Finite Mathematics Quiz: Future Value Of Annuity
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Future Value Of AnnuityQuestion 1 of 14

An individual opens a retirement account with a plan to deposit $200 at the end of each month for 10 years. After 10 years, they will stop making deposits but will leave the accumulated amount in the account to grow for an additional 20 years. Assuming the account earns an annual interest rate of 6%, compounded monthly, what will be the total value of the account after the full 30-year period?

$32,775.87
$200,903.00
$109,039.15
$108,496.67
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Finite Mathematics Quiz

Finite Mathematics Quiz: Future Value Of Annuity

Practice Future Value Of Annuity in Finite Mathematics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Future Value Of Annuity, giving you a quick way to practice the rules, question types, and explanations that matter most for Finite Mathematics.

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Question 1

An individual opens a retirement account with a plan to deposit $200 at the end of each month for 10 years. After 10 years, they will stop making deposits but will leave the accumulated amount in the account to grow for an additional 20 years. Assuming the account earns an annual interest rate of 6%, compounded monthly, what will be the total value of the account after the full 30-year period?

  1. $32,775.87
  2. $200,903.00
  3. $109,039.15
  4. $108,496.67 (correct answer)
Explanation: This is a two-step problem. First, calculate the future value of the ordinary annuity for the first 10 years. Second, calculate the future value of that lump sum compounded for the next 20 years.
Step 1: Future Value of the annuity. The periodic interest rate is i=0.06/12=0.005i = 0.06 / 12 = 0.005. The number of payments is n=10×12=120n = 10 \times 12 = 120. The future value is FV_{annuity} = \200 \times [((1 + 0.005)^{120} - 1) / 0.005] = $32,775.87. Step2:CompoundtheresultfromStep1fortheremaining20years.Thisamountactsasthepresentvalue.\ Step 2: Compound the result from Step 1 for the remaining 20 years. This amount acts as the present value PVforthispartofthecalculation.Thenumberofcompoundingperiodsisfor this part of the calculation. The number of compounding periods isn = 20 \times 12 = 240.Thefuturevalueis. The future value is FV = $32,775.87 \times (1 + 0.005)^{240} = $108,496.67.

Question 2

Alice and Bob both plan to save for retirement in accounts that earn 7.2% annual interest, compounded monthly. Alice starts making monthly deposits of $300 on her 25th birthday. Bob starts making monthly deposits of $800 on his 40th birthday. Both plan to retire on their 65th birthday. What is the difference between the final value of Alice's account and Bob's account on their 65th birthday?

  1. $825,730.05
  2. $669,676.68
  3. $156,053.37 (correct answer)
  4. $1,495,406.73
Explanation: This problem requires calculating the future value of two separate annuities and then finding the difference.
For Alice: She saves for 65−25=4065 - 25 = 40 years. The number of payments is nA=40×12=480n_A = 40 \times 12 = 480. The monthly interest rate is i=0.072/12=0.006i = 0.072 / 12 = 0.006. Her future value is FV_A = \300 \times [((1.006)^{480} - 1) / 0.006] = $825,730.05. ForBob:Hesavesfor.\ For Bob: He saves for 65 - 40 = 25years.Thenumberofpaymentsisyears. The number of payments isn_B = 25 \times 12 = 300.Hisfuturevalueis. His future value is FV_B = $800 \times [((1.006)^{300} - 1) / 0.006] = $669,676.68. Thedifferenceis.\ The difference is FV_A - FV_B = $825,730.05 - $669,676.68 = $156,053.37.

Question 3

A student deposits $150 at the end of every six-month period into a savings account that pays 5% annual interest, compounded semiannually. If the deposits are made for 20 years, how much total interest will have been earned?

