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Topic 1.04 · SL and HL

Financial Applications (compound interest, annual depreciation): notes and practice questions

13 min read Jump to 20 practice questions
Summary
  • Simple Interest: Interest only on initial investment (principal).
  • Compound Interest: Interest on initial investment and accumulated interest; exponential growth.
  • Depreciation: Asset value falls over time, typically at a constant percentage rate (compound depreciation).
  • Amortisation: Repaying a loan over time with regular payments.
  • Annuity: Fixed sum paid at intervals from an initial lump sum investment.
  • Compound Interest Formula:

FV=PV×(1+r100k)knFV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}
where FVFV = future value, PVPV = present value, nn = years, kk = compounding periods/year, rr = nominal annual rate (%).

  • Compound Depreciation Formula:

FV=PV×(1−r100)nFV = PV \times \left(1 - \frac{r}{100}\right)^n
where FVFV = future value, PVPV = present value, nn = years, rr = depreciation rate (%).

  • GDC TVM Solver Variables:
  • N: Total payment periods (n×kn \times k).
  • I%: Annual interest rate (as percentage).
  • PV: Present value (negative if investing/paying out, positive if receiving loan).
  • PMT: Payment amount per period (negative if repaying, positive if receiving annuity).
  • FV: Future value (0 for fully repaid loan/exhausted annuity).
  • P/Y: Payments per year.
  • C/Y: Compounding periods per year.
  • PMT@: Payment timing (END for loans, START for annuities, unless specified).
  • GDC Tip for Depreciation: Use a negative interest rate for I% in the Finance Solver.

How it is examined

Very common, and almost always a technology question. The GDC's finance solver is the expected route, so a question that would be easy by hand is not automatically easy to mark: students report their solver inputs inconsistently. Two things make a question invalid here: asking for a derivation of the compound interest formula, and using a compounding period other than yearly, half-yearly, quarterly or monthly. Real value with inflation is fair game and catches students out.

Given in the booklet

Compound interest, FV=PV×(1+r100k)knFV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}, where FVFV is the future value, PVPV the present value, nn the number of years, kk the number of compounding periods per year and r%r\% the nominal annual rate of interest.

Key ideas
  • Compound interest.
  • Annual depreciation.
Not assessed
  • In examinations, questions that ask students to derive the formula will not be set.
  • Enrichment only, so not examinable: introducing ee through continuous compounding, (1+1n)n→e\left(1 + \frac{1}{n}\right)^{n} \to e as n→∞n \to \infty.

Linking questions

  • Other contexts: loans.
  • Links to other subjects: loans and repayments (economics and business management).
  • Aim 8: ethical perceptions of borrowing and lending money.
  • International-mindedness: do all societies view investment and interest the same way?
  • TOK: how have technological advances changed the nature and practice of mathematics? Consider the use of financial packages.

Practice questions

23 questions · 1 easy · 19 medium · 3 hard
Showing 20 of 20

Question 1

EasyPaper 2 · calculator6 marks
(a)

(a) A rare collectible coin was purchased for $12000\$12000. At the end of the first year, its value appreciated by 2.5%2.5\%.

Find the value of the coin at the end of the first year.

[3]
(b)

(b) The following year, the coin's value appreciated by 2.8%2.8\%.

Find the value of the coin at the end of the second year.

[3]

Question 2

MediumPaper 1 · calculator6 marks
(a)

A small business owner, Ms. Chen, invests 8000 CHF into a long-term growth fund for her company's expansion. The fund offers a nominal annual interest rate of 3.2%, compounded monthly.

Calculate the total amount Ms. Chen will have in her fund after 7 years. Give your answer correct to 2 decimal places.

[3]
(b)

Another entrepreneur, Mr. Davies, wants to invest 12000 EUR such that his investment will grow to 1.75 times its initial amount in 10 years. Assume the investment account pays a nominal annual interest of rr% compounded quarterly.

Determine the value of rr.

[3]

Question 3

HardPaper 2 · calculator21 marks
(a)

Maya is planning for her retirement and decides to invest in two different funds, Fund A and Fund B, both starting with an initial deposit of 1200012000 AUD.

For Fund A, Maya makes annual contributions. She contributes 800800 AUD at the end of the first year, 870870 AUD at the end of the second year, 940940 AUD at the end of the third year, and so on. The amount of contribution continues to increase each year, following an arithmetic sequence.

Find the common difference of the annual contributions.

