Marco considers two five-year loans of €24 000. Loan A has a nominal annual interest rate of $7.2\%$, 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.

A retirement account receives a deposit of €4000 at the end of every year for 12 years. Interest is paid at $3.6\%$ per year, compounded annually. Inflation averages $2.2\%$ per year over the same period.

A credit-card balance is €4200. Interest of $1.45\%$ 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.

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 $4.8\%$, compounded monthly. Inflation is $2.5\%$ per year.

A loan of €32 000 has a nominal annual interest rate of $5.8\%$, compounded quarterly. The loan is repaid by equal payments at the end of every month for three years.

Elijah borrows €15 000 at a nominal annual interest rate of $8.4\%$, compounded monthly. He pays €600 at the end of each month until the loan is repaid.

Noah compares two six-year investment accounts. In Account A, interest is paid at a nominal annual rate of $4.6\%$, 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 $4.35\%$, compounded monthly, with no fees. Noah initially invests €10 000 in either account.

A community orchestra invests €18 000 in an account paying a nominal annual interest rate of $3.6\%$, compounded quarterly. The annual inflation rate is expected to remain at $2.1\%$. The investment is held for five years.

A bakery buys an industrial oven for €64 000. Its value depreciates by $18\%$ at the end of each year. A replacement oven currently costs €78 000, and its price is expected to increase by $3.2\%$ each year.

A theatre has €24 000 available for four years. Account S pays simple interest at $5.2\%$ per year. Account C pays a nominal annual interest rate of $4.8\%$, compounded quarterly.

A housing cooperative borrows €285 000 at a nominal annual interest rate of $5.4\%$, compounded monthly. The loan is to be repaid by equal payments at the end of each month over 20 years.

Leonie deposits €320 at the end of every month into a travel account. The account pays a nominal annual interest rate of $4.2\%$, compounded monthly. She makes no initial deposit.

A farm installs solar equipment costing €92 000. The equipment depreciates by $12\%$ 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 $2.8\%$ each year. For part (c), the replacement fund earns a nominal annual interest rate of $4.5\%$, compounded monthly, and equal deposits are made at the end of each month for seven years.

A student borrows €48 000. Interest is charged at $6\%$ per year, compounded annually. The student pays €9000 at the end of each year. Let $B_q$ be the balance immediately after payment $q$, where $B_0=48000$.

A community centre deposits €7500 at the end of each year into a reserve account paying $3.25\%$ interest per year, compounded annually. It makes 12 deposits. It then stops depositing and withdraws €9000 at the end of each subsequent year.

An environmental trust invests €25 000 in an account paying a nominal annual rate of $4.1\%$, compounded annually. Inflation is predicted to remain at $2.7\%$ per year. Both rates are assumed constant.

A manufacturer buys a machine for €96 000. Its value depreciates by $18\%$ at the end of each year. An equivalent replacement machine currently costs €96 000, and its price is expected to increase by $3\%$ each year. The machine will be replaced after six years.

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 $4.2\%$, compounded monthly.

A theatre borrows €80 000. Interest is charged at a nominal annual rate of $7.2\%$, 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.

Two retirement strategies use an account paying a nominal annual interest rate of $5.0\%$, 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.

A scholarship fund invests €15 000 for six years at $5\%$ interest per year, compounded annually. Inflation is $2\%$ in each of years 1, 3 and 5, and $4\%$ in each of years 2, 4 and 6.

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 $4.8\%$, compounded monthly. At the end of five years, one US dollar can be exchanged for $4.50$ units of the local currency.

A company has €310 000 outstanding on a mortgage. The current mortgage has a nominal annual interest rate of $6.1\%$, compounded monthly, and 18 years remain. The company is offered refinancing at $4.9\%$, compounded monthly, over 15 years. A refinancing fee of €4200 will be added to the new loan.

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 $5.2\%$, compounded monthly. Inflation averages $2.4\%$ per year.

A research vessel is purchased using a loan of €175 000. Interest is quoted at a nominal annual rate of $6.0\%$, compounded quarterly. The loan agreement requires payments of €1400 at the end of every month for eight years, followed by a balloon payment.

A credit-card balance is €6800. Interest of $1.7\%$ is charged monthly. A payment of €240 is made at the end of each month after interest has been charged.

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 $4.8\%$, compounded monthly. Domestic inflation is $2.7\%$ per year.

A community organization borrows €120 000 at a nominal annual interest rate of $6.3\%$, 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.

A credit-card account has an initial balance of €6200. Interest of $1.8\%$ is charged monthly. At the end of each month, the required payment is the greater of €150 or $4\%$ of the balance after interest has been charged.

A housing cooperative borrows €260 000 for 20 years. The nominal annual interest rate is $5.4\%$, compounded quarterly, but repayments are made monthly at the end of each month.

A clinic has €185 000 outstanding on a loan with seven years remaining. The loan has a nominal annual interest rate of $6.6\%$, compounded monthly. A different bank offers to refinance the balance for the same remaining term at $5.1\%$, compounded monthly. Refinancing requires an immediate fee of €2800.

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.

A graduate owes €45 000 on a student loan charging a nominal annual interest rate of $6.0\%$, 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.

A cooperative deposits €6000 at the end of every year into an account earning $4.6\%$ interest per year. Immediately after each deposit, an administration fee equal to $1\%$ of the entire account balance is deducted. Let $B_q$ be the balance immediately after the fee at the end of year $q$, with $B_0=0$.

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 $5.6\%$, 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.

A laboratory borrows €120 000 using a ten-year loan at a nominal annual interest rate of $4.2\%$, compounded monthly. Equal payments are made at the end of each month. After 24 payments, the annual rate rises to $6.6\%$, compounded monthly.

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 $4.0\%$ 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 $2.5\%$.

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.

An investor places €20 000 into one of two accounts. Account A earns $4.5\%$ interest per year with no fee. Account B earns $4.9\%$ interest per year, but €120 is deducted at the end of every year after interest is added. Let $A_n$ and $B_n$ be the balances after $n$ years and after any fee has been deducted.