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IB Mathematics AI HL - Questionbank

Financial Mathematics

Compound Interest, Depreciation, Loans & Amortization, Annuities, Using Finance Solver on the Calculator...

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Question 1

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easy

[Maximum mark: 6]

Jeremy invests $8000\$8000 into a savings account that pays an annual interest rate of 5.55.5 %, compounded annually.

  1. Write down a formula which calculates that total value of the investment after nn years. [2]

  2. Calculate the amount of money in the savings account after:

    1. 11 year;

    2. 33 years. [2]

  3. Jeremy wants to use the money to put down a $10000\$10\hspace{0.15em}000 deposit on an apartment. Determine if Jeremy will be able to do this within a 55-year timeframe.[2]

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Question 2

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easy

[Maximum mark: 6]

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

Mia deposits 40004000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 66 %, compounded semi-annually.

  1. Find the amount of interest that Mia will earn over the next 2.52.5 years. [3]

Ella also deposits AUD into a bank account. Her bank pays a nominal annual interest\text{interest} rate of 44 %, compounded monthly. In 2.52.5 years, the total amount in Ella's account will be 40004000 AUD.

  1. Find the amount that Ella deposits in the bank account. [3]

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Question 3

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easy

[Maximum mark: 6]

Hannah buys a car for $24900\$24\hspace{0.15em}900. The value of the car depreciates by 1616 % each year.

  1. Find the value of the car after 1010 years. [3]

Patrick buys a car for $12000\$12\hspace{0.15em}000. The car depreciates by a fixed amount each year, and after 66 years it is worth $6200\$6200.

  1. Find the annual rate of depreciation of the car. [3]

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Question 4

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easy

[Maximum mark: 6]

Edward wants to buy a new car, and he decides to take out a loan of 7000070\hspace{0.15em}000 Australian dollars from a bank. The loan is for 66 years, with a nominal annual interest rate of 7.2%7.2\%, compounded monthly. Edward will pay the loan in fixed monthly instalments.

  1. Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]

  2. Calculate the amount Edward will still owe the bank at the end of the third year. [3]

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Question 5

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easy

[Maximum mark: 6]

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

Elena invests in a retirement plan in which equal payments of €15001500 are made at the beginning of each year. Interest is earned on each payment at a rate of 2.492.49 % per year, compounded annually.

  1. Find the value of the investment after 2525 years. [3]

  2. Find the amount of interest Elena will earn on the investment over 2525 years.[3]

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Question 6

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easy

[Maximum mark: 6]

Maria invests $25000\$25\hspace{0.15em}000 into a savings account that pays a nominal annual interest rate of 4.254.25 %, compounded monthly.

  1. Calculate the amount of money in the savings account after 33 years. [3]

  2. Calculate the number of years it takes for the account to reach $40000\$40\hspace{0.15em}000. [3]

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Question 7

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easy

[Maximum mark: 6]

Isabella and Charlotte both receives 8000080\hspace{0.15em}000 Australian dollars (AUD) on their 1818th birthday to invest for later in their life.

Isabella deposits her 8000080\hspace{0.15em}000 AUD in a bank account that pays a nominal annual interest rate of xx %, compounded monthly.

  1. The amount in a bank account after 66 years will be 100000100\hspace{0.15em}000 AUD. Find the nominal annual interest rate. Give your answer correct to two decimal places.[3]

Charlotte uses her 8000080\hspace{0.15em}000 AUD to buy a house that increases in value at a rate of 33 % per year.

  1. Find the house price after 66 years. [3]

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Question 8

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easy

[Maximum mark: 6]

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

Charlie deposits 80008000 Canadian dollars (CAD) into a bank account. The bank pays a nominal annual interest rate of 55 %, compounded semi-monthly.

  1. Find the amount of interest that Charlie will earn over the next 22 years. [3]

Oscar also deposits CAD into a bank account. His bank pays a nominal annual interest rate of 66 %, compounded quarterly. In 22 years, the total amount in Oscar's account will be $8000\$8000 CAD.

