Sandro has a superannuation fund of \$600\,000. He withdraws \$40\,000 at the beginning of each year as an income stream in his retirement. The interest compounds monthly on the remaining balance at 3\% per annum.
Calculate the balance in the account at the end of the first year, correct to the nearest dollar.
By how much did the balance decrease in the first year?
Alex has a superannuation fund of \$500\,000. He withdraws \$20\,000 at the beginning of each year as an income stream in his retirement. The interest compounds monthly on the remaining balance at 3\% per annum.
Calculate the balance in the account at the end of the first year.
By how much did the balance decrease at the end of the first year?
Calculate the interest he received in the first year.
Rochelle invests \$190\,000 at a rate of 7\% per annum compounded annually, and wants to work out how much she can withdraw each year to ensure the investment lasts 20 years.
Write an expression for A_{1}, the balance of the account after 1 year. Use x to represent the amount to be withdrawn each year.
Write an expression for:
A_{2}
A_{3}
Hence, write an expression for A_{n}.
Hence, determine Rochelle's annual withdrawal amount.
Xanthe invests \$30\,000 at a rate of 1.5\% per month compounded monthly. Each month, she withdraws \$600 from her investment after the interest is paid and the balance is reinvested in the account.
Calculate the balance of the account after 1 month.
Find the balance of the account after:
2 months
3 months
Hence, write an expression for A_{n}.
Calculate the amount Xanthe has saved after 2 years.
Lisa invests \$30\,000 at a rate of 1.5\% per month compounded monthly. Each month, she withdraws \$600 from her investment after the interest is paid and the balance is reinvested in the account.
Write an expression for A_{1}, the balance of the account after 1 month.
Write an expression for:
A_{2}
A_{3}
Hence, write an expression for A_{n}.
Calculate the amount Lisa has saved after 3 years.
Jenny invests \$110\,000 at a rate of 6\% per annum compounded monthly, and wants to work out how much she can withdraw each month to ensure the investment lasts 30 years.
Write an expression for A_{1}, the balance of the account after 1 month. Use x to represent the amount to be withdrawn each month.
Write an expression for:
A_{2}
A_{3}
Hence, write an expression for A_{n}.
Calculate Jenny's monthly withdrawal amount.
Ryan invests \$110\,000 at a rate of 16\% per annum compounded quarterly. He wants to find the quarterly withdrawal amount needed if he wants the investment to last 25 years.
Write an expression for A_{1}, the balance of the account after the 1st quarter. Use x to represent the amount to be withdrawn each month.
Write an expression for:
A_{2}
A_{3}
Hence, write an expression for A_{n}.
Calculate Ryan's quarterly withdrawal amount.
Emma invests \$190\,000 at a rate of 3\% per annum compounded monthly, and wants to work out how much she can withdraw each month to ensure the investment lasts 30 years.
Write an expression for A_{1}, the balance of the account after 1 month. Use x to represent the amount to be withdrawn each month.
Continue the process and write expressions for A_{2} and A_{3}. Do not simplify the expression.
A_{2}
A_{3}
Hence, write an expression for A_{n}.
Calculate Emma's monthly withdrawal amount.
Skye received an inheritance of \$150\,000. She invests the money at 5\% per annum with interest compounded annually at the end of the year. After the interest is paid, Skye withdraws \$10\,000 and the amount remaining in the account is invested for another year.
Calculate the balance of the account at the end of the first year, A_1.
Write an expression for:
A_{2}
A_{3}
Hence, write an expression for A_{n}.
At the end of which year will the annuity have run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$150\,000?
If Skye instead withdraws \$17\,000 each year, at the end of which year will the annuity have run out?
Christa received an inheritance of \$180\,000. She invests the money at 8\% per annum with interest compounded annually at the end of the year. After the interest is paid, Christa withdraws \$15\,000 and the amount remaining in the account is invested for another year.
Calculate the balance of the account at the end of the first year, A_1.
A_{2}
A_{3}
Hence, write an expression for A_{n}.
