This content can't be printed from the browser. Use Export PDF with Pro for an offline copy.
Question 1
Mina invests €5000 for three years. The account pays interest at a nominal annual rate of , compounded quarterly.
Calculate the value of Mina's investment after three years.
[3]
Find how much more this is than the value obtained using simple interest at per year.
[2]
Question 2
A small business borrows €7500 at an annual interest rate of , compounded annually. A payment of €1800 is made at the end of each year. Let be the balance immediately after payment , where .
Write down a recurrence relation for .
[1]
Calculate the balance immediately after the second payment.
[2]
Find the amount of the second payment that reduces the principal.
[1]
Question 3
A delivery company buys a van for €28 500. Its value depreciates by at the end of each year.
Calculate the value of the van after four years.
[2]
Determine the first whole number of years after purchase for which the value of the van is less than €10 000.
[3]
Question 4
Sofia invests €12 000 in an account paying interest per year, compounded annually. Inflation is expected to be per year, compounded annually. Both rates remain constant for six years.
Calculate the nominal value of the investment after six years.
[2]
Calculate the real value of the investment after six years, expressed in today's money.
[2]
State whether the purchasing power of the investment has increased or decreased.
[1]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 5
Ravi wants to invest €8000 for five years. Bank A offers a nominal annual rate of , compounded monthly. Bank B offers a nominal annual rate of , compounded quarterly.
Calculate the value of the investment after five years at Bank A.
[2]
Calculate the value of the investment after five years at Bank B.
[2]
State which bank gives the greater return and find the difference between the two final values, using the unrounded values.
[1]
Question 6
Lucas borrows €18 000 to buy a car. The loan has a nominal annual interest rate of , compounded monthly. It is repaid by equal monthly payments made at the end of each month for four years.
Using a financial package, calculate the monthly repayment.
[3]
Find the total interest paid over the four years, using the unrounded monthly repayment.
[2]
Question 7
Nadia deposits €250 at the end of every month into a savings account. The account pays a nominal annual interest rate of , compounded monthly. She makes deposits for seven years.
Using a financial package, find the value of the account immediately after the final deposit.
[3]
Calculate the interest earned during the seven years.
[2]
Question 8
An investment of €6500 grows to €8000 in three years. Interest is compounded monthly at a constant nominal annual rate of .
Determine .
[3]
Find the annual simple interest rate that would grow €6500 to €8000 in three years.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 9
Aisha wants to accumulate €60 000 by making equal deposits at the end of each month for eight years. Her account pays a nominal annual interest rate of , compounded monthly. She makes no initial deposit.
Using a financial package, determine the required monthly deposit.
[3]
Using the unrounded monthly deposit, calculate the total interest earned.
[2]
Question 10
Marco considers two five-year loans of €24 000. Loan A has a nominal annual interest rate of , compounded monthly, and an administration fee of €350. Loan A is repaid by equal payments at the end of each month. Loan B requires 60 monthly payments of €485 and has no fee.
Using a financial package, calculate the monthly repayment for Loan A.
[3]
Find the total amount paid for Loan A, including the fee.
[2]
Determine which loan has the lower total cost.
[1]
Question 11
A retirement account receives a deposit of €4000 at the end of every year for 12 years. Interest is paid at per year, compounded annually. Inflation averages per year over the same period.
Using a financial package, calculate the nominal value of the account immediately after the final deposit.
[3]
Calculate the real value of the account in today's money.
[2]
Determine whether the account achieves a target real value of €45 000.
[1]
Question 12
A credit-card balance is €4200. Interest of is charged monthly. A payment of €180 is made at the end of each of the first six months. No payment is made at the end of the seventh month.
Write down a recurrence relation for the balance immediately after payment during the first six months.
[1]
Determine the balance immediately after the sixth payment.
[2]
Calculate the balance at the end of the seventh month after interest is charged.
[1]
State how much greater this balance is than it would have been if the seventh payment had been made.
[1]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 13
At the start of a ten-year savings plan, Priya invests €5000. She then deposits €300 at the end of every month. The account pays a nominal annual interest rate of , compounded monthly. Inflation is per year.
Calculate the future value of the initial €5000 after ten years.
[2]
Using a financial package, find the total nominal value of the savings plan immediately after the final monthly deposit.
[2]
Calculate the real value of the savings plan in today's money.
[2]
Question 14
A loan of €32 000 has a nominal annual interest rate of , compounded quarterly. The loan is repaid by equal payments at the end of every month for three years.