  1. Find the amount that Oscar deposits in the bank account. [3]

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Question 9

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easy

[Maximum mark: 6]

Michael buys a second hand Tesla car for $18000\$18\hspace{0.15em}000. The value of the car depreciates by 10%10\% each year.

  1. Find the total amount the car will depreciate after 5 years, giving your answer correct to the nearest dollar. [3]

The price of a different used car depreciates by 5%5\% each year.

  1. Find the value reduction of this car after 44 years as a percentage, when compared to the original purchase price. [3]

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Question 10

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easy

[Maximum mark: 6]

Charles plans to invest in a retirement plan for 3030 years. In this plan, he will deposit 10001000 British pounds (GBP) at the end of every month and receive a 6.5%6.5\hspace{0.05em}\% interest rate per annum, compounded monthly.

  1. Find the future value of the investment at the end of the 3030 years. Give your answer correct to the nearest pound.[3]

After the 3030-year period, Charles will start receiving regular monthly payments of 15001500 GBP.

  1. Calculate the number of years it will take Charles's monthly retirement payments\text{payments} to match the total amount originally invested. [3]

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Question 11

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easy

[Maximum mark: 6]

At the beginning of each year, Jack invests $5000\$5000 in a savings account that pays 4%4\hspace{0.05em}\% annual interest, compounded quarterly

  1. Find the number of years it will take until Jack has $100000\$100\hspace{0.15em}000 in his account. [3]

At the beginning of each year, John invests $6000\$6000 in a savings account that pays an annual interest rate, compounded semi-annually. After 2020 years John has $200000\$200\hspace{0.15em}000 in his account.

  1. Find the annual interest rate. [3]

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Question 12

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easy

[Maximum mark: 6]

Mike wants to deposit part of his savings in a bank account that pays an annual interest rate of 4.1%compounded annually4.1\,\%\, \textbf{compounded annually}. The annual inflation rate is expected to be 3%3\% per year throughout the following 88 years. Mike wants his initial deposit to have a real value of $5000\$5\,000 after 88 years, compared to current values.

The bank gives Mike two proposals:

Proposal 1:A one-time investment at the start of the 8-year period.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textbf{Proposal 1:} \,\,\,\text{A one-time investment at the start of the 8-year period.}

Proposal 2:Invest $2000 at the start of the 8-year period and make payments of $ x at the end of each year.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textbf{Proposal 2:} \,\,\,\text{Invest \$2\,000 at the start of the 8-year period and make payments of \$ $x$ at the end of each year.}

  1. Find the minimum amount Mike should deposit if he accepts proposal 1. Round your answer to the nearest dollar. [3]

  2. Find the minimum value of the annual payments, xx, if Mike accepts proposal 2. Round your answer to the nearest dollar. [3]

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Question 13

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easy

[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

Benjamin spends € 3200032\hspace{0.15em}000 buying bitcoin mining hardware for his cryptocurrency mining\text{mining} business. The hardware depreciates by 1616 % each year.

  1. Find the value of the hardware after two years. [3]

  2. Find the number of years it will take for the hardware to be worth less than 8000\text{\euro\hspace{0.05em}\(8000\)}. [3]

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Question 14

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easy

[Maximum mark: 6]

Ali bought a car for $18000\$18\hspace{0.15em}000. The value of the car depreciates by 10.510.5 % each year.

  1. Find the value of the car at the end of the first year. [2]

  2. Find the value of the car after 44 years. [2]

  3. Calculate the number of years it will take for the car to be worth exactly half its original value. [2]

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Question 15

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easy

[Maximum mark: 6]

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

George invests in a retirement plan in which equal payments of $2750\$2750 are made at the end of each year. Interest is earned on each payment at a rate of 33 % per year, compounded semi-annually.

  1. Find the value of the investment after 2020 years. [3]

  2. Find the amount of interest George will earn on the investment over 2020 years.[3]

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Question 16

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easy

[Maximum mark: 6]

Alex invests an amount of USD into a savings account which pays 3.3% (p.a.) interest, compounded monthly. After 5 years Alex has 80008\hspace{0.15em}000 USD in the account.

