IB Mathematics AI SL - Questionbank
Financial Mathematics
Compound Interest, Depreciation, Loans & Amortization, Annuities, Using Finance Solver on the Calculator...
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Question 1
[Maximum mark: 6]
Jeremy invests into a savings account that pays an annual interest rate of %, compounded annually.
-
Write down a formula which calculates that total value of the investment after years. [2]
-
Calculate the amount of money in the savings account after:
-
year;
-
years. [2]
-
-
Jeremy wants to use the money to put down a deposit on an apartment. Determine if Jeremy will be able to do this within a -year timeframe.[2]
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Question 2
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Mia deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded semi-annually.
-
Find the amount of interest that Mia will earn over the next years. [3]
Ella also deposits AUD into a bank account. Her bank pays a nominal annual rate of %, compounded monthly. In years, the total amount in Ella's account will be AUD.
- Find the amount that Ella deposits in the bank account.
[3]
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Question 3
[Maximum mark: 6]
Hannah buys a car for . The value of the car depreciates by % each year.
-
Find the value of the car after years. [3]
Patrick buys a car for . The car depreciates by a fixed amount each year, and after years it is worth .
- Find the annual rate of depreciation of the car.
[3]
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Question 4
[Maximum mark: 6]
Edward wants to buy a new car, and he decides to take out a loan of Australian dollars from a bank. The loan is for years, with a nominal annual interest rate of , compounded monthly. Edward will pay the loan in fixed monthly instalments.
-
Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]
-
Calculate the amount Edward will still owe the bank at the end of the third year. [3]
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Question 5
[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 € are made at the beginning of each year. Interest is earned on each payment at a rate of % per year, compounded annually.
-
Find the value of the investment after years. [3]
-
Find the amount of interest Elena will earn on the investment over years.[3]
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Question 6
[Maximum mark: 6]
Maria invests into a savings account that pays a nominal annual interest rate of %, compounded monthly.
-
Calculate the amount of money in the savings account after years. [3]
-
Calculate the number of years it takes for the account to reach . [3]
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Question 7
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Charlie deposits Canadian dollars (CAD) into a bank account. The bank pays a nominal annual interest rate of %, compounded semi-monthly.
-
Find the amount of interest that Charlie will earn over the next years. [3]
Oscar also deposits CAD into a bank account. His bank pays a nominal annual interest rate of %, compounded quarterly. In years, the total amount in Oscar's account will be CAD.
- Find the amount that Oscar deposits in the bank account.
[3]
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Question 8
[Maximum mark: 6]
At the beginning of each year, Jack invests in a savings account that pays annual interest, compounded quarterly
-
Find the number of years it will take until Jack has in his account. [3]
At the beginning of each year, John invests in a savings account that pays an annual interest rate, compounded semi-annually. After years John has in his account.
- Find the annual interest rate. [3]
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Question 9
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
Benjamin spends € buying bitcoin mining hardware for his cryptocurrency business. The hardware depreciates by % each year.
-
Find the value of the hardware after two years. [3]
-
Find the number of years it will take for the hardware to be worth less than . [3]
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Question 10
[Maximum mark: 6]
Ali bought a car for . The value of the car depreciates by % each year.
-
Find the value of the car at the end of the first year. [2]
-
Find the value of the car after years. [2]
-
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 11
[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 are made at the end of each year. Interest is earned on each payment at a rate of % per year, compounded semi-annually.
-
Find the value of the investment after years. [3]
-
Find the amount of interest George will earn on the investment over years.[3]
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Question 12
[Maximum mark: 6]
Isabella and Charlotte both receives Australian dollars (AUD) on their th birthday to invest for later in their life.
Isabella deposits her AUD in a bank account that pays a nominal annual interest rate of %, compounded monthly.
-
The amount in a bank account after years will be AUD. Find the nominal annual interest rate. Give your answer correct to two decimal places.[3]
Charlotte uses her AUD to buy a house that increases in value at a rate of % per year.
- Find the house price after years. [3]
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Question 13
[Maximum mark: 6]
Michael buys a second hand Tesla car for . The value of the car depreciates by each year.
-
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 each year.
- Find the value reduction of this car after years as a
percentage, when compared to the original purchase price.
[3]
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Question 14
[Maximum mark: 6]
Charles plans to invest in a retirement plan for years. In this plan, he will deposit British pounds (GBP) at the end of every month and receive a interest rate per annum, compounded monthly.