By the end of which year will the annuity have run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$180\,000?
If Christa was only able to invest the money at 5 \% per annum, but still withdrew \$15\,000 each year, by the end of which year will the annuity have run out?
Ellie and Mark worked out that they would save \$300\,000 in five years by depositing all their combined monthly salary of \$ x at the beginning of each month into a savings account and withdrawing \$4000 at the end of each month for living expenses. The savings account paid interest at the rate of 3\% p.a. compounded monthly.
Write an expression for A_{2}, the balance of their savings account at the end of the second month.
Write an expression for the balance in their account at the end of five years.
Find their combined monthly salary.
Anita opens a savings account. At the start of each month, she deposits \$ X into the savings account. At the end of each month, after the interest is added into the savings account, the bank withdraws \$3000 from the savings account as a loan repayment. The savings account pays interest of 3.6\% per annum compounded monthly.
Let M_{n} be the amount in the savings account after the nth withdrawal.
Write an expressison for M_{2}, the amount in the savings account after the second withdrawal.
Find the value of X so that the amount in the savings account is \$80\,000 after the last withdrawal of the fourth year.
Kylo invests \$ P at 7\% p.a. compounded annually. He plans to withdraw \$8000 at the end of each year for 7 years to cover university fees.
Write an expression for A_{1}, the balance of the account after the first withdrawal.
Write an expression for A_{3}, the balance of the account after the third withdrawal.
How much does Kylo need to invest if the account balance is to be \$ 0 at the end of seven years?
Ivan takes out a car loan for \$24\,000. He is charged 8.1\% per annum interest, compounded monthly. Ivan makes repayments of \$450 at the end of each month.
Complete the table below:
| Month | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| 1 | 24\,000 | 162 | 450 | 23\,712 |
| 2 | ||||
| 3 |
Kate takes out a personal loan for \$40\,000. The interest on the loan is charged quarterly. Kate makes repayments of \$900 at the end of each quarter. Consider the following table showing the balance and payments for the first year:
| Quarter | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| 1 | 40\,000 | 400 | 900 | 39\,500 |
| 2 | ||||
| 3 |
Find the quarterly interest rate Kate is charged for this loan as a percentage.
State the annual interest rate of this loan as a percentage.
Complete the table.
How much of the loan has Kate repaid in the first three quarters of the year?
Xanthe takes out a car loan. The last few months of Xanthe's repayments are shown below:
| Month | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| 69 | 868.48 | 4.34 | 250 | 622.82 |
| 70 | 622.82 | 3.11 | 250 | 375.93 |
| 71 | 375.93 | 1.88 | 250 | 127.81 |
| 72 |
Calculate the monthly interest rate charged on this loan. Give your answer as a percentage to one decimal place.
Complete the last row of the table.
How many years did it take for her to pay off the loan?
Calculate her total repayments.
If her loan was for \$15\,000, calculate the total interest paid on the loan.
Mrs. Strife borrows \$50\,000 at 15\% p.a. monthly reducible interest for 20 years and agrees to repay the loan in equal monthly instalments.
Calculate the value of each monthly instalment.
Calculate the total amount of interest paid on this loan.
A couple borrows \$50\,000 in order to purchase a new house. They agree to repay the loan in equal quarterly instalment over 15 years. Calculate the amount of each instalment if the interest is charged at the rate of 12\% p.a. compounded quarterly on the balance owing.
When Kim entered University, her parents borrowed \$35\,000 to pay for her education. They plan to repay the loan by making 48 equal monthly instalments. Interest is charged at the rate of 1.2\% per month on the balance owing.
Write an expression for A_{2}, the balance owing right after the second payment. Use x as the monthly instalment.
Calculate the value of each monthly instalment.
Karina borrows \$100\,000 to start a business. She plans to repay the loan in equal monthly instalments of \$ M at 0.8 \% per month reducible interest. Interest is calculated and charged just before each repayment.
Let A_{n} be the amount owing after n repayments.
Write an expression for A_{3}, the amount owing after 3 repayments.