  1. Find the amount of USD initially invested, rounding your answer to two decimal places.[3]

With this money, Alex purchases a used car for 50005\hspace{0.15em}000 dollars, and sells it 3 years later for 42004\hspace{0.15em}200.

  1. Find the rate at which the car depreciates per year over the 3 year period.[3]

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Question 17

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easy

[Maximum mark: 6]

Smith has saved $5000\$5\,000 from working a part-time job and wants to invest this money so that it grows over time. His bank offers a savings account that pays an annual interest rate of 4.2%4.2\%, compounded quarterly.

  1. Find how many years it will take for Smith's investment amount to double in value, rounding your answer to the nearest integer. [3]

Smith wants his money to grow faster than this first option. His wants to invest the money so that it will double in value in 55 years. He considers an high-growth, higher-risk option, which pays an annual interest of r%r\%, compounding half-yearly.

  1. Determine the value of rr required in this option, rounding your answer to two decimal places. [3]

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Question 18

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easy

[Maximum mark: 6]

Greg has saved 20002000 British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of 8%\text{\(8\)\hspace{0.05em}\%}, compounded monthly, for two years.

  1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.

  1. Calculate the nominal annual interest rate the second bank offers. [3]

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Question 19

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[Maximum mark: 7]

On January 1st 2023, Virgil deposits 1500 Canadian dollars (CAD) into a savings account which pays a nominal annual interest rate of 4.6%4.6\% compounded monthly. At the end of each month, Virgil deposits an extra CAD1000\,1\,000 into the savings account.

After kk months, Virgil will have enough money to withdraw CAD25000\,25\,000.

  1. Find the smallest possible value for kk, given that kk is a whole number.[4]

At this time, kk months, annual interest rates have improved. Virgil withdraws CAD25000\,25\,000 and re-invests the remaining money in the same account with the new nominal annual interest rate for 24 months, making no further deposits. After 24 months, Virgil has CAD800\,800 in the account.

  1. Determine the new nominal annual interest rate.[3]

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Question 20

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[Maximum mark: 6]

Emily deposits 20002000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 44 %, compounded monthly.

  1. Find the amount of money that Emily will have in her bank account after 55 years. Give your answer correct to two decimal places. [3]

Emily will withdraw the money back from her bank account when the amount reaches 30003000 AUD.

  1. Find the time, in full years, until Emily withdraws the money from her bank account. [3]

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Question 21

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[Maximum mark: 6]

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

Stella receives a loan of € 3200032\hspace{0.15em}000 for her flower shop business at an interest rate 5.295.29 % per year, compounded monthly. She agrees to pay back the loan in 6060 equal installments, made at the end of each month over the next five years.

  1. Calculate the amount of monthly installment. [3]

Four years after she starts repaying the loan, Stella decides to repay the rest in a final single installment.

  1. Calculate the amount owing at the end of the four years. [3]

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Question 22

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[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

Michelle takes out a loan of $12000\$12\hspace{0.15em}000. The unpaid balance on the loan has an interest rate of 4.34.3 % per year, compounded annually.

  1. The loan is to be repaid in payments of $1500\$1500 made at the end of each year.

    1. Find the number of years it will take to repay the loan.

    2. Calculate the total amount that has been paid in amortising the loan.[3]

  2. The loan is to be amortised over 55 years.

    1. Find the annual payment made at the end of each year.

    2. Calculate the total amount that has been paid in amortising the loan.[3]

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Question 23

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[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of 7400074\hspace{0.15em}000 Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of 5.55.5 %, compounded monthly.

  1. Find the amount of AUD Julia needs to invest to reach 7400074\hspace{0.15em}000 AUD after 88 years. Give your answer correct to the nearest dollar. [3]

Julia's parents offer to add 50005000 AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of 3.53.5 %, compounded quarterly.

  1. Find the number of years it would take Julia to save the 7400074\hspace{0.15em}000 AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]

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Question 24

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[Maximum mark: 6]

Olivia takes a mortgage (loan) of $250000\$250\hspace{0.15em}000 to buy an apartment in Sydney. Interest\text{Interest} on the loan accumulates at the rate of $3.49\$3.49 % per year, compounded semi-annually. Olivia agrees with the bank to amortise the loan in monthly payments, made at the beginning of each month.

