State whether the following are true about compound interest:
Interest is earned on the principal.
The interest in any time period is calculated using only the original principal.
Interest is earned on any accumulated interest.
The amount of interest earned in any time period changes from one period to the next.
A \$2090 investment earns interest at 4.2\% p.a. compounded annually over 17 years. Use the compound interest formula to calculate the value of this investment.
Dave's investment of \$6000 earns interest at 2\% p.a, compounded annually over 3 years.
Find the value of the investment after 3 years.
Find the amount of interest earned.
Sally's investment of \$8950 earns interest at 4\% p.a. compounded annually over 4 years. Calculate the amount of interest earned.
\$392 is invested at 2\% p.a. compounded annually for 7 years. After this time the principal plus interest is reinvested at 4\% p.a. compounded annually for 5 more years. Find the final value of the investment.
Calculate the value of a \$1330 brand new desk after 7 years when the annual inflation rate is 11\%.
Find the value of a wine collection, currently valued at \$4790, after 10 years if its rate of appreciation is 9\% p.a.
The value of a piece of land is expected to grow by 6.8\% each year for the next 5 years. If it currently costs \$460 per square metre, calculate its value at the end of 5 years.
Scott wants to have \$1500 at the end of 5 years. If the bank offers 2.3\% p.a. compounded annually, find the amount he should invest now.
An investment of \$2000 earns compound interest of 3\% per annum. Find the value of the investment after one year if the interest is compounded:
Annually
Quarterly
Monthly
Weekly
An investment of \$5000 earns compound interest of 6\% per annum. Find the value of the investment after 3 years if the interest is compounded:
Annually
Quarterly
Monthly
Weekly
Tina has \$900 in a savings account which earns compound interest at a rate of 2.4\% p.a. If interest is compounded monthly, find the total interest Tina earns in 17 months.
Noah has \$700 in a savings account which earns compound interest at a rate of 1.6\% p.a. Calculate the amount in Noah's account after 5 years, if interest is compounded quarterly.
Hannah puts \$600 in a savings account for a period of 21 months. If interest is compounded quarterly at a rate of 1.1\% p.a., calculate the amount that is now in Hannah's account.
The balance of an investment earning interest compounded annually is modelled by the recurrence relation:V_{n + 1} = 1.06 V_{n}, V_{0} = 2000
Where V_{n} is the value of the investment after n years.
State the annual interest rate, as a percentage.
Determine the balance at the end of the first year.
Use the sequences facility on your calculator to find the balance at the end of 15 years.
The balance of an investment earning interest compounded monthly is modelled by the recurrence relation:V_{n + 1} = 1.025 V_{n}, V_{0} = 4000
Where V_{n} is the value of the investment n months after the initital investment at the begining of January 2010.
State the monthly interest rate, as a percentage.
Use the sequences facility on your calculator to find the balance at the end of the first year.
Use the compound interest formula to find the balance at the end of the first year and confirm the answer from the previous part.
Use the sequences facility on your calculator to find at the end of which month and year the investment is first worth double the initial amount invested.
\$4000 is invested at the beginning of the year in an account that earns 4\% per annum interest, compounded annually.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the beginning of year \left(n + 1\right).
\$5000 is invested at the beginning of the year in an account that earns 3\% per annum interest, compounded annually.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{n+1}, that gives the balance in the account at the end of year \left(n+1\right).
\$2000 is invested at the beginning of the year in an account that earns 8\% per annum interest, compounded quarterly.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{\left(n + 1\right)}, that gives the balance in the account at the end of the \left(n + 1\right) \text{th} quarter.
\$2000 is invested at the beginning of the year in an account that earns 4\% per annum interest, compounded quarterly.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the beginning of the \left(n + 1\right) \text{th} quarter.
\$3000 is invested at the beginning of the year in an account that earns 6\% per annum interest, compounded monthly.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the end of the \left(n + 1\right) \text{th} month.
\$3000 is invested at the beginning of the year in an account that earns 18\% per annum interest, compounded monthly.