Write an expression for A_{n}.
If she wishes to repay the loan in 10 years, calculate the amount of each instalment.
If she can only repay \$1000 per month, how long will it take her to repay the loan? Round your answer to the nearest whole year.
Ross decides to borrow \$200\,000 to buy a luxury car. For a 10-year loan, interest is compounded monthly on the balance still owing at a rate of 12\% p.a. The loan is to be repaid in equal monthly repayments of \$ M at the end of each month.
Let A_{n} be the amount owing at the end of n months.
Write an expression for A_{2}, the amount owing at the end of the second month.
Write an expression for A_{n}.
Find the value of M.
At the end of the 5th year, the interest rate changes to 9.6\% p.a., compounding monthly. Assuming he does not change his repayments, how much sooner will he be able to pay off the loan? Round your answer to the nearest whole month.
Geoff has found a beach house and needs to borrow \$700\,000 from the bank to finance his purchase. The loan and interest is to be repaid in equal monthly instalments of \$ M , at the end of each month, over a 20-year period. The reducible interest will be charged at 4.8\% p.a. and compounded monthly. As an extra inducement, the bank agrees that the first 6 months of the loan can be interest free, although Geoff will begin making repayments from the end of the first month.
Let A_{n} be the amount owing to the bank at the end of n months.
Given that A_{6} = 700\,000 - 6 M, write an expression for:
A_{7}
A_{8}
Hence, write an expression for A_{n}.
Calculate the monthly instalment, M, Geoff will need to pay in order to repay the loan on time.
Scott borrowed \$100\,000 to buy a bread shop. He agreed to repay the loan at 1\% monthly reducible interest over 5 years by making 10 equal instalments of \$ P at 6 monthly intervals.
Write an expression for A_{1}, the amount Scott owes after he made his first repayment.
Write an expression for A_{2}, the amount Scott owes after he made his second repayment.
Hence, write an expression for A_{n}.
Calculate the value of Scott's repayments, P.
Beth and Adam borrowed \$80\,000 to buy a dream house. The bank is going to charge them 0.9\% monthly reducible interest and they decided they would repay the bank \$2000 per month.
How much will Beth and Adam owe the bank immediately after they make their first repayment?
Write an expression for A_{n}, the amount they owe after their nth repayment.
Hence, find the number of repayments Beth and Adam will make.
Robert borrows \$300\,000 to buy a house. The interest rate is 6\% p.a. compounding monthly. He agrees to repay the loan in 25 years with equal monthly repayment of \$ M.
Let A_{n} be the amount owing after the nth repayment.
If the amount owing after two repayments A_{2} is \$299\,132.04, solve for M.
Write an expression for A_{n}.
After how many months will the amount owing be less than \$150\,000?
Sam borrows \$110\,000 to repaid at a reducible interest rate of 0.6\% per month.
Let A_{n} be the amount owing at the end of n months and \$ M be the monthly repayment.
Write an expression for A_{2}.
Write an expression for A_{n}.
If Sam makes monthly repayments of \$780, find the amount owing after making 120 repayments.
Immediately after making the 120th repayment, Sam makes a one-off payment, reducing the amount owing to \$48\,500. The interest rate and monthly repayment remain unchanged. After how many more months will the amount owing be completely repaid?
A country borrowed \$300 million on January 1, 2013 from the International Monetary Fund to build infrastructure. The country agreed to pay \$1.2 million at the end of each month after a compounded interest of 0.6\% per month has been added to the debt.
Write an expression for A_{2}, the amount owed by the country on March 1, 2013.
On November 30, 2013, the country decided that their monthly payment for December would be \$2\,585\,562.38. Calculate their debt on January 1, 2014.
The new government elected in the country decided to further increase their monthly payment to \$5 million beginning from the payment of January 2014 until the debt is paid. How long will it take for the debt to be paid? Give your answer in years and months.
At the completion of her degree, Manpreet had a Higher Educational Loan Payment (HELP) debt of \$60\,000. She plans to repay this in equal monthly repayments of \$ M. Interest is charged at a rate of 0.4\% per month.
