Neil and John both inherit \$12\,000 and put their money in compound interest-bearing accounts for a period of 5 years.
Neil places his money in an account with an interest rate of 2.75\% p.a. compounded monthly.
Complete the table for Neil’s account showing the value of the investment at the end of each month:
| \text{Number of months} | 1 | 2 | 3 | 60 |
|---|---|---|---|---|
| \text{Value of investment }(\$) | 12\,027.50 | 13\,766.65 |
Write a recursive rule for Neil's investment V_{n + 1} in terms of V_n, where V_n describes the value of the investment after the nth month, in exact form.
John places his money in an account which earns interest compounded daily. At the end of the five years, John’s balance is the same as Neil’s balance. Calculate the interest rate per annum for John’s investment account as a percentage.
Round your answer to three decimal places. Assume there are 365 days in a year.
Does the difference in compounding periods mean that John’s interest rate per annum is higher or lower than Neil’s?
Roxanne turns 53 today and is saving for her planned retirement at 65 years of age. She currently has \$298\,000 in her superannuation account. She plans to have \$750\,000 in her account at retirement from which she will receive an annuity each year. The interest rate on her superannuation account is 5.75\% p.a. compounded monthly, plus she makes a monthly deposit into the account.
Calculate the value of the monthly deposit required in order for Roxanne to meet her retirement savings goal.
Frank wins the lottery and decides to deposit the winnings in a high interest savings account. He has the following two choices:
Option A: 5.25\% p.a. compounded monthly
Option B: 5.3\% p.a. compounded weekly
Determine the effective annual interest rate for option A as a percentage. Round your answer to three decimal places.
Determine the effective annual interest rate for option B as a percentage. Round your answer to three decimal places. Assume there are 52 weeks in a year.
Another bank offers option C with an interest rate of 5.35\% p.a. If the effective rate is also 5.35\%, how many times a year must the interest be compounded?
Which option should Frank choose? Explain your answer.
Elizabeth is given \$3500 as a 21st birthday present and decides to invest the money in an account where interest is compounded quarterly. She decides to also make a \$75 per quarter deposit into the account.
The table shows the balance of Elizabeth’s account over the first 5 quarters:
| \text{Quarter} | \text{Balance at start of quarter} | \text{Interest} | \text{Deposit} | \text{Balance at end of quarter} |
|---|---|---|---|---|
| 1 | 3500.00 | 119.00 | 75 | 3694.00 |
| 2 | 3694.00 | 125.60 | 75 | 3894.60 |
| 3 | 3894.60 | 132.42 | 75 | 4102.02 |
| 4 | 4102.02 | 139.47 | 75 | 4316.49 |
| 5 | 4316.49 | 146.76 | 75 | 4538.25 |
Calculate the annual interest rate for Elizabeth’s account as a percentage. Round your answer to one decimal place.
Find the recursive rule, where P_n describes Elizabeth's balance after the nth quarter.
After four quarters, Elizabeth realises she is not going to make her savings goal which is to have \$8000 at then end of three years. By only depositing \$75 per quarter, by how much is Elizabeth short of her goal?
What deposit does Elizabeth need to make each quarter to reach her goal of \$8000 at the end of three years?
The table below gives the account balance of an investment after each week:
Calculate the weekly interest rate as a percentage.
Write the recursive rule for the investment B_{n + 1} in terms of B_n, where B_n describes the value of the investment after the nth week.
Hence find the values of X and Y.
If interest was to be calculated weekly and added to the account monthly, would the balance be higher, lower, or stay the same after 4 years?