Calculate the effective monthly interest rate.
[2]
Using a financial package with the given payment and compounding frequencies, calculate the monthly repayment.
[3]
Find the total interest paid over the loan term.
[1]
Question 15
Elijah borrows €15 000 at a nominal annual interest rate of , compounded monthly. He pays €600 at the end of each month until the loan is repaid.
Using a financial package, determine the number of payments indicated for the loan.
[2]
State the actual number of monthly payments required.
[1]
Calculate the outstanding balance immediately after the twenty-seventh payment.
[2]
Hence calculate the amount of the final payment and the total interest paid over the loan term.
[2]
Question 16
Noah compares two six-year investment accounts. In Account A, interest is paid at a nominal annual rate of , compounded quarterly. A fee of €40 is deducted at the end of each year after that year's interest has been added. Account B pays a nominal annual rate of , compounded monthly, with no fees. Noah initially invests €10 000 in either account.
Calculate the effective annual growth factor for Account A.
[1]
Using a recurrence or spreadsheet, calculate the value of Account A after six years.
[2]
Calculate the value of Account B after six years and determine which account gives the greater final value.
[2]
Determine the minimum initial investment in Account A that would give the same final value as investing €10 000 in Account B.
[1]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 17
A community orchestra invests €18 000 in an account paying a nominal annual interest rate of , compounded quarterly. The annual inflation rate is expected to remain at . The investment is held for five years.
Consider the value of the investment after five years.
I.
Calculate its nominal value.
[2]
II.
Calculate its real value in today's money.
[2]
Determine the percentage increase in purchasing power over the five years.
[2]
The orchestra needs the nominal balance to exceed €25 000. Determine the minimum number of complete quarters for which the money must remain invested, and express this as years and months.
[3]
Question 18
A bakery buys an industrial oven for €64 000. Its value depreciates by at the end of each year. A replacement oven currently costs €78 000, and its price is expected to increase by each year.
Consider the value of the industrial oven.
I.
Calculate its value after three years.
[2]
II.
Determine the first year in which its value is less than €40 000.
[2]
The bakery plans to replace the oven after three years and use the depreciated oven as a trade-in.
I.
Calculate the expected price of the replacement oven at that time.
[2]
II.
Hence calculate the additional amount the bakery must provide after using the old oven as a trade-in.
[2]
Question 19
A theatre has €24 000 available for four years. Account S pays simple interest at per year. Account C pays a nominal annual interest rate of , compounded quarterly.
Calculate the value after four years in each account.
I.
Find the value in Account S.
[2]
II.
Find the value in Account C.
[2]
State which account has the greater value after four years and calculate the difference between the account values.
[2]
Determine the first whole number of years for which Account C has a greater value than Account S.
[2]
Question 20
A housing cooperative borrows €285 000 at a nominal annual interest rate of , compounded monthly. The loan is to be repaid by equal payments at the end of each month over 20 years.
Use a financial package for the original loan.
I.
Determine the monthly repayment.
[2]
II.
Calculate the total interest paid if the loan continues for the full 20 years.
[2]
Determine the outstanding balance immediately after the 72nd payment.
[2]
Immediately after the 72nd payment, the cooperative increases its monthly repayment by €250. If you did not obtain the balance in part (b), use €228 863.15.
I.
Determine the number of additional monthly payments required.
[2]
II.
Determine how many months earlier the loan will be repaid than originally planned.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 21
Leonie deposits €320 at the end of every month into a travel account. The account pays a nominal annual interest rate of , compounded monthly. She makes no initial deposit.
Suppose Leonie continues making deposits of €320 for six years.
I.
Find the account value immediately after the 72nd deposit.
[2]
II.
Calculate the interest earned during the six years.
[2]
Determine the minimum number of monthly deposits of €320 required for the account value to exceed €35 000.
[2]
Instead, Leonie deposits €320 for the first 36 months. She wants the account to contain €35 000 immediately after the 72nd deposit. Determine the equal monthly deposit required for the final 36 months.
[3]
Question 22
A farm installs solar equipment costing €92 000. The equipment depreciates by each year. The farm plans to replace it after seven years. Equivalent new equipment currently costs €110 000, and its price is expected to increase by each year. For part (c), the replacement fund earns a nominal annual interest rate of , compounded monthly, and equal deposits are made at the end of each month for seven years.
Calculate the expected values after seven years.
I.
Find the trade-in value of the existing equipment.
[2]
II.