Calculate the amount of money in the account at the end of the first year.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the beginning of the \left(n + 1\right) \text{th} month.
\$3000 is invested at 4\% p.a., compounded annually. The table below tracks the growth of the investment over three years:
| \text{Time period} \\ n \text{ (years)} | \text{Value at beginning} \\ \text{of time period} | \text{Interest earned} \\ \text{in time period} | \text{Value at end} \\ \text{of time period} |
|---|---|---|---|
| 1 | \$3000.00 | A | B |
| 2 | C | D | \$3244.80 |
| 3 | \$3244.80 | E | \$3374.59 |
Calculate the value of:
Find the total interest earned over the three years.
Maria invested \$1400 at 10\% p.a., compounded annually. The table below was created to track the growth of the investment over three years:
| \text{Year} | \text{Value at beginning} \\ \text{year} | \text{Interest} | \text{Value at end} \\ \text{of the year} |
|---|---|---|---|
| 1 | 1400.00 | ||
| 2 | |||
| 3 |
Complete the given table.
Find the total amount of interest earned over the 3 years.
State the total interest as a percentage of the initial investment, to two decimal places.
Over the same period of time \$1400 is invested at 10\% p.a. simple interest. Calculate the intertest earned over 3 years.
For the 3 years, calculate the additional amount of interest acculumated using the compound interest investment over the simple interest investment.
Neil 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:
| \text{Month} | \text{Balance } (\$) |
|---|---|
| \text{March} | 3825.00 |
| \text{April} | 3901.50 |
| \text{May} | 3979.53 |
Calculate the monthly interest rate, r, of his investment, as a percentage.
Calculate the nominal annual interest rate of his investment, as a percentage.
Find the amount Neil deposited into this savings account when he opened it.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the beginning of the \left(n + 1\right) \text{th} month.
Sally opened a savings account at the beginning of March 2012, where the interest is compounded quarterly. Her statements for June, September and December show her account balance at the beginning of each quarter:
| \text{Month} | \text{Balance } (\$) |
|---|---|
| \text{June} | 7800.00 |
| \text{September} | 8112.00 |
| \text{December} | 8436.48 |
Calculate the quarterly interest rate, r, of her investment, as a percentage.
Calculate the nominal annual interest rate of her investment, as a percentage.
Find the amount Sally deposited into this savings account when she opened it.
Write a recursive rule, V_{n + 1}, that gives the balance in the account at the end of the \left(n + 1\right) \text{th} quarter.
The following spreadsheet shows the balance (in dollars) in a savings account in 2013, where interest is compounded monthly.
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Month} | \text{Balance at the beginning} \\ \text{ of the month} | \text{Interest} | \text{Balance at the end} \\ \text{of the month} |
| 2 | \text{July} | 6000.00 | 120.00 | X |
| 3 | \text{August} | 6120.00 | 122.40 | 6242.40 |
| 4 | \text{September} | 6242.40 | Y | 6367.25 |
| 5 | \text{October} | Z | 127.35 | 6494.60 |
| 6 | \text{November} | 6494.60 | 129.89 | 6624.49 |
Calculate the value of X.
Calculate the monthly interest rate, as a percentage.
Calculate the value of Y.
Calculate the value of Z.
Write a recursive rule, B_{n + 1}, that gives the balance at the end of the \left(n + 1\right) \text{th} month, with July being the first month.
Write an explicit rule for B_{n}, the balance at the end of the n\text{th} month, with July being the first month.
The following spreadsheet shows the balance (in dollars) in a savings account in 2011, where interest is compounded quarterly.
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Quarter} | \text{Balance at the beginning of quarter} | \text{Interest} | \text{Balance at end of quarter} |
| 2 | 1 | Z | 240.00 | 8240.00 |
| 3 | 2 | 8240.00 | Y | 8487.20 |
| 4 | 3 | 8487.20 | 254.62 | X |
| 5 | 4 | 8741.82 | 262.25 | 9004.07 |
Calculate the value of X.
Calculate the value of Y.
Calculate the value of Z.
Calculate the quarterly interest rate, as a percentage.