If the interest rate was doubled, what would the effect be on the amount of interest earned?
| \text{Week }n | \text{Balance} |
|---|---|
| 0 | 15\,000 |
| 1 | 15\,675 |
| 2 | 16\,380 |
| 3 | 17\,117.49 |
| 4 | 17\,887.78 |
| 5 | 18\,692.73 |
| 6 | X |
| 45 | Y |
Beth wishes to compare the rates offered by different banks so she constructs the following table which shows the effective interest rate for nominal rates of 3.5\%, 4\% ,4.5\% and 5\% per annum:
| \text{Yearly rate} | 3.5\% | 4\% | 4.5\% | 5\% |
|---|---|---|---|---|
| \text{Quarterly rate} | 3.546\% | 4.060\% | 4.577\% | 5.095\% |
| \text{Monthly rate} | 3.557\% | 4.074\% | 4.594\% | 5.116\% |
| \text{Weekly rate} | 3.561\% | 4.079\% | 4.601\% | 5.125\% |
Beth wishes to borrow \$12\,000 for a car and the bank offers her a rate of 4.5\% p.a. with the choice of quarterly, monthly or weekly compounding. Which compounding period should she choose if she plans to pay the entire loan plus interest off in one year?
If Beth wishes to invest \$20\,000 in an account which offers a nominal rate of 3.5\% p.a., which compounding period should she choose?
How much interest would Beth earn if she invested \$1000 for one year in an account offering 4\% p.a. compounded monthly?
Paul has 15 years until retirement and wishes to have a savings of \$165\,000 in that time by making regular monthly payments into an account paying 4.5\% compounded daily. Assume there are 365 days in a year.
Complete the table showing the settings required for the financial application on your calculator.
Hence, determine the monthly payment for Paul to reach the required amount if he currently has \$2500 in the account.
| Value | |
|---|---|
| N | |
| I\% | |
| Pmt | |
| PV | |
| FV | |
| P/Y | |
| C/Y |
Sharon has \$25\,000 that she wishes to invest for a period of time without touching it. She chooses to invest this money in an account offering 4.25\% p.a. compound interest.
Calculate the value of the investment in dollars after 7 years if interest is compounded monthly.
Calculate the number of years required to double her investment if interest is compounded daily. Assume there are 365 days in a year. Round your answer to two decimal places.
Michael is deciding between two investment options:
Option A: Interest rate of 3.5\% p.a. compounded quarterly.
Option B: Interest rate of 3.4\% p.a. compounded monthly.
Calculate the effective annual interest rate for option B for Michael as a percentage. Round your answer to three decimal places.
Which option should Michael choose? Explain your answer.
Aaron has saved \$750\,000 for his retirement and sets up an annuity fund on January 1st which pays 3.5\% p.a compounded quarterly. He makes a \$9000 per month withdrawal.
Determine the number of whole months that he will be able to withdraw from this annuity.
If Aaron wishes his annuity to last for 10 years, what monthly withdrawal should he make?
Brad is about to retire and invests the \$560\,000 he has saved into an annuity plan which pays interest of 5.4\% per annum compounded monthly. He plans to make a \$3500 withdrawal each month.
Write the recursive rule for the annuity B_{n + 1} in terms of B_n, where B_n describes the value of the investment after the nth month.
For how many whole years will Brad be able to receive his full annuity of \$3500 per month?
What amount should Brad withdraw each month if the annuity is to become a perpetuity and the balance remains stable?
Annie sets up her pension fund on October 19th, 2020 with a principal of \$820\,000. The fund advertises a growth rate of 5.75\% per annum, compounded monthly and she intends to withdraw \$65\,000 each year on October 19th, starting in 2021.
Calculate the balance after the withdrawal on October 19th in 2025.
The investment fund changed its rate to 7.5\% per annum, compounded monthly, starting on October 19th 2025. If Annie continues to withdraw \$65\,000 per year, calculate the balance after the withdrawal on October 19th 2031.
Calculate the yearly amount that Annie could withdraw, starting on October 19th, 2025 if she wished her balance to remain the same.
Avril, a successful business owner wishes to set up a perpetuity of \$3000 per year to be awarded to a deserving young athlete from her local district. The perpetuity is to be paid at the end of the year in one payment. A bank offers to provide an account for this perpetuity with a fixed interest rate of 7.45\% per annum, compounded weekly.