Find the price of the replacement equipment.
[2]
Hence determine the amount that must be available in addition to the trade-in value.
[2]
The farm creates a replacement fund paying a nominal annual interest rate of , compounded monthly. Equal deposits are made at the end of each month for seven years. If you did not obtain the amount in part (b), use €95 900.
I.
Determine the required monthly deposit.
[2]
II.
Calculate the interest earned by the replacement fund.
[2]
Question 23
A student borrows €48 000. Interest is charged at per year, compounded annually. The student pays €9000 at the end of each year. Let be the balance immediately after payment , where .
Consider the first two annual payments.
I.
Write down a recurrence relation for the balance and calculate .
[2]
II.
Calculate the amount of the second payment that reduces the principal.
[2]
Use a financial package to determine the actual number of annual payments required to repay the loan.
[2]
The first six payments are €9000 each.
I.
Calculate the outstanding balance immediately after the sixth payment.
[2]
II.
Hence determine the amount of the final payment made one year later.
[1]
Question 24
A community centre deposits €7500 at the end of each year into a reserve account paying interest per year, compounded annually. It makes 12 deposits. It then stops depositing and withdraws €9000 at the end of each subsequent year.
Consider the accumulation phase.
I.
Determine the value immediately after the 12th deposit.
[2]
II.
Calculate the interest earned during the accumulation phase.
[2]
Determine the number of complete withdrawals of €9000 that can be made.
[2]
If you did not obtain the account value in part (a), use €108 000.
I.
Calculate the balance immediately after the 15th withdrawal.
[2]
II.
Hence calculate the final partial withdrawal available one year later.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 25
An environmental trust invests €25 000 in an account paying a nominal annual rate of , compounded annually. Inflation is predicted to remain at per year. Both rates are assumed constant.
I.
Calculate the nominal value of the investment after nine years.
[2]
II.
Calculate its real value after nine years, expressed in today's money.
[2]
Show that the annual real growth factor is approximately .
[2]
Determine the first whole number of years after which the real value exceeds €30 000. If you did not obtain the factor in part (b), use .
[2]
Question 26
A manufacturer buys a machine for €96 000. Its value depreciates by at the end of each year. An equivalent replacement machine currently costs €96 000, and its price is expected to increase by each year. The machine will be replaced after six years.
I.
Calculate the trade-in value of the original machine after six years.
[2]
II.
Calculate the price of the replacement machine after six years.
[2]
Hence calculate the amount that must be provided by a replacement fund after the trade-in value is used.
[2]
Equal deposits are made into the replacement fund at the end of each year. The fund earns interest per year. Using a financial package, determine the annual deposit required. If you did not obtain the amount in part (b), use €85 400.
[2]
Question 27
Leonie saves for a studio. She deposits €500 at the end of each month for three years, followed by €650 at the end of each month for four years. Her account pays a nominal annual interest rate of , compounded monthly.
I.
Calculate the value at the end of seven years of the deposits made during the first three years.
[2]
II.
Calculate the value at the end of seven years of the deposits made during the final four years.
[2]
Hence determine the total nominal value of Leonie's account after seven years.
[2]
Inflation averages per year over the seven years. Calculate the real value of the account in today's money and comment on its numerical comparison with Leonie's undiscounted total deposits of €49 200. This total is not an inflation-adjusted benchmark because the deposits were made at different times. If you did not obtain the value in part (b), use €56 600.
[2]
Question 28
A theatre borrows €80 000. Interest is charged at a nominal annual rate of , compounded quarterly. Under Plan A, the theatre pays €4500 at the end of every quarter for five years and then clears any outstanding balance with a balloon payment.
I.
Write down a recurrence relation for the balance immediately after payment .
[2]
II.
Calculate the balloon payment due immediately after the twentieth regular payment.
[2]
Calculate the total interest paid under Plan A.
[2]
Under Plan B, the loan is repaid by twenty equal quarterly payments with no balloon payment. Using a financial package, determine which plan has the lower total repayment and by approximately how much.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 29
Two retirement strategies use an account paying a nominal annual interest rate of , compounded monthly. Under Strategy E, €700 is deposited at the end of every month for eight years and the accumulated amount is then left invested for twelve years. Under Strategy L, no deposits are made for eight years, followed by deposits of €700 at the end of every month for twelve years.
I.
Calculate the final value under Strategy E.
[2]
II.
Calculate the final value under Strategy L.
[2]
Calculate the difference between the total amounts deposited under the two strategies.
