5.01 Compound interest
\$3500 is invested for three years at a rate of 10\% p.a., compounded annually.
Complete the table below to determine the final value of the investment.
| Balance at beginning of year | Interest earned | |
|---|---|---|
| First year | \$3500 | \$350 |
| Second year | \$3850 | \$385 |
| Third year | ||
| Fourth year | - |
Calculate the total interest earned over the three years.
\$ 3200 is invested for three years at a rate of 6\% p.a., compounded annually.
Complete the table below to determine the final value of the investment.
| Balance at beginning of year | Interest earned | |
|---|---|---|
| First year | \$3200 | \$192 |
| Second year | \$3392 | \$203.52 |
| Third year | ||
| Fourth year | - |
Calculate the total interest earned over the three years.
Tara borrows \$5000 at a rate of 4.5\% p.a., compounded annually.
After 3 years, Tara repays the loan all at once. How much money does she pay back in total?
How much interest was generated on the loan over the three years?
Ivan borrows \$3000 at a rate of 6.9\% p.a., compounded annually.
After 6 years, Ivan repays the loan all at once. How much money does he pay back in total?
How much interest was generated on the loan over the six years?
Valerie borrows \$1250 at a rate of 7.6\% p.a, compounding annually.
Write an expression that represents the amount Valerie must repay after 15 years, assuming that she hasn't paid anything back.
How much interest was generated on the loan over the fifteen years? Round your answer to two decimal places.
Tom borrows \$500\,000 at a rate of 1.2\% p.a, compounding annually.
Write an expression that represents the amount Tom must repay after 24 years, assuming that he hasn't paid anything back.
How much interest was generated on the loan over the twenty four years? Round your answer to two decimal places.
Rochelle is planning to make an investment which accumulates monthly compound interest at a rate of 4.8\% p.a. If she aims to have her investment reach \$90\,000 after 11 years, find the principal amount that she should invest.
Carl has a savings account which accumulates compound interest yearly at a rate of 2.9\% p.a. If he aims to have \$60\,000 in his account after 4 years, find the principal amount that he should invest.
Danielle is planning to invest \$3000 into a savings account which accumulates interest yearly. If she aims to have her investment double after 6 years, what interest rate per annum, r, will she need? Round your answer to four decimal places.
Calculate the future value, A, of the following investments:
A \$5420 investment earns interest at 5\% p.a. compounded monthly over 4 years.
Mae's investment of \$5970 earns interest at 2.1\% p.a. compounded annually over 17 years. Calculate the amount of interest earned.
Calculate the interest that the following investments accumulate:
A \$5460 investment earns interest at 2.8\% p.a. compounded monthly over 10 years.
A \$6240 investment earns interest at 2.1\% p.a., compounded weekly over 6 years.
Bart is planning to invest \$110\,000 into a saving scheme which accumulates interest weekly. If he aims to have his investment reach \$120\,000 after 2 years, what interest rate per annum, r, will he need? Round your answer to four decimal places.
The following spreadsheet shows the balance (in dollars) in a savings account in 2015, where interest is compounded quarterly:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Quarter} | \text{Balance at beginning} \\ \text{ of quarter} | \text{Interest} | \text{Balance at end} \\ \text{of quarter} |
| 2 | 1 | 2000 | 20 | 2020 |
| 3 | 2 | 2020 | 20.20 | 2040.20 |
| 4 | 3 | 2040.20 | 20.40 | 2060.60 |
| 5 | 4 |
Calculate the quarterly interest rate.
Complete the table for quarter 4.
Han opened a savings account at the beginning of February 2011, where the interest is compounded monthly. His statements for March, April and May show his account balance at the beginning of each month:
Find the monthly interest rate, r, for his investment.
How much did Han deposit into this savings account when he opened it?
| Month | Balance |
|---|---|
| \text{March} | \$3825 |
| \text{April} | \$3901.50 |
| \text{May} | \$3979.53 |
Victoria borrows \$35\,000 at a rate of 4.8\% p.a., compounded monthly.
After 4 months, Victoria repays the loan all at once. How much money does she pay back in total?
How much interest was generated on the loan over the four months?
Consider the following graph showing two investments, one with simple interest (Investment A) and another with compound interest (Investment B):
Which investment has a higher principal amount?
Which investment has a higher final amount after 10 years?
After how many years will the investments be equal in value?
State whether the following graphs could represent a savings account that earns compound interest over 50 years:
The value of a painting valued at \$700 appreciates in value at a rate of 9\% per year.
Sketch the relationship between the number of years passed and the painting's value on a number plane.
How many years will it take for the painting's value to reach a total value of \$2100?
Nadia invests \$600 into an account which accumulates interest at a rate of 9\% p.a., compounding annually.
Sketch the relationship between the number of years passed and the Nadia's account balance on a number plane.
How many years will it take for the account to reach a total value of \$2800?