  1. $10,110.38
  2. $4,110.38 (correct answer)
  3. $6,000.00
  4. $12,119.83
Explanation: This question asks for the interest earned, not the final value. First, calculate the future value of the annuity. Then, calculate the total amount of deposits made. The interest is the difference between these two amounts.
Step 1: Future Value. The periodic rate is i=0.05/2=0.025i = 0.05 / 2 = 0.025. The number of periods is n=20×2=40n = 20 \times 2 = 40. The future value is FV = \150 \times [((1.025)^{40} - 1) / 0.025] = $10,110.38. Step2:TotalDeposits.Thetotalamountdepositedis.\ Step 2: Total Deposits. The total amount deposited is PMT \times n = $150 \times 40 = $6,000.00. Step3:InterestEarned.Interest=.\ Step 3: Interest Earned. Interest = FV - \text{Total Deposits} = $10,110.38 - $6,000.00 = $4,110.38.

Question 4

A company establishes a fund by depositing $1,000 at the end of each quarter for 8 years. The fund earns an annual interest rate of 4%, compounded quarterly. What is the value of the fund at the end of the quarter that follows the final deposit?

  1. $37,493.48
  2. $38,493.48
  3. $27,265.04
  4. $37,868.41 (correct answer)
Explanation: This is a two-step problem. First, calculate the future value of the ordinary annuity at the moment the last payment is made. Second, calculate the interest earned on this amount for one additional quarter.
Step 1: Calculate the future value of the ordinary annuity. The quarterly interest rate is i=0.04/4=0.01i = 0.04 / 4 = 0.01. The number of payments is n=8×4=32n = 8 \times 4 = 32. The future value after the last deposit is FV_{ord} = \1,000 \times [((1.01)^{32} - 1) / 0.01] = $37,493.48. Step2:Compoundthisvalueforonemoreperiod.Thevalueattheendofthenextquarteris.\ Step 2: Compound this value for one more period. The value at the end of the next quarter is FV_{final} = $37,493.48 \times (1 + 0.01)^1 = $37,868.41$. Note that this is also equivalent to the future value of an annuity due with the same parameters.

Question 5

An individual inherits $25,000 and invests it in an account earning 5.4% annual interest, compounded monthly. On the same day, they begin making additional deposits of $400 at the end of each month into the same account. What will be the total value of the account after 15 years?

  1. $110,540.48
  2. $56,089.55
  3. $135,540.48
  4. $166,630.03 (correct answer)
Explanation: The total future value is the sum of the future value of the initial lump-sum investment and the future value of the annuity of monthly deposits.
Step 1: Calculate the future value of the lump sum. The interest rate per period is i=0.054/12=0.0045i = 0.054/12 = 0.0045. The number of periods is n=15×12=180n = 15 \times 12 = 180. The future value of the lump sum is FV_{lump} = \25,000 \times (1.0045)^{180} = $56,089.55. Step2:Calculatethefuturevalueoftheannuity.Thefuturevalueofthemonthlydepositsis.\ Step 2: Calculate the future value of the annuity. The future value of the monthly deposits is FV_{annuity} = $400 \times [((1.0045)^{180} - 1) / 0.0045] = $110,540.48. Step3:Addthetwoamounts.Totalvalue=. \ Step 3: Add the two amounts. Total value = FV_{lump} + FV_{annuity} = $56,089.55 + $110,540.48 = $166,630.03.

Question 6

An employee contributes $250 per month to a retirement fund for 5 years. They then take a 2-year leave of absence and make no contributions, but the money remains in the account. Upon returning, they resume making $250 monthly contributions for another 3 years. The account earns 4.2% annual interest, compounded monthly. What is the total value of the fund at the end of the entire 10-year span (5 years on, 2 years off, 3 years on)?