[2]
Explain why Strategy E produces the greater final value despite having fewer deposits.
[2]
Question 30
A scholarship fund invests €15 000 for six years at interest per year, compounded annually. Inflation is in each of years 1, 3 and 5, and in each of years 2, 4 and 6.
Year | Fund growth factor | Inflation growth factor |
|---|---|---|
1 | 1.05 | 1.02 |
2 | 1.05 | 1.04 |
3 | 1.05 | 1.02 |
4 | 1.05 | 1.04 |
5 | 1.05 | 1.02 |
6 | 1.05 | 1.04 |
I.
Calculate the nominal value of the investment after six years.
[2]
II.
Calculate the cumulative inflation growth factor over the six years.
[2]
Hence calculate the real value of the investment after six years, expressed in today's money.
[2]
A manager proposes replacing the changing inflation rates by their arithmetic mean of per year. Explain why this does not give exactly the same real value.
[2]
Question 31
A worker living abroad deposits 1200 units of local currency at the end of every month for five years. The account pays a nominal annual interest rate of , compounded monthly. At the end of five years, one US dollar can be exchanged for units of the local currency.
I.
Using a financial package, calculate the value of the account in local currency immediately after the final deposit.
[2]
II.
Convert this value to US dollars using the final exchange rate.
[2]
Local inflation averages per year, compounded annually. Calculate the account's real value in local currency measured at the start of the five years.
[2]
Explain why converting the nominal balance into US dollars does not, by itself, measure the investment's change in purchasing power.
[2]
Question 32
A company has €310 000 outstanding on a mortgage. The current mortgage has a nominal annual interest rate of , compounded monthly, and 18 years remain. The company is offered refinancing at , compounded monthly, over 15 years. A refinancing fee of €4200 will be added to the new loan.
Determine the monthly repayment for each option.
I.
Find the monthly repayment if the current mortgage is retained.
[2]
II.
Find the monthly repayment after refinancing.
[2]
Calculate the total remaining payments under each option, and hence determine the approximate saving from refinancing.
[4]
Suppose the refinancing fee is instead € and is added to the new loan.
I.
Explain why the new monthly repayment is proportional to .
[2]
II.
Determine the largest refinancing fee, to the nearest €100, for which refinancing gives a lower total payment.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 33
A retirement account initially contains €20 000. For 15 years, Amara deposits €600 at the end of each month. The account pays a nominal annual interest rate of , compounded monthly. Inflation averages per year.
Consider the account immediately after the final deposit.
I.
Calculate the future value of the initial €20 000.
[2]
II.
Determine the total nominal account value, including the monthly deposits.
[3]
Calculate the real value of the account in today's money.
[2]
At retirement, Amara stops making deposits. She makes a fixed nominal withdrawal of €1800 at the end of each month, and the account continues to earn the nominal annual interest rate of , compounded monthly. If you did not obtain the nominal value in part (a), use €207 000.
I.
Determine the number of complete withdrawals of €1800 that can be made.
[2]
II.
Interpret this duration in years and months.
[2]
Question 34
A research vessel is purchased using a loan of €175 000. Interest is quoted at a nominal annual rate of , compounded quarterly. The loan agreement requires payments of €1400 at the end of every month for eight years, followed by a balloon payment.
The payment and compounding frequencies are different.
I.
Calculate the effective monthly interest rate.
[2]
II.
Explain why using as the monthly rate would be inappropriate.
[2]
Determine the balloon payment due immediately after the 96th monthly payment of €1400.
[2]
For an alternative loan with the same principal and interest rate as the original loan, and with monthly payments, fully repaid by 96 equal end-of-month payments with no balloon payment, determine the equal monthly repayment.
I.
Determine the equal monthly repayment for the alternative loan.
[2]
II.
Calculate the total interest paid under the original balloon-payment agreement.
[2]
Question 35
A credit-card balance is €6800. Interest of is charged monthly. A payment of €240 is made at the end of each month after interest has been charged.
For the first 12 months, the scheduled monthly payment is €240.
I.
Write down a recurrence relation for the balance immediately after payment .
[2]
II.
Calculate the balance immediately after the 12th payment, assuming all payments are made.
[2]
The payments at the ends of months 7 and 8 are missed, but all other scheduled payments are made. Determine the balance immediately after month 12.
[3]
From month 13 onward, the cardholder pays €320 at the end of each month. If you did not obtain the balance in part (b), use €5680.
I.