  1. Given that the loan is to be amortised over 3030 years, find:

    1. the monthly payment amount;

    2. the total amount paid in amortising the loan. [4]

  2. Olivia has the capacity to increase her monthly payments by $85\$85. Justify to Olivia why this may be a smart financial choice. [2]

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Question 25

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[Maximum mark: 5]

Phil's phone shop sells Azura smartphones for $1499\$1\hspace{0.05em}499 and Bellson smartphones for $850\$850. It is expected that a Bellson smartphone will depreciate at a rate of 20%20\hspace{0.05em}\% per year.

After 2 years, an Azura smartphone is worth approximately $735\$735.

  1. Show that the expected annual depreciation rate of an Azura smartphone is 30%. [2]

An Azura smartphone and a Bellson smartphone will have the same value nn years after they were purchased.

  1. Find the value of nn. [2]

  2. Comment on the validity of your answer to part (b). [1]

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Question 26

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[Maximum mark: 14]

Bruce goes into a car dealership to purchase a new vehicle. The one he wants to buy costs $16000\$16\hspace{0.15em}000, however he doesn't have that much money in his bank. The salesman offers him a financing option of a 3030 % deposit followed by 1212 monthly payments of $1150\$1150.

  1. Find the amount of the deposit. [1]

  2. Calculate the total cost of the loan under this financing option. [2]

Bruce's father generously offers him an interest free loan of $16000\$16\hspace{0.15em}000 to buy the car to avoid the expensive loan repayments. They agree that Bruce will repay the loan by paying his father $x\$\hspace{0.05em}x in the first month and $y\$\hspace{0.05em}y every following month until the $16000\$16\hspace{0.15em}000 is repaid.

The total amount Bruce's father receives after 1212 months is $5200\$5200. This can be expressed by the equation x+11y=5200x + 11y = 5200. The total amount that Bruce's father receives after 2424 months is $10600\$10\hspace{0.15em}600.

  1. Write down a second equation involving xx and yy. [1]

  2. Determine the value of xx and the value of yy. [2]

  3. Calculate the number of months it will take Bruce's father to receive
    the $16000\$16\hspace{0.15em}000. [3]

Bruce decides to buy a cheaper car for $12000\$12\hspace{0.15em}000 and invest the remaining $4000\$4000. He is considering two investment options over four years.

Option A: Compound interest at an annual rate of 6.56.5 %.

Option B: Compound interest at a nominal annual rate of 66 %, compounded monthly.

Express each answer in part (f) to the nearest dollar.

  1. Calculate the value of each investment option after four years.

    1. Option A.

    2. Option B. [5]

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Question 27

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[Maximum mark: 5]

Tom takes out a loan of $80000\$80\hspace{0.15em}000 to purchase some new machinery for his farming business. He agrees to pay the bank $1200\$1200 at the end of every month to amortise the loan. Interest accumulates on the balance at a rate of 5.655.65 % per year, compounded monthly.

  1. Find the number of years and months it takes to pay back the loan. [2]

  2. Calculate the total amount that Tom pays in amortising the loan. [1]

  3. Tom decides to increase the monthly payment to $1500\$1500. How much interest will Tom save in comparison to the former payment schedule.[2]

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Question 28

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[Maximum mark: 19]

Nathan receives a lump sum inheritance of $55000\$ 55\,000 and invests the money into a savings account with an annual interest rate of 7.5%7.5 \%, compounded quarterly.

  1. Calculate the value of Nathan's investment after 5 years, rounding your answer to the nearest dollar. [3]

After mm months, the amount in the savings account has increased to more than $70000\$70\,000.

  1. Find the minimum value of mm, where mNm\in N.[4]

Nathan is saving to purchase a property. The price of the property is $150000\$150\,000. Nathan puts down a 15%15\% deposit and takes out a loan for the remaining amount.

  1. Write down the loan amount.[1]

The loan duration is for eight years, compounded monthly, with equal monthly payments of $1500\$1500 made by Nathan at the end of each month.