Let A_{n} be the amount owing at the end of the nth month.
Write an expression for A_{3}, the amount owing after 3 months.
If Manpreet decides that she would like to pay off her loan by the end of 8 years, find her monthly repayment.
If Manpreet decides that she can only repay \$450 each month, how long will it take her to repay the loan? Give your answer in years and months.
Kate invests \$70\,000 at a rate of 0.5\% per month compounded monthly. Each month, she withdraws \$550 from her investment after the interest is paid and the balance is reinvested in the account.
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n months and an initial condition A_{0}.
If the interest rate was higher and the withdrawals were the same, would the annuity end sooner or later?
If the interest rate remained the same and the withdrawals were larger, would the annuity end sooner or later?
David received an inheritance of \$100\,000. He invests the money at 7\% per annum with interest compounded annually at the end of the year. After the interest is paid, David withdraws \$10\,000 and the amount remaining in the account is invested for another year.
Calculate the balance of the account at the end of the first year.
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n months and an initial condition A_{0}.
Calculate the value of the investment after 3 years.
Amelia wins a prize of \$60\,000. She invests the money at 6\% per annum with interest compounded monthly at the end of each month. At the start of each month, before interest is earned, Amelia withdraws \$600 and the amount remaining in the account is invested.
Calculate the interest earned in the first month.
Calculate the balance of the account at the end of the second month.
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n months, and an initial condition A_{0}.
Calculate the value of the investment after 4 months.
Gwen received an inheritance of \$150\,000. She invests the money at 6\% per annum with interest compounded annually at the end of the year. After the interest is paid, Gwen withdraws \$10\,000 and the amount remaining in the account is invested for another year.
Calculate the amount of the account at the end of the first year.
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_{0}.
What amount should be withdrawn at the end of each year so that the balance remains at \$150\,000?
Tara received an inheritance of \$100\,000. She invests the money at 6\% per annum with interest compounded annually at the end of the year. After the interest is paid, Tara withdraws \$8000 and the amount remaining in the account is invested for another year.
How much is in the account at the end of the first year?
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_{0}.
At the end of which year will the annuity have run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$100\,000?
If Tara instead withdraws \$12\,000 each year, at the end of which year will the annuity have run out?
Lachlan received an inheritance of \$100\,000. He invests the money at 8\% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws \$9000 and the amount remaining in the account is invested for another year.
How much is in the account at the end of the first year?
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_{0}.
Calculate the value of the investment after 3 years.
Christa wins a prize of \$80\,000. She invests the money at 12\% per annum with interest compounded monthly at the end of each month. At the end of each month, before interest is earned, Christa withdraws \$1100 and the amount remaining in the account is invested.
Calculate the interest earned in the first month.
Find the balance of the account at the end of the second month.
Write a recursive rule for A_{n} in terms of A_{n - 1} that gives the value of the account after n months, and an initial condition A_{0}.
Calculate the value of the investment after 4 months.
Harry takes out a mortgage to purchase an investment property. A portion of his payments and balances are shown in the table below in dollars:
| Month | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| 1 | x | 2800 | 348\,425 | |
| 2 | 348\,425 | 1219.4875 | 2800 | |
| 3 | y |
Calculate the monthly interest rate charged on this loan.
Calculate the annual interest rate charged on this loan.
Write an equation in terms of x that will determine the initial amount borrowed on this mortgage.
Hence, find the value of x.
Hence, find the value of y correct to two decimal places.
Maximilian takes out a mortgage to purchase an investment property. A portion of his payments and balances are shown in the table below in dollars:
| Month | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| 1 | x | 3500 | 417\,970 | |
| 2 | 417\,970 | 1462.895 | 3500 | |
| 3 | y |
Calculate the monthly interest rate charged on this loan.
State the annual interest rate charged on this loan.
Write an equation in terms of x that will determine the initial amount borrowed on this mortgage.
Hence, find the value of x.
Hence, find the value of y correct to two decimal places.