Determine the number of further payments required to reduce the balance to zero.
[2]
II.
Explain why the final payment will be less than €320.
[2]
Question 36
A university creates an overseas study fund with an initial deposit of €45 000 and deposits of €350 at the end of every month for nine years. The fund pays a nominal annual interest rate of , compounded monthly. Domestic inflation is per year.
Calculate the fund value after nine years.
I.
Find the future value of the initial deposit.
[2]
II.
Find the total nominal value including all monthly deposits.
[2]
Calculate the real value of the fund in today's domestic currency.
[2]
Today, one euro buys units of the destination currency. The number of destination-currency units per euro is expected to decrease by per year. Inflation in the destination country is expected to be per year. Assume that tuition fees increase at the destination-country inflation rate.
I.
Calculate the nominal fund value in the destination currency after nine years.
[2]
II.
Determine whether the fund will have enough purchasing power to pay tuition currently costing 60 000 destination-currency units.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 37
A community organization borrows €120 000 at a nominal annual interest rate of , compounded monthly. The loan is initially arranged to be repaid by equal end-of-month payments over ten years. After the thirty-sixth payment, the organization receives a donation.
I.
Using a financial package, calculate the original monthly repayment.
[2]
II.
Calculate the outstanding balance immediately after the thirty-sixth payment.
[2]
Immediately after the thirty-sixth payment, €15 000 is paid from the donation. The remaining term is unchanged. Calculate the revised monthly repayment.
[2]
Determine whether making the additional payment reduces the organization's total future payments, including the €15 000 payment, and calculate the saving. Use unrounded calculator values.
[2]
Question 38
A credit-card account has an initial balance of €6200. Interest of is charged monthly. At the end of each month, the required payment is the greater of €150 or of the balance after interest has been charged.
I.
Show that while the percentage payment is greater than €150, the balances immediately after successive payments form a geometric sequence with ratio .
[2]
II.
Calculate the balance immediately after the twelfth payment.
[2]
Determine the first payment number for which the fixed payment of €150 applies.
[2]
Explain why describing this arrangement only as a “minimum payment of €150” could be misleading to a borrower.
[2]
Question 39
A housing cooperative borrows €260 000 for 20 years. The nominal annual interest rate is , compounded quarterly, but repayments are made monthly at the end of each month.
I.
Calculate the effective monthly interest rate.
[2]
II.
Using a financial package with the stated payment and compounding frequencies, calculate the monthly repayment.
[2]
An employee incorrectly divides by to obtain the monthly rate. Calculate the monthly repayment produced by this error.
[2]
Evaluate the employee's method, referring to the two repayments and the rate conversion.
[2]
Question 40
A clinic has €185 000 outstanding on a loan with seven years remaining. The loan has a nominal annual interest rate of , compounded monthly. A different bank offers to refinance the balance for the same remaining term at , compounded monthly. Refinancing requires an immediate fee of €2800.
I.
Calculate the monthly repayment if the clinic keeps the original loan.
[2]
II.
Calculate the monthly repayment after refinancing.
[2]
Calculate the net saving over the full seven years after including the refinancing fee.
[2]
Determine the first whole number of months after which the cumulative reduction in monthly payments exceeds the €2800 fee.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 41
A research institute wants to accumulate €100 000 by depositing €900 at the end of every month for eight years. The account has a constant nominal annual interest rate, compounded monthly, but the rate is not known.
I.
State the values of , , and that may be entered into a financial package, using a consistent cash-flow sign convention.
[2]
II.
Determine the nominal annual interest rate required.
[2]
If inflation averages per year, calculate the real value of the target after eight years in today's money.
[2]
The institute requires purchasing power equivalent to €100 000 today. Determine the nominal target that should replace €100 000 if inflation remains at per year.
[2]
Question 42
A graduate owes €45 000 on a student loan charging a nominal annual interest rate of , compounded monthly. Under Plan A, no payments are made during an 18-month grace period. The resulting balance is then repaid by equal end-of-month payments over eight years.
I.
Calculate the balance at the end of the grace period.
[2]
II.
Using a financial package, calculate the monthly repayment after the grace period.
[2]
Under Plan B, the graduate pays the monthly interest during the grace period, so the balance remains €45 000. Calculate the interest-only monthly payment and the subsequent monthly repayment over eight years.
[2]
Compare the total amounts paid under the two plans and determine which plan costs less.