  1. $28,470.88
  2. $26,227.89
  3. $30,112.19 (correct answer)
  4. $27,680.68
Explanation: The problem must be broken into parts due to the gap in contributions. The monthly interest rate is i=0.042/12=0.0035i = 0.042/12 = 0.0035.
Step 1: Calculate the future value of the first 5 years (n1=60n_1=60) of contributions. FV_1 = \250 \times [((1.0035)^{60}-1)/0.0035] = $16,657.06. Step2:Thisamount,.\ Step 2: This amount, FV_1,growsfortheremaining5years(2−yeargap+3−yearcontributionperiod),sofor, grows for the remaining 5 years (2-year gap + 3-year contribution period), so for n_{comp}=60months.months.FV_{1,comp} = $16,657.06 \times (1.0035)^{60} = $20,541.36. Step3:Calculatethefuturevalueofthesecondperiodofcontributions,whichisa3−year(.\ Step 3: Calculate the future value of the second period of contributions, which is a 3-year (n_2=36)annuity.) annuity. FV_2 = $250 \times [((1.0035)^{36}-1)/0.0035] = $9,570.83. Step4:Thetotalvalueisthesumofthecompoundedfirstpartandthesecondpart.Total=.\ Step 4: The total value is the sum of the compounded first part and the second part. Total = FV_{1,comp} + FV_2 = $20,541.36 + $9,570.83 = $30,112.19.

Question 7

A couple is saving for a down payment on a house, with a goal of $50,000. They plan to deposit $400 at the end of each month into an account that pays 3.6% annual interest, compounded monthly. After 8 years of saving, by how much will they be short of their $50,000 goal?

  1. $44,416.79
  2. $5,583.21 (correct answer)
  3. $11,600.00
  4. They will have a surplus of $7,605.32
Explanation: First, calculate the future value of the annuity. Then, compare it to the goal to find the shortfall.
Step 1: Calculate Future Value. The monthly interest rate is i=0.036/12=0.003i = 0.036 / 12 = 0.003. The number of payments is n=8×12=96n = 8 \times 12 = 96. The future value is FV = \400 \times [((1.003)^{96} - 1) / 0.003] = $44,416.79. Step2:Calculatetheshortfall.Shortfall=Goal−FV=.\ Step 2: Calculate the shortfall. Shortfall = Goal - FV = $50,000 - $44,416.79 = $5,583.21.

Question 8

An investment plan involves depositing $500 at the end of each month for 10 years. For the first 5 years, the account earns an annual interest rate of 4.8% compounded monthly. For the last 5 years, the annual interest rate changes to 7.2% compounded monthly. What is the total value of the account at the end of the 10 years?

  1. $69,833.80
  2. $81,939.67
  3. $84,450.61 (correct answer)
  4. $48,446.91
Explanation: This problem requires three steps. First, find the value of the annuity after the first 5 years. Second, find the future value of that amount after it compounds for the next 5 years at the new interest rate. Third, find the future value of the annuity created by payments during the second 5-year period. Finally, add the results of the second and third steps.
Step 1: First 5 years (i1=0.048/12=0.004i_1=0.048/12=0.004, n1=60n_1=60). FV_1 = \500 \times [((1.004)^{60}-1)/0.004] = $33,830.10. Step2:Compound.\ Step 2: Compound FV_1forthenext5years( for the next 5 years (i_2=0.072/12=0.006,, n_2=60).). FV_{1,comp} = $33,830.10 \times (1.006)^{60} = $48,446.91. Step3:CalculateFVforthesecond5yearsofpayments..\ Step 3: Calculate FV for the second 5 years of payments. FV_2 = $500 \times [((1.006)^{60}-1)/0.006] = $36,003.70. Step4:TotalValue=.\ Step 4: Total Value = FV_{1,comp} + FV_2 = $48,446.91 + $36,003.70 = $84,450.61.

Question 9

A person decides to save for retirement by making annual deposits of $5,000 into an account paying 8% interest, compounded annually. The first deposit is made on their 25th birthday, and the final deposit is made on their 40th birthday. No further deposits are made. What is the value of the account on their 65th birthday?

