interest.
Art, age 45, dreams of buying a retirement home at age 65. To prepare, he deposits $600 per month into an investment account earning 7.2% annual interest, compounded monthly, for 20 years.
- How much money will Tyler have saved at retirement?
- How much total interest has he earned?
- If his target amount is $250,000, will he reach it?
- Answer
-
- A = $320,257.40
- I = $176,257.40
- Yes, with an extra $70,257.40. (Tyler earns more interest than he contributed; a sign of strong compounding over 20 years).
A Roth IRA is a retirement savings account where you put in money you've already paid taxes on (after‑tax income), so your investments can grow tax‑free. You can withdraw the money in retirement without paying any taxes on your contributions or your earnings. Linda, age 25, wants to start saving early. She deposits $150 at the end of every month into a Roth IRA earning 8% annually, compounded monthly. She continues this plan for 40 years.
- What is the future value of her Roth IRA after 40 years?
- How much of the final amount is her own contributions?
- How much comes from interest/growth?
- Answer
-
- A = $523,651.17
- Total Contributions = $72,000
- I = $451,651.17
Doug, age 30, wants to have $500,000 saved by age 65. His investment account earns 6.9% annual interest, compounded monthly. He has 35 years to save.
- What monthly deposit must Derek make to reach his goal of $500,000?
- How much total money will he contribute over 35 years?
- How much of the final balance will come from interest?
- Answer
-
- R = $284.30
- Total Contributions = $119,405
- I = $380,595
Mariana, age 50, wants to retire at age 65 with $300,000 saved in her investment account. Her account earns 6.3% annual interest, compounded monthly, and she has 15 years to save.
- What monthly deposit must Michelle make to reach her goal of $300,000?
- How much will she contribute in total over 15 years?
- How much of her final balance will come from interest?
- Answer
-
- R = $1,005.45
- Total Contributions = $180,981.17
- I = $119,018.83