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

A new car is purchased for 10000 euros. The value of the car depreciates at a nominal annual rate of 10%, with the depreciation calculated semi-annually.

The value of the car after two years can be written as $10000(1 - k)^4$.

(a) Write down the value of $k$.

(b) Expand and simplify $(1 - x)^4$.

(c) Hence or otherwise, find the value of the car after two years, giving your answer correct to the nearest euro.

A research organization purchases a new deep-sea submersible for 250,000 Australian Dollars (AUD).

The value of the submersible depreciates at a nominal annual rate of 8%, compounded semi-annually.

(a) In this question, give all answers correct to two decimal places.

(i) Calculate the value of the submersible after 6 years.

(ii) Find the total amount of depreciation of the submersible after 6 years.

(b) The company's insurance policy covers the full amount of depreciation. The policy pays out 51,737.66 British Pounds (GBP). The exchange rate is 1 AUD = $x$ GBP. Calculate the value of $x$.

The endowment fund of a university has an initial capital of $150\,000$. The fund invests this amount in a conservative bond fund that offers an interest rate of 6.5% per annum compounded annually.

(a) (i) Calculate the value of this investment after 8 years, giving your answer to the nearest hundred dollars.

(a) (ii) Determine the minimum number of years required for the investment to reach $300\,000$.

(b) The university considers investing in a more aggressive equity fund. Determine the minimum annual interest rate, $r\%$, compounded quarterly, required for the initial $150\,000$ investment to reach $300\,000$ after 7 years. Give your answer to two decimal places.

(c) A generous donor pledges to contribute to the endowment fund annually. The first contribution is $50\,000$, and each subsequent contribution is $\frac{3}{4}$ of the previous year's contribution.

(i) Show that the total amount contributed by the donor will never reach $250\,000$.

(ii) Find the amount the donor would need to contribute initially for the total contributions to reach $180\,000$ after 6 years. Give your answer to the nearest dollar.

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

Anya invests $2500 in a long-term savings bond that pays a nominal annual rate of interest of 3.2%, compounded quarterly. Anya makes no further payments to, or withdrawals from, this bond.

Find the amount that Anya will have in her account after 8 years.

Ben also invests $2500 in a different savings account that pays an annual rate of interest of r%, compounded yearly. Ben makes no further payments or withdrawals from this account.

(b) Find the value of r required so that the amount in Ben's account after 8 years will be equal to the amount in Anya's account.

(c) Find the total interest Ben will earn over the 8 years.

A rare vintage comic book, 'The Cosmic Crusader #1', was valued at $8000 on January 1st 2015. Its value is projected to increase by 3.5% on January 1st each year.

Find the projected value of 'The Cosmic Crusader #1' for the year 2025, to the nearest dollar.

Another rare comic book, 'Galactic Guardian #1', has had its value tracked over several years. The values for various years are shown in the following table.

Assuming 'Galactic Guardian #1''s annual value can be approximately modelled by the equation $V = ax + b$, use your GDC to show that 'Galactic Guardian #1' is projected to have a higher value than 'The Cosmic Crusader #1' in the year 2024, according to the model.

A rare collectible coin was purchased for $\$7200$. Its value is expected to appreciate at an annual rate of $3.2\%$ compounded yearly.

(a) Determine the value of the coin after $6$ years.

(b) Calculate, to the nearest year, how long it will take for the coin's value to reach $\$15000$.

A school is organizing a field trip to the local science museum. The cost to rent a bus for the day is $\$350$.

Additionally, there is an entry fee of $\$12.75$ per student.

(a) Write down a formula connecting the total cost of the trip ($T$) with the number of students attending ($s$).

(b) Explain why $T = f(s)$ is a function.

(c) Derive an expression for $s$ in terms of $T$.

The school has a maximum budget of $\$900$ for the field trip.

(d) Hence, calculate the greatest number of students that can attend the trip.

(e) Given that only $20$ students attend the trip, calculate how much each student should be charged so that the school covers its costs.

(a) A cutting-edge drone is purchased by a photography studio for $\$45000$. The value of the drone depreciates at an annual rate of $12\%$.

Find the value of the drone after eight years. Give your answer to two decimal places.

(b) An individual invests $\$80000$ into a retirement fund that offers a compound interest rate of $1.2\%$ per month.

Over the same period, the average inflation rate is $0.5\%$ per month.

Find the number of months required for the real value of the investment to first exceed $\$95000$.

At the start of 2024, Mateo receives a gift of $\$12\,000$. He wants to buy a vintage car which costs $\$22\,000$ so he decides to invest his money.

On 1 January 2024 Mateo invests his money in a bank account which pays interest at a nominal annual rate of $5.2\%$, compounded monthly. The interest is paid into his account on the last day of each month. He makes no further deposits to, or withdrawals from, the account.

Find the amount of money Mateo will have in his bank account on 1 January 2029. Give your answer correct to the nearest dollar.

Show that Mateo will first have more than $\$22\,000$ in his bank account during the year 2035.

The cost of the vintage car at the start of 2024 was $\$22\,000$, and it depreciates at a constant annual rate of $r\%$. After one year the cost of the car is $\$20\,790$.

Find the value of $r$.

Due to this depreciation, Mateo will be able to buy the car before 2035. He will buy the car as soon as he has enough money in his bank account to pay for it.

Determine the year during which Mateo will buy the car.

An investor, Mr. Chen, started an investment portfolio on January 1st, 2015, with an initial capital of $\$60000$. His portfolio's value is expected to increase by $3\%$ on January 1st each year after 2015. Calculate the value of Mr. Chen's investment portfolio for the year 2025, to the nearest dollar.

Another investor, Ms. Davies, started a different investment portfolio. Her portfolio's value for several years is shown in the following table:

Assuming Ms. Davies' investment value can be approximately modelled by the equation $S=ax+b$, show that Ms. Davies' investment portfolio had a higher value than Mr. Chen's in the year 2025, according to the model.

A tech startup buys a high-end 3D printer for 125,000 US Dollars (USD).

The value of the printer depreciates at an annual rate of 20%.

(a) (i) Find the value of the 3D printer after 3 years.

(ii) Find the total amount of depreciation of the printer after 3 years.

(b) The startup sells the printer after 3 years to a European company for 32,000 Euros (EUR). The exchange rate is 1 USD = $y$ EUR. Calculate the value of $y$.

A company purchases a new piece of specialized printing equipment for 160,000 USD.

The value of the equipment depreciates at a nominal annual rate of 20%, compounded semi-annually.

(a) (i) Find the value of the equipment after 2 years.

(ii) Find the total amount of depreciation of the equipment after 2 years.

(b) The company's insurance policy covers the full amount of depreciation. The policy pays out 27,512 Canadian Dollars (CAD). The exchange rate is 1 USD = $x$ CAD. Calculate the value of $x$.