-
Find the future value of the investment at the end of the years. Give your answer correct to the nearest pound.[3]
After the -year period, Charles will start receiving regular monthly payments of GBP.
- Calculate the number of years it will take Charles's
monthly retirement to match the total
amount originally invested. [3]
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Question 15
[Maximum mark: 6]
Mike wants to deposit part of his savings in a bank account that pays an annual interest rate of . The annual inflation rate is expected to be per year throughout the following years. Mike wants his initial deposit to have a real value of after years, compared to current values.
The bank gives Mike two proposals:
-
Find the minimum amount Mike should deposit if he accepts proposal 1. Round your answer to the nearest dollar. [3]
-
Find the minimum value of the annual payments, , if Mike accepts proposal 2. Round your answer to the nearest dollar. [3]
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Question 16
[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 USD in the account.
- Find the amount of USD initially invested, rounding your answer to
two decimal places.[3]
With this money, Alex purchases a used car for dollars, and sells it 3 years later for .
- Find the rate at which the car depreciates per year over
the 3 year period.[3]
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Question 17
[Maximum mark: 6]
Smith has saved 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 , compounded quarterly.
- 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 years. He considers an high-growth, higher-risk option, which pays an annual interest of , compounding half-yearly.
- Determine the value of required in this option, rounding your
answer to two decimal places. [3]
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Question 18
[Maximum mark: 6]
Greg has saved 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 , compounded monthly, for two years.
-
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.
- Calculate the nominal annual interest rate the second bank
offers. [3]
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Question 19
[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 compounded monthly. At the end of each month, Virgil deposits an extra CAD into the savings account.
After months, Virgil will have enough money to withdraw CAD.
- Find the smallest possible value for , given that is a whole number.[4]
At this time, months, annual interest rates have improved. Virgil withdraws CAD 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 CAD in the account.
- Determine the new nominal annual interest rate.[3]
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Question 20
[Maximum mark: 6]
Emily deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of money that Emily will have in her bank account after years. Give your answer correct to two decimal places. [3]
Emily will withdraw the money back from her bank account when the amount reaches AUD.
- Find the time, in full years, until Emily withdraws the
money from her bank account. [3]
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Question 21
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Stella receives a loan of € for her flower shop business at an interest rate % per year, compounded monthly. She agrees to pay back the loan in equal installments, made at the end of each month over the next five years.
-
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.
- Calculate the amount owing at the end of the four years.
[3]
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Question 22
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
Michelle takes out a loan of . The unpaid balance on the loan has an interest rate of % per year, compounded annually.
-
The loan is to be repaid in payments of made at the end of each year.
-
Find the number of years it will take to repay the loan.
-
Calculate the total amount that has been paid in amortising the loan.[3]
-
-
The loan is to be amortised over years.
-
Find the annual payment made at the end of each year.
-
Calculate the total amount that has been paid in amortising the loan.[3]
-
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Question 23
[Maximum mark: 6]
Julia wants to buy a house that requires a deposit of Australian dollars (AUD).
Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of AUD Julia needs to invest to reach AUD after years. Give your answer correct to the nearest dollar. [3]
Julia's parents offer to add 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 %, compounded quarterly.
- Find the number of years it would take Julia to save the
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
[Maximum mark: 6]
Olivia takes a mortgage (loan) of to buy an apartment in Sydney. on the loan accumulates at the rate of % per year, compounded semi-annually. Olivia agrees with the bank to amortise the loan in monthly payments, made at the beginning of each month.
-
Given that the loan is to be amortised over years, find:
-
the monthly payment amount;
-
the total amount paid in amortising the loan. [4]
-
-
Olivia has the capacity to increase her monthly payments by . Justify to Olivia why this may be a smart financial choice. [2]
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Question 25
[Maximum mark: 5]
Phil's phone shop sells Azura smartphones for and Bellson smartphones for . It is expected that a Bellson smartphone will depreciate at a rate of per year.
After 2 years, an Azura smartphone is worth approximately .
- 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 years after they were purchased.
-
Find the value of . [2]
-
Comment on the validity of your answer to part (b). [1]
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Question 26
[Maximum mark: 14]
Bruce goes into a car dealership to purchase a new vehicle. The one he wants to buy costs , however he doesn't have that much money in his bank. The salesman offers him a financing option of a % deposit followed by monthly payments of .