  1. For this loan, find
    1. the amount of interest paid by Nathan over the life of the loan.

    2. the annual interest rate of the loan, correct to two decimal places. [5]

After 55 years of paying this loan, Nathan decides to pay the outstanding loan amount in one final payment.

  1. Find the amount of the final payment after 55 years, rounding your answer to the nearest dollar. [3]

  2. Find the amount Nathan saved by making this final payment after 55 years, rounding your answer to the nearest dollar.[3]

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Question 29

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[Maximum mark: 7]

Ray takes out a loan of $200000\$200\hspace{0.15em}000 to purchase a house. He agrees to pay the bank $1250\$1250 at the end of every month to amortise the loan, and interest accumulates on the balance at a rate of 3.793.79 % per year, compounded monthly.

  1. Find the number of years and months it takes to pay back the loan. [2]

  2. Calculate the total amount that Ray has paid in amortising the loan. [2]

  3. Ray decides to increase the monthly payment to $1500\$1500. Justify this decision.[3]

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Question 30

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[Maximum mark: 14]

In this question, give all answers correct to the nearest whole number.

Ann is considering investing $85000\$85\hspace{0.15em}000 into a term deposit in one of two banks. The first bank offers an annual interest rate of 33 %, compounding monthly. The second bank offers a fixed payment amount of $250\$250 per month.

  1. Calculate:

    1. the amount that would be in the account in the first bank at the end of the first year;

    2. the amount that would be in the account in the second bank at the end of the first year. [4]

  2. Write down an expression for:

    1. the amount in the account in the first bank at the end of the nnth year;

    2. the amount in the account in the second bank at the end of the nnth year. [4]

  3. Calculate the year in which the amount in the first bank account becomes
    greater than the amount in the second bank. [2]

  4. Calculate:

    1. the interest that Ann would earn if she invested in the first bank for 1010 years;

    2. the interest that Ann would earn if she invested in the second bank for 1010 years. [4]

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Question 31

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[Maximum mark: 12]

Lily and Eva both receive 5000050\hspace{0.15em}000 Australian dollars (AUD) on their 1818th birthday. Lily deposits her 5000050\hspace{0.15em}000 AUD into a bank account. The bank pays an annual interest rate of 55 %, compounded yearly. Eva invests her 5000050\hspace{0.15em}000 AUD into a high-yield mutual fund that returns a fixed amount of 30003000 AUD per year.

  1. Calculate:

    1. the amount in Lily's bank account at the end of the first year;

    2. the total amount of Eva's funds at the end of the first year. [2]

  2. Write down an expression for:

    1. the amount in Lily's bank account at the end of the nnth year;

    2. the total amount of Eva's funds at the end of the nnth year. [4]

  3. Calculate the year in which the amount in Lily's bank account becomes
    greater than the amount in Eva's fund. [2]

  4. Calculate:

    1. the interest amount that Lily earns if invested for 1212 years, giving your answer correct to two decimal places;

    2. the amount of funds that Eva earns for her investment if invested for 1212 years. [4]

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Revision Village is ranked the #1 IB Math Resources by IB Students & Teachers.

34% Grade Increase

Revision Village students scored 34% greater than the IB Global Average in their exams (2021).

80% of IB Students

More and more IB students are using Revision Village to prepare for their IB Math Exams.

Frequently Asked Questions

The IB Math Applications and Interpretation (AI) HL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AI HL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AI Higher Level course.

The AI HL Questionbank is designed to help IB students practice AI HL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AI HL Question bank. For example, if you want to practice AI HL exam style questions involving Matrices, you can go to AI HL Topic 1 (Number & Algebra) and go to the Matrices area of the question bank. On this page there is a carefully designed set of IB Math AI HL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.

The AI HL Questionbank is perfect for revising a particular topic or concept, in-depth. For example, if you wanted to improve your knowledge of Complex Numbers, there is a designed set of full length IB Math AI HL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AI HL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AI HL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.

With an extensive and growing library of full length IB Math Applications and Interpretation (AI) HL exam style questions in the AI HL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AI HL Question bank, then a popular option would be to go to the AI HL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AI HL exam.