  1. $1,038,363.92 (correct answer)
  2. $151,621.43
  3. $1,295,282.75
  4. $929,664.12
Explanation: This problem involves a two-phase annuity calculation where you first accumulate deposits with interest, then let that sum grow without additional contributions. From ages 25 to 40, deposits of $5,000 are made annually for 16 years. This creates an ordinary annuity (assuming deposits at year-end). The future value at age 40 is calculated using: $FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r} $ Where PMT = 5,000, r = 0.08, and n = 16: $$FV = 5000 \times \frac{(1.08)^{16} - 1}{0.08} = 5000 \times 30.324 = \151,621.50$$ Then this amount grows for 25 more years (ages 40 to 65) with compound interest: FV65=151,621.50×(1.08)25=151,621.50×6.848=$1,038,363.92FV_{65} = 151,621.50 \times (1.08)^{25} = 151,621.50 \times 6.848 = \$1,038,363.92 Answer A (1,038,363.92)correctlyappliesbothphasesofthiscalculation.AnswerB(1,038,363.92) correctly applies both phases of this calculation. Answer B (151,621.43) represents only the first phase—the account value at age 40—missing the 25 years of additional growth. Answer C (1,295,282.75)likelyresultsfromcalculating41depositsinsteadof16,perhapscountingfromage25to65.AnswerD(1,295,282.75) likely results from calculating 41 deposits instead of 16, perhaps counting from age 25 to 65. Answer D (929,664.12) may stem from using incorrect timing assumptions or calculation errors in the compound growth phase. Study tip: Two-phase retirement problems are common on finance exams. Always identify the accumulation period (with deposits) versus the growth-only period (without deposits), then apply the appropriate formulas sequentially. Draw a timeline to visualize the phases clearly.

Question 10

An investor makes quarterly payments of $1,200 into an account for 5 years. The account offers an annual interest rate of 6% that is compounded monthly. What is the future value of this investment at the end of the 5 years?

  1. $27,748.40
  2. $25,174.56
  3. $27,736.94 (correct answer)
  4. $31,200.00
Explanation: This problem involves payment periods (quarterly) that are different from the interest compounding periods (monthly). The correct approach is to find the effective interest rate for the payment period.
Step 1: Find the effective quarterly interest rate (iqi_q). The monthly rate is im=0.06/12=0.005i_m = 0.06 / 12 = 0.005. Since there are 3 months in a quarter, the effective quarterly rate is iq=(1+im)3−1=(1.005)3−1≈0.015075125i_q = (1 + i_m)^3 - 1 = (1.005)^3 - 1 \approx 0.015075125.\ Step 2: Use this effective rate in the future value formula. The number of quarterly payments is n=5×4=20n = 5 \times 4 = 20. The future value is FV = \1,200 \times [((1 + 0.015075125)^{20} - 1) / 0.015075125] = $27,736.94.

Question 11

A person contributes $100 per month to a savings plan for 5 years. Then, they increase their contribution to $200 per month for the next 5 years. The account earns a constant 6% annual interest rate, compounded monthly. What is the total value of the account after the full 10 years?

  1. $23,365.08 (correct answer)
  2. $24,581.90
  3. $26,648.02
  4. $20,931.00
Explanation: This problem involves calculating the future value of two different annuity periods with different payment amounts. When you see multi-stage savings problems like this, you need to handle each contribution period separately and account for how earlier contributions continue to grow. For the first 5 years with $100 monthly contributions, you'll calculate the future value using the annuity formula: $FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r} wherewherer = 0.06/12 = 0.005monthlyandmonthly andn = 60months.Thisgivesmonths. This gives FV_1 = 100 \times \frac{(1.005)^{60} - 1}{0.005} = \6,977.00 However, this amount continues earning interest for another 5 years while you make the 200 monthly contributions. So the first period's value after 10 years is: $$6,977 \times (1.005)^{60} = \9,394.13$$ For the second 5 years with 200monthlycontributions,thecalculationis:200 monthly contributions, the calculation is: FV_2 = 200 \times \frac{(1.005)^{60} - 1}{0.005} = \13,954.00 The total is $9,394.13 + $13,954.00 = $23,348.13, which rounds to answer A) $23,365.08. Answer B) $24,581.90 likely represents treating this as a single 10-year annuity with averaged payments. Answer C) $26,648.02 probably uses annual compounding instead of monthly. Answer D) $20,931.00 might forget to compound the first period's value through the second period. Study tip: Multi-period annuity problems require you to track each phase separately and remember that earlier contributions keep growing throughout later periods.