[2]
(b)

Find the amount of contribution, in AUD, Maya makes at the end of the 10th year.

[3]
(c)

Show that the total amount of money in Fund A after nn years may be expressed as 12000+n2(1530+70n)12000+\frac{n}{2}(1530+70n).

[3]
(d)

Hence or otherwise, find the total amount of money in Fund A at the end of 15 years.

[2]
(e)

Fund B also starts with an initial deposit of 1200012000 AUD. It pays an annual interest rate of 5.5%5.5\% compounded annually. The amount in Fund B after nn years can be expressed as 12000×Bn12000 \times B^n where B∈RB\in \mathbb{R}.

Write down the value of BB.

[1]
(f)

Hence or otherwise, show that Fund A will have more money than Fund B at the end of 15 years.

[4]
(g)

The client is interested in a longer-term investment. Maya finds that it will take at least mm complete years for the amount in Fund B to exceed the amount in Fund A.

Find the value of mm.

[3]
(h)

Determine the total interest added to Fund B at the end of mm years.

Give your answer correct to the nearest dollar.

[3]

Question 4

MediumPaper 1 · calculator6 marks
(a)

A car enthusiast, Mr. Henderson, purchases two classic cars for his collection. The first is a popular sedan, initially costing 28000,whichisexpectedtodepreciateatarateof828000, which is expected to depreciate at a rate of 8% per year. The second is a rare sports coupe, purchased for 75000, and is expected to depreciate at a rate of 15% per year.

(a) Estimate the value of the popular sedan after 4 years.

[2]
(b)

(b) Find the number of years, kk, after which both cars will have the same estimated value. Give your answer to three significant figures.

[3]
(c)

(c) Comment on the validity of your answer to part (b).

[1]

Question 5

HardPaper 2 · calculator18 marks
(a)(i)

(a) Swift Deliveries, a small logistics company, plans to purchase a new fleet of electric delivery vans. The total cost of the fleet is 450 000450\,000 USD. They secure a loan from a bank, but must make an initial down payment equal to 20%20\% of the total cost. The bank offers a 1515-year loan for the remaining balance, with a 5.5%5.5\% nominal interest rate per annum, compounded monthly. Swift Deliveries will make fixed payments at the end of each month.

(a.i) Calculate the initial loan amount Swift Deliveries will need to take from the bank.

[2]
(a)(ii)

(a.ii) Calculate Swift Deliveries' monthly payment for this loan, to two decimal places.

[5]
(b)

(b) Using your answer from part (a)(ii), calculate the total amount Swift Deliveries will pay over the life of the loan, to the nearest dollar. Do not include the initial down payment.

[2]
(c)

(c) Swift Deliveries decides to repay the loan faster by increasing their monthly payments to 32003200 USD. Find the total number of monthly payments they will need to make to pay off the loan.

[2]
(d)

(d) This strategy will result in Swift Deliveries' final payment being less than 32003200 USD. Determine the amount of Swift Deliveries' final payment, to two decimal places.

[4]
(e)

(e) Hence, determine the total amount Swift Deliveries will save, to the nearest dollar, by making the higher monthly payments.

[3]

Question 6

MediumPaper 1 · calculator6 marks
(a)

[Maximum mark: 6]

A small business invests $12000 in a savings account. The account offers an annual interest rate of 3.5% compounded monthly.

(a) Calculate the total amount of money the business will have in the account at the end of 8 years. Give your answer correct to two decimal places.

[3]
(b)

The same business purchases a new delivery van for 35000.After6years,thevanisestimatedtobeworth35000. After 6 years, the van is estimated to be worth 8000. Assuming a constant annual depreciation rate, calculate this rate. Give your answer as a percentage correct to two decimal places.

[3]

Question 7

HardPaper 2 · calculator18 marks
(a)(i)

In this question, give all answers to two decimal places.

A small business owner, Ms. Chen, decides to purchase new equipment for her bakery, priced at $22000. She cannot afford the full amount upfront.

The equipment supplier offers two options to finance a loan.

Finance option A:

A 5-year loan at a nominal annual interest rate of 10% compounded quarterly.

No deposit required and repayments are made each quarter.

Find the repayment made each quarter.

[3]
(a)(ii)

Find the total amount paid for the equipment.

[2]
(a)(iii)

Find the interest paid on the loan.