Paul used repeated applications of simple interest to calculate how much an investment of \$2000 would grow over 3 years if it earned compound interest at a rate of 22\% p.a., compounded annually:
| Year | Balance + interest | Value | Interest earned |
|---|---|---|---|
| \text{Start of investment, } 0 | \qquad \quad \enspace - | \quad \quad \$ 2000.00 | \qquad \$440 |
| \text{First year, } 1 | \quad \$ 2000.00 + \$440.00 | \qquad \$ 2440.00 | \qquad \$ 536.80 |
| \text{Second year, } 2 | \quad \$ 2440.00 \, + \$ 536.80 | \qquad \$ 2976.80 | \qquad \$654.90 |
| \text{Third year, }3 | \quad \$ 2976.80 + \$654.90 | \qquad \$ 3631.70 | \qquad\quad - |
Sketch the relationship between the number of years passed and the value of the investment on a number plane.
Is the growth of the investment linear or non-linear?
Amy used repeated applications of simple interest to calculate how much an investment of \$300 would grow over 3 years if it earned compound interest at a rate of 23\% p.a., compounded annually:
| Year | Balance + interest | Value | Interest earned |
|---|---|---|---|
| \text{Start of investment, }0 | \qquad \quad \enspace - | \quad \quad \$ 300.00 | \qquad \$69.00 |
| \text{First year, }1 | \quad \$ 300.00 + \$ 69.00 | \qquad \$ 369.00 | \qquad \$ 84.87 |
| \text{Second year, }2 | \quad \$ 369.00 \, + \$ 84.87 | \qquad \$ 453.87 | \qquad \$104.39 |
| \text{Third year, }3 | \quad \$ 453.87 + \$104.39 | \qquad \$ 558.26 | \qquad\quad - |
Sketch the relationship between the number of years passed and the value of the investment on a number plane.
Is the growth of the investment linear or non-linear?
The price of train tickets in 1996 was \$3. If the value inflated at an average rate of 6.3\% per annum, what would the price have been in 2005?
A one year magazine subscription currently costs \$416. Calculate the cost in 2 years’ time if the inflation rate is on average 3.3\% per annum. Round your answer correct to the nearest dollar.
In terms of today's dollar, find the value of \$15 in 8 years' time if the inflation rate is:
2.7\%
4.7\%
A piece of artwork appreciates at 4.4\% per annum and is currently valued at \$61\,000. Calculate its value in 6 years. Round your answer correct to the nearest dollar.
A house was valued 9 years ago to be worth \$755\,000. Its value appreciated at 4.4\% p.a. Calculate its appreciated value. Round your answer correct to the nearest dollar.
An item costs \$14\,000 today. Find its price, P, 6 years ago if the inflation rate was an average of 2.5\% per annum. Round your answer to the nearest dollar.
Vanessa has just won \$30\,000. When she retires in 21 years, she wants to have \$57\,000 in her fund which earns 9\% interest per annum. How much of her winnings, does she need to invest now to achieve this?
Dylan wants to put a deposit on a house in 5 years. In order to finance the \$19\,000 deposit, he decides to put some money into a high interest savings account that pays 11\% p.a. interest compounded monthly. If P is the amount of money that he must put into his account now to accumulate enough for the deposit, find P.
A piece of art that costs \$6000, appreciates at approximately 6.2\% p.a. After how many full years, n, will the value of the piece of art be over \$15\,000?
Luke is looking to purchase a car that he is planning to re-sell in 4 years. A new luxury car is going to cost \$10\,000, which depreciates at 18\% p.a. A used reliable car costs \$6000 and depreciates at 7\% p.a.
Which car has the higher resale value? Explain your answer.
Which car has the smallest decrease in value? Explain your answer.
A camera, originally purchased for \$4000, depreciates at 16\% p.a. Calculate its expected value after 9 years.
A laptop depreciated by 17\% p.a. and was valued at \$600 after 10 years. What was its original price, P?
A microwave selling for \$900, depreciates at 17\% p.a.
What is the percentage of the original value that will remain after one year?
What is the percentage of the original value that will remain after two years? Write your answer as a percentage to two decimal places.
What is the percentage of the original value that will remain after three years? Write your answer as a percentage to two decimal places.
How many full years will it take for the microwave to lose at least half its original value?
How many years will it take for the microwave to lose 90\% of its original value?
A swimming pool is losing water at the rate of 2\% per week due to evaporation. The pool currently holds 850\text{ kL} of water.
How much water will be lost in the next 27 days? Round your answer correct to two decimal places.
How many weeks will it take for the pool to lose at least half its water?
A ball dropped from a height of 27\text{ m} will bounce back off the ground to 50\% of the height of the previous bounce (or the height from which it is dropped when considering the first bounce).
Write a function, y, to represent the height of the nth bounce.
Find the height of the fifth bounce, correct to two decimal places.