Calculate the amount that Avril must invest in the perpetuity to the nearest \$100. Assume there are 52 weeks in a year.
Tom buys a motorbike with a purchase price of \$23\,000 but is told that it will depreciate. The value of the motorbike after t years is given by the recursive rule: a_{t + 1} = 0.82 a_t, a_0 = 23\,000.
State the rate of depreciation per annum of Tom’s motorbike as a percentage.
Complete table of values below, showing the value of the motorbike at the end of each year.
| \text{Year }(t) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Value of motorbike }(\$) | 23\,000 |
Luke buys a motorbike with a purchase price of \$23\,000 but is told that it will depreciate. The value of the motorbike after t years, a_t, is given by the recursive rule: a_{t + 1} = 0.82 a_t, a_0 = 23\,000
Determine an explicit rule for the t th term of the sequence.
Determine the value of Luke’s motorbike after 7 years.
When the bike drops to one third of its original value, Luke plans to sell it. After how many whole years should he sell the motorbike?
Charlie is an electrician. Exactly two months ago he bought a new van for his business at a cost of \$25\,000. He paid a deposit of \$4500 and borrowed the rest of the money from a building society at a reducible business loan rate of 9.6\% per annum compounded monthly. He organised to make payments of \$1200 at the end of each month.
Write a recursive rule for the annuity B_{n + 1} in terms of B_n, where B_n describes the value of the investment after the nth month.
Determine the current balance of the loan.
Determine how many whole months from now it will take Charlie to pay off the loan.
Calculate the final payment on the loan.
Tom the tradie buys a shed for \$15\,400 and after 4 years finds that it is valued at \$6500. Determine the average annual rate of depreciation as a percentage. Round your answer to two decimal places.
Homer the electrician borrowed \$20\,500 from a building society at a reducible business loan rate of 9.6\% per annum compounded monthly to buy a new van. He organised to make payments of \$1200 at the end of each month. The building society recommends he considers two following options:
Option A: increase his payments by \$100 per month.
Option B: he takes a lower rate of 9.2\% per annum compounded monthly.
Complete the table showing the time taken to pay each option and the total amount paid:
| Option A | Option B | |
|---|---|---|
| Months to pay loan | 17 | |
| Total amount paid | \$22\,089.10 |
Which option should Homer take? Explain your answer.
Maddy borrows \$9000 to buy a car. The finance department at the car yard offers a special deal of 1.5\% p.a. interest added monthly for the first 18 months, followed by a rate of 3.5\% per annum added monthly for the duration of the loan. She plans to make payments of \$350 per month.
Write the recursive rule for the loan A_n in terms of A_{n - 1}, where A_n describes the value of the loan after the nth month in the first 18 months.
Calculate how much Maddy still owes after 18 months.
How much does Maddy owe after 2 years?
What is the final repayment for the car?
Calculate the total cost of the car.
A young couple take out a personal loan for \$28\,500 with reducible interest of 9.6\% p.a. calculated monthly. Repayments of \$750 per month are made. The table below charts the progress of the loan for the first four months:
| \text{Month } | \text{Amount owing } \\ \text{at start of month} | \text{Interest} | \text{Repayment} | \text{Amount owing} \\ \text{at end of month} |
|---|---|---|---|---|
| 1 | 28\,500 | 228 | 750 | 27\,978 |
| 2 | 27\,978 | 223.82 | 750 | 27\,451.82 |
| 3 | 27\,451.82 | 219.61 | 750 | 26\,921.44 |
| 4 | 26\,921.44 | X | 750 | Y |
Find the recursive rule that describes this situation, where A_n is the amount owing after the nth month.
Calculate the values of X and Y.
Calculate how many whole months it takes to pay off the loan.
Calculate the total amount of interest paid during the loan.
If the couple increase their payments to \$1000 per month, how many whole months will it take until the loan is paid off?
If the couple wishes to pay off the loan in 4 years exactly, what monthly payment should they make?