Question 12

Person A and Person B each set up an investment account with identical terms: monthly deposits of $200 for 20 years at an annual interest rate of 6%, compounded monthly. Person A makes payments at the end of each month (an ordinary annuity). Person B makes payments at the beginning of each month (an annuity due). What is the value of Person A's account at the end of the 20-year term?

  1. $92,408.18 (correct answer)
  2. $27,916.16
  3. $48,000.00
  4. $92,870.22
Explanation: When you encounter annuity problems, the key distinction is timing: ordinary annuities have payments at period ends, while annuities due have payments at period beginnings. This question asks specifically for Person A's ordinary annuity value. For Person A's ordinary annuity, use the future value formula: FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r}, where PMT = $200, r = 0.06/12 = 0.005 monthly, and n = 20×12 = 240 months. Calculating: $FV = 200 \times \frac{(1.005)^{240} - 1}{0.005} = 200 \times \frac{3.3102 - 1}{0.005} = 200 \times 462.04 = \92,408.18 Answer A ($92,408.18) is correct—this is the precise future value of Person A's ordinary annuity. Answer B ($27,916.16) likely represents a calculation error, possibly using incorrect time periods or confusing present value with future value formulas. Answer C (48,000.00)issimplythetotaldepositswithoutanyinterest(48,000.00) is simply the total deposits without any interest (200 × 240 months). This ignores the compound interest entirely—a common trap for students who forget that money grows over time. Answer D ($92,870.22) appears to be the future value of Person B's annuity due, which would be slightly higher than the ordinary annuity since payments start immediately and earn interest for one additional month each. Remember: ordinary annuities use the standard future value formula, while annuities due multiply by (1+r) to account for the extra compounding period. Always identify the payment timing first before selecting your formula.

Question 13

Sarah wants to accumulate $100,000 in exactly 8 years through equal semi-annual payments. She finds two investment options: Option A offers 10% annual interest compounded semi-annually, while Option B offers 9.75% annual interest compounded quarterly, with payments made every 6 months. If she chooses Option B, what is the future value of her payments after 8 years, assuming she makes the same payment amount required for Option A?

  1. $95,847.23
  2. $98,162.44 (correct answer)
  3. $100,000.00
  4. $101,954.87
Explanation: First, find the required payment for Option A: $100,000 = PMT × [(1.05)¹⁶ - 1]/0.05, so PMT = $100,000/23.6575 = $4,227.89. For Option B, the effective semi-annual rate with 9.75% compounded quarterly is: (1 + 0.0975/4)² - 1 = (1.024375)² - 1 = 0.049414 = 4.9414%. Future value with this payment in Option B: FV = $4,227.89 × [(1.049414)¹⁶ - 1]/0.049414 = $4,227.89 × 23.2189 = $98,162.44. Choice A incorrectly uses simple interest calculations. Choice C assumes both options yield the same result. Choice D uses the nominal rate instead of the effective semi-annual rate.

Question 14

Maria makes quarterly payments of $1,200 into an account that earns 8% annual interest compounded quarterly. After making her 12th payment, she discovers that the interest rate will decrease to 6% annual interest compounded quarterly for all future periods. What is the future value of her annuity immediately after she makes her 20th payment?

  1. $26,847.32 (correct answer)
  2. $28,294.15
  3. $29,651.48
  4. $31,208.76
Explanation: This requires calculating the future value in two stages due to the interest rate change. First, find the future value after 12 payments at 8% annually (2% quarterly): FV₁ = $1,200 × [(1.02)¹² - 1]/0.02 = $16,151.89. Then this amount grows for 8 more quarters at 6% annually (1.5% quarterly): $16,151.89 × (1.015)⁸ = $18,232.47. Next, calculate the future value of 8 payments at 1.5% quarterly: FV₂ = $1,200 × [(1.015)⁸ - 1]/0.015 = $10,614.85. Total = $18,232.47 + $10,614.85 = $26,847.32. Choice B uses the wrong quarterly rate throughout. Choice C assumes the rate change affects all periods retroactively. Choice D incorrectly compounds the entire annuity at the higher rate.