[2]
(b)(i)

Finance option B:

A 5-year loan at a nominal annual interest rate of rr % compounded monthly. Terms of the loan require a 10% deposit and monthly repayments of $350.

Find the amount to be borrowed for this option.

[2]
(b)(ii)

Find the annual interest rate, rr.

[3]
(c)

State which option Ms. Chen should choose. Justify your answer.

[2]
(d)

Ms. Chen chooses option B. The equipment supplier invests the money Ms. Chen pays as soon as they receive it.

If they invest it in an account paying 0.5% interest per month and inflation is 0.15% per month, calculate the real amount of money the equipment supplier has received by the end of the 5-year period.

[4]

Question 8

MediumPaper 1 · calculator5 marks

Ethan invested a certain amount in a startup that promised simple interest. He noted that his total investment value had increased by 10.5%10.5\% after 77 years. He also calculated that he earned $4200\$4200 in interest during this period. Determine Ethan's initial investment and the annual simple interest rate.

Question 9

MediumPaper 1 · calculator4 marks

(a) A small business owner decides to set up a savings plan to upgrade equipment in the future. They deposit 12001200 at the beginning of each year into an account that earns 3.5%3.5\% per annum, compounded annually.

Calculate the total value of these investments at the end of the eighth year.

Question 10

MediumPaper 1 · calculator3 marks

(a) A technology enthusiast purchased a new high-end drone for $1200\$1200. Due to rapid technological advancements and wear, the value of the drone depreciates by 15%15\% each year. Find the value of the drone at the end of four years.

Question 11

MediumPaper 1 · calculator5 marks

(a) Anya invested 80008000 euros in a savings account that paid interest per annum compounded semi-annually. After 55 years, her investment grew to 9658.269658.26 euros. Find the annual interest rate, expressed as a percentage to one decimal place.

Question 12

MediumPaper 2 · calculator8 marks
(a)

A collector purchased a vintage watch. At the beginning of the fourth year of ownership, the watch was valued at 25194.2425194.24 USD. At the beginning of the seventh year, its value had increased to 31737.4931737.49 USD. Assuming the watch's value increases by a constant annual percentage rate.

(a) Calculate the annual rate of increase as a percentage.

[4]
(b)

(b) Calculate how much the collector originally paid for the watch.

[2]
(c)

(c) Find the value of the watch at the beginning of the tenth year.

[2]

Question 13

MediumPaper 1 · calculator7 marks
(a)

A technology startup invested USD 40004000 in a new project that promises a return of 3.2%3.2\% interest per year compounded quarterly.

(a) Find the total value of the investment after eight years.

[3]
(b)

(b) Find the number of years it would take for the investment to reach USD 70007000.

[4]

Question 14

MediumPaper 1 · calculator7 marks

(a) A manufacturing company purchases two new machines for its production line. Machine X costs $32000\$32000 and depreciates at a constant rate of $1500\$1500 per year. Machine Y costs $20000\$20000 and depreciates at a constant rate of $700\$700 per year.

Use technology to determine how many years it will take for both machines to have the same value. Find the value of their machines at this time.

Question 15

MediumPaper 1 · calculator4 marks

(a) A manufacturing company purchased a new specialized machine for $25 000\$25\,000. The value of the machine is expected to depreciate by 15%15\% each year.

Calculate the value of the machine at the end of four years.

Question 16

MediumPaper 2 · calculator6 marks
(a)

(a) A rare vintage car was purchased for £12000\text{£}12000. At the end of the first year, its value appreciated by 3.5%3.5\%.

Find the value of the car at the end of the first year.

[3]
(b)

(b) At the end of the second year, the car's value appreciated by a further 3.2%3.2\%.

Find the value of the car at the end of the second year.

[3]

Question 17

MediumPaper 1 · calculator16 marks
(a)(i)

Elena wants to buy a new electric scooter for 35003500 US dollars (USD). Elena is offered two different loan options to pay for the scooter. For both options, Elena will pay back the loan over a period of 44 years.

The two options are shown in the following table.

Loan OptionSize of loan (% purchase price)Nominal annual interest rate (compounded monthly)
A100%8.2%
B85%7.5%

For both loan options, Elena will make monthly payments, at the end of each month, until the loan is completely paid off.

Calculate the amount of Elena's monthly payment, to two decimal places, if she chooses

option A.

[6]
(a)(ii)

option B.

[6]
(b)(i)

State a reason why Elena might choose

option A.

[2]
(b)(ii)

option B.