Calculate the nominal annual interest rate, expressed as a percentage.
Write a recursive rule, B_{\left(n + 1\right)}, that gives the balance at the beginning of the \left(n + 1\right) \text{th} quarter.
Write an explicit rule for B_{n}, the balance at the beginning of the n \text{th} quarter.
The following spreadsheet shows the balance (in dollars) in a savings account in 2013, where interest is compounded monthly:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Month} | \text{Balance at the } \\ \text{beginning of month} | \text{Interest} | \text{Balance at end of the month} |
| 2 | \text{January} | 2000.00 | 20.00 | 2020.00 |
| 3 | \text{February} | 2020.00 | 20.20 | 2040.20 |
| 4 | \text{March} | 2040.20 | 20.40 | 2060.60 |
| 5 | \text{April} | 2060.60 | 20.61 | 2081.21 |
| 6 | \text{May} |
Calculate the monthly interest rate, as a percentage.
Complete the sixth row of the table.
Write a recursive rule, B_{n + 1}, that gives the balance at the beginning of the \left(n + 1\right) \text{th} month of 2013.
Use the sequences facility on your calculator to find the balance at the end of this year.
Calculate the total amount of interest earned over the year.
The following spreadsheet shows the balance (in dollars) in a savings account in 2012, where interest is compounded quarterly:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Quarter} | \text{Balance at the } \\ \text{beginning of quarter} | \text{Interest} | \text{Balance at end of quarter} |
| 2 | 1 | 8000.00 | 160.00 | 8160.00 |
| 3 | 2 | 8160.00 | 163.20 | 8323.20 |
| 4 | 3 | 8323.20 | 166.46 | 8489.66 |
| 5 | 4 |
Calculate the quarterly interest rate, as a percentage.
Comple the fifth row of the table.
Write a recursive rule, B_{n + 1}, that gives the balance at the end of the \left(n + 1\right) \text{th} quarter after the beginning of 2012.
Use the sequences facility on your calculator to find the balance at the beginning of the third year.
State the number of quarters, after the beginning of 2012, when the balance will reach at least double the initial investment of \$8000.
The following graph shows both an investment with simple interest, and one with compound interest, labelled Investment A and Investment B respectively:
Which investment has a higher principal amount?
Which investment has a higher final amount after 10 years?
After how many years are the investments equal in value?
Jack wants to invest some money and is considering three different compound investment options:
Which compounding investment rate yields the same as the simple interest investment shown in this graph after 10 years?
Which compounding investment rate yields the better investment overall after 12 years?
Sketch a graph that could represent a savings account that earns compound interest over 50 years.
When \$2700 is deposited, a bank offers two types of savings accounts:
Account A: After depositing \$2700 the account earns 2.9\% simple interest per annum
Account B: After depositing \$1650 the account earns 2.8\% compound interest per annum
Write an equation for the future value, FV, of Account A after n years.
Write an equation for the future value, FV, of Account B after n years.
Which account would have a greater balance in the 11th year?
Which account would have a greater balance in the 25th year?
Will Account B ever have a greater balance than Account A? Explain your answer.
When \$2550 is deposited, a bank offers two types of savings accounts:
Account A: After depositing \$2550 the account earns 2.9\% simple interest per annum.
Account B: After depositing \$2400 the account earns 2.8\% compound interest per annum.
Write an equation for the future value, FV, of Account A after n years.
Write an equation for the future value, FV, of Account B after n years.
Construct a graph that represents the equation found in part (a).
Construct a graph that represents the equation found in part (b).
Hence, which account will have a greater balance in the 25th year?
When \$4000 is deposited, a bank offers two types of savings accounts:
Account A: After depositing \$4000 the account earns 4\% simple interest per annum.
Account B: After depositing \$3500 the account earns 4\% compound interest per annum.
Write an equation for the future value, FV, of Account A after n years.
Write an equation for the future value, FV, of Account B after n years.
Construct a graph which represents equation found in part (a).
Construct a graph which represents equation found in part (b).
Hence, which account will have a greater balance in the 2nd year?