[2]
Question 43
A cooperative deposits €6000 at the end of every year into an account earning interest per year. Immediately after each deposit, an administration fee equal to of the entire account balance is deducted. Let be the balance immediately after the fee at the end of year , with .
I.
Show that .
[2]
II.
Use the recurrence to calculate the balance after ten years.
[2]
Calculate the balance after ten years if no administration fee is charged.
[2]
Explain why it would be incorrect to describe the account simply as earning a net rate of per year.
[2]
Question 44
A tour operator needs an electric shuttle for five years. It can buy the shuttle for €96 000 by paying a €20 000 deposit and borrowing the remainder at a nominal annual interest rate of , compounded monthly. The loan is repaid monthly over five years. Alternatively, it can lease the shuttle by paying €6000 initially and €1150 at the end of each month for five years.
Part (a): Consider the purchase option.
I.
Determine the monthly loan repayment.
[2]
II.
Calculate the total interest paid on the loan.
[3]
Part (b): For the purchase option, assume that the shuttle depreciates by of its value each year and that maintenance costs €1200 at the end of each year; maintenance is included in the lease.
I.
Calculate the resale value of the shuttle after five years.
[2]
II.
Calculate the net five-year cost of buying, after allowing for maintenance and resale.
[2]
Evaluate which option is cheaper over five years, and determine the annual depreciation rate at which the two options would have equal net cost.
[3]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
Question 45
A laboratory borrows €120 000 using a ten-year loan at a nominal annual interest rate of , compounded monthly. Equal payments are made at the end of each month. After 24 payments, the annual rate rises to , compounded monthly.
Consider the loan before the rate change.
I.
Determine the original monthly repayment.
[2]
II.
Determine the outstanding balance immediately after the 24th payment.
[2]
Determine the revised monthly repayment and the increase in the monthly repayment. If you did not obtain the balance in part (a), use €99 800.
[3]
Instead, the laboratory keeps paying the original monthly amount after the rate rises.
I.
If the original monthly repayment is maintained after the rate rises, determine how many additional months beyond the original term are required.
[2]
II.
Explain why keeping the original payment leads to greater total interest than accepting the revised payment.
[2]
Question 46
A scholarship endowment begins with €250 000. A donor then contributes €15 000 at the end of each year for ten years. The fund earns interest per year, compounded annually. Immediately after the tenth contribution, donations stop. Scholarships are then paid at the end of each year. The first scholarship is €30 000, and each subsequent scholarship increases by .
Determine the value of the endowment immediately after the tenth contribution.
I.
Calculate the future value of the initial €250 000.
[2]
II.
Calculate the total endowment value including all ten contributions.
[2]
The present value, at the start of the scholarship phase, of the first scholarships is
I.
Explain why is the common ratio when calculating the present value of the scholarships.
[2]
II.
Determine the greatest number of complete scholarships that the endowment can fund.
[2]
Assess the situation after the 22nd complete scholarship. If you did not obtain the initial scholarship-phase balance, use €550 000.
I.
Calculate the amount available for a partial 23rd scholarship one year later.
[2]
II.
Calculate the scheduled value of the 23rd scholarship and explain why the scholarship programme cannot continue indefinitely.
[2]
Question 47
A short-term lender advertises a loan as “€3000 today, repaid by twelve monthly payments of €310”. An arrangement fee of €180 is deducted from the amount transferred to the borrower. The first repayment is made one month after the loan is issued.
I.
Calculate the amount actually received by the borrower and the total amount repaid.
[2]
II.
Using a financial package, determine the monthly interest rate that makes the present value of the repayments equal to the amount actually received.
[2]
Calculate the nominal annual rate and the effective annual rate corresponding to the monthly rate found in part (a). If you did not obtain the monthly rate, use .
[2]
Explain why the advertised statement may conceal the true cost of borrowing.
[2]
Question 48
An investor places €20 000 into one of two accounts. Account A earns interest per year with no fee. Account B earns interest per year, but €120 is deducted at the end of every year after interest is added. Let and be the balances after years and after any fee has been deducted.
I.
Write down recurrence relations for and .
[2]
II.
Calculate both balances after ten years.
[2]
Determine the first whole number of years for which Account B has a greater balance than Account A.
[2]
Account B considers changing its annual fee. Determine the maximum annual fee for which its balance after 15 years is at least the balance of Account A after 15 years. Give your answer to the nearest euro.
[2]
Build a Practice Exam — Math AI HL
Test your knowledge with a custom paper containing questions from selected topics.
The Ultimate All-in-One
IB Platform
All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.