-
Find the amount of the deposit. [1]
-
Calculate the total cost of the loan under this financing option. [2]
Bruce's father generously offers him an interest free loan of to buy the car to avoid the expensive loan repayments. They agree that Bruce will repay the loan by paying his father in the first month and every following month until the is repaid.
The total amount Bruce's father receives after months is . This can be expressed by the equation . The total amount that Bruce's father receives after months is .
-
Write down a second equation involving and . [1]
-
Determine the value of and the value of . [2]
-
Calculate the number of months it will take Bruce's father to receive
the . [3]
Bruce decides to buy a cheaper car for and invest the remaining . He is considering two investment options over four years.
Option A: Compound interest at an annual rate of %.
Option B: Compound interest at a nominal annual rate of %, compounded monthly.
Express each answer in part (f) to the nearest dollar.
-
Calculate the value of each investment option after four years.
-
Option A.
-
Option B. [5]
-
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Question 27
[Maximum mark: 5]
Tom takes out a loan of to purchase some new machinery for his farming business. He agrees to pay the bank at the end of every month to amortise the loan. Interest accumulates on the balance at a rate of % per year, compounded monthly.
-
Find the number of years and months it takes to pay back the loan. [2]
-
Calculate the total amount that Tom pays in amortising the loan. [1]
-
Tom decides to increase the monthly payment to . How much interest will Tom save in comparison to the former payment schedule.[2]
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Question 28
[Maximum mark: 19]
Nathan receives a lump sum inheritance of and invests the money into a savings account with an annual interest rate of , compounded quarterly.
- Calculate the value of Nathan's investment after 5 years, rounding
your answer to the nearest dollar. [3]
After months, the amount in the savings account has increased to more than .
- Find the minimum value of , where .[4]
Nathan is saving to purchase a property. The price of the property is . Nathan puts down a deposit and takes out a loan for the remaining amount.
- Write down the loan amount.[1]
The loan duration is for eight years, compounded monthly, with equal monthly payments of made by Nathan at the end of each month.
- For this loan, find
-
the amount of interest paid by Nathan over the life of the loan.
-
the annual interest rate of the loan, correct to two decimal places. [5]
-
After years of paying this loan, Nathan decides to pay the outstanding loan amount in one final payment.
-
Find the amount of the final payment after years, rounding your answer to the nearest dollar. [3]
-
Find the amount Nathan saved by making this final payment after years, rounding your answer to the nearest dollar.[3]
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Question 29
[Maximum mark: 7]
Ray takes out a loan of to purchase a house. He agrees to pay the bank at the end of every month to amortise the loan, and interest accumulates on the balance at a rate of % per year, compounded monthly.
-
Find the number of years and months it takes to pay back the loan. [2]
-
Calculate the total amount that Ray has paid in amortising the loan. [2]
-
Ray decides to increase the monthly payment to . Justify this decision.[3]
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Question 30
[Maximum mark: 14]
In this question, give all answers correct to the nearest whole number.
Ann is considering investing into a term deposit in one of two banks. The first bank offers an annual interest rate of %, compounding monthly. The second bank offers a fixed payment amount of per month.
-
Calculate:
-
the amount that would be in the account in the first bank at the end of the first year;
-
the amount that would be in the account in the second bank at the end of the first year. [4]
-
-
Write down an expression for:
-
the amount in the account in the first bank at the end of the th year;
-
the amount in the account in the second bank at the end of the th year. [4]
-
-
Calculate the year in which the amount in the first bank account becomes
greater than the amount in the second bank. [2]
-
Calculate:
-
the interest that Ann would earn if she invested in the first bank for years;
-
the interest that Ann would earn if she invested in the second bank for years. [4]
-
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Question 31
[Maximum mark: 12]
Lily and Eva both receive Australian dollars (AUD) on their th birthday. Lily deposits her AUD into a bank account. The bank pays an annual interest rate of %, compounded yearly. Eva invests her AUD into a high-yield mutual fund that returns a fixed amount of AUD per year.
-
Calculate:
-
the amount in Lily's bank account at the end of the first year;
-
the total amount of Eva's funds at the end of the first year. [2]
-
-
Write down an expression for:
-
the amount in Lily's bank account at the end of the th year;
-
the total amount of Eva's funds at the end of the th year. [4]
-
-
Calculate the year in which the amount in Lily's bank account becomes
greater than the amount in Eva's fund. [2]
-
Calculate:
-
the interest amount that Lily earns if invested for years, giving your answer correct to two decimal places;
-
the amount of funds that Eva earns for her investment if invested for years. [4]
-
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