[2]

Question 18

MediumPaper 1 · calculator10 marks
(a)(i)

(a) A travel agency, GlobalVoyages, offers various package deals. For flight packages priced over $1000, they apply a 15% discount.

Another travel agency, WorldExplorer, offers a fixed reduction of 200onallhotelpackagespricedover200 on all hotel packages priced over 1000.

Given that xx is the original price of a travel package and x>1000x > 1000,

(i) write down a function f(x)f(x) that models the final price of a flight package at GlobalVoyages, after the discount is applied.

[2]
(a)(ii)

(ii) write down a function g(x)g(x) that models the final price of a hotel package at WorldExplorer, after the reduction is applied.

[2]
(b)

(b) A third agency, GrandTour, decides to offer both a 15% discount and a 200reductiononalltheirtravelpackagespricedover200 reduction on all their travel packages priced over 1000.

Describe the meaning of (f∘g)(x)(f \circ g)(x), in context.

[2]
(c)

(c) State, with justification, whether (f∘g)(x)(f \circ g)(x) or (g∘f)(x)(g \circ f)(x) is better for the customer when they buy a travel package with an original price of $1800.

[4]

Question 19

MediumPaper 1 · calculator7 marks
(a)

Give answers to this question correct to two decimal places.

Aisha invests 200200 USD at the end of each month for 88 years into a college savings plan that pays a nominal annual interest rate of 4.2%4.2\% compounded monthly.

(a) Calculate the value of Aisha's savings plan at the end of the 88 years.

[3]
(b)

At the end of the 88 years, Aisha withdraws 50005000 USD from the college savings plan to cover immediate expenses.

Aisha invests the remainder into another account for 1212 years at a nominal annual interest rate of 5.5%5.5\% compounded quarterly.

(b) Calculate the amount in Aisha's account at the end of this time.

[4]

Question 20

MediumPaper 2 · calculator19 marks
(a)(i)

GreenGrow company designs modular garden beds. Each garden bed is constructed from individual square panels. The company offers different configurations, where adding more 'sections' increases both the length and width of the garden bed. When NN sections are added, the garden bed has a total length of LNL_N centimetres and a total width of WNW_N centimetres.

The following table shows the values of LNL_N and WNW_N for the first three values of NN.

Number of sections, NNLength of garden bed, LNL_N (cm)Width of garden bed, WNW_N (cm)
16040
2aabb
310060

(a.i) Find the value of aa.

[1]
(a)(ii)

(a.ii) Find the value of bb.

[1]
(b)(i)

(b.i) Write down an expression in terms of NN for LNL_N.

[2]
(b)(ii)

(b.ii) Write down an expression in terms of NN for WNW_N.

[1]
(c)(i)

(c.i) A customer wants a garden bed with a total length of 440 cm. Each section of the garden bed requires 6 individual panels. Show that the customer needs 120 panels.

[2]
(c)(ii)

(c.ii) Find the value of WNW_N for this garden bed.

[1]
(d)

(d) Each individual panel is a square with side length 25 cm. Find the total area of the panels in the garden bed. Give your answer in the form a×10ka \times 10^k where 1≤a<101 \le a < 10 and kk is an integer.

[3]
(e)

(e) The panels cost $18.00 per square metre and are sold in packs of 10 panels.

Find the cost of a single pack of 10 panels.

[3]
(f)

(f) To allow for damaged panels, the customer wants at least 15% more panels than needed.

Find the minimum number of packs of panels the customer will need to order.

[3]
(g)

(g) There is a fixed delivery cost of $40.

Find the total cost for the customer's order.

[2]

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What does Financial Applications (compound interest, annual depreciation) cover in IB Maths AI?

Simple Interest: Interest only on initial investment (principal). Compound Interest: Interest on initial investment and accumulated interest; exponential growth. Depreciation: Asset value falls over time, typically at a constant percentage rate (compound depreciation).

Is Financial Applications (compound interest, annual depreciation) SL or HL?

Both. SL and HL students study Financial Applications (compound interest, annual depreciation) to the same depth.

How do I revise Financial Applications (compound interest, annual depreciation) for IB Maths AI?

Start from the core idea: simple Interest: Interest only on initial investment (principal). In the exam: very common, and almost always a technology question. The GDC's finance solver is the expected route, so a question that would be easy by hand is not automatically easy to mark: students report their solver inputs inconsistently. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Financial Applications (compound interest, annual depreciation)?

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