Question 1
Two friends Amelia and Bill, each set themselves a target of saving $20 000. They each have $9000 to invest.
(a) Amelia invests her $9000 in an account that offers an interest rate of 7 % per annum compounded annually.
(i) Find the value of Amelia’s investment after 5 years to the nearest hundred dollars.
(ii) Determine the number of years required for Amelia’s investment to reach the target.
(b)
Bill invests his $9000 in an account that offers an interest rate of r % per annum compounded monthly, where r is set to two decimal places.
Find the minimum value of r needed for Bill to reach the target after 10 years.
(c) A third friend Chris also wants to reach the $20 000 target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.
(i) Show that Chris will never reach the target if his initial deposit is $9000.
(ii) Find the amount Chris needs to deposit initially in order to reach the target after 5 years. Give your answer to the nearest dollar.
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Question 2
In a local weekly lottery, tickets cost $2 each. In the first week of the lottery, a player will receive $D for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving $10 is 0.03. The grand prize in the first week of the lottery is $1000.
(a) Find the value of c.
(b) Determine whether this lottery is a fair game in the first week. Justify your answer.
If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be $2000 in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.
(c) Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of n for the value of the grand prize in the nth week of the lottery.
The wth week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the wth week, his expected profit is $p
(d) Find the value of p .
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Question 3
In this question, give all answers correct to 2 decimal places.
Raul and Rosy want to buy a new house and they need a loan of 170 000 Australian dollars (AUD) from a bank. The loan is for 30 years and the annual interest rate for the loan is 3.8 %, compounded monthly. They will pay the loan in fixed monthly instalments at the end of each month.
(a) Find the amount they will pay the bank each month.
(b)
(i) Find the amount Raul and Rosy will still owe the bank at the end of the first 10 years.
(ii) Using your answers to parts (a) and (b) (i), calculate how much interest they will have paid in total during the first 10 years.
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Question 4
Sam invests $ 1700 in a savings account that pays a nominal annual rate of interest of 2.74 %, compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.
(a) Find the amount that Sam will have in his account after 10 years. David also invests $ 1700 in a savings account that pays an annual rate of interest of r % , compounded yearly. David makes no further payments or withdrawals from this account.
(b) Find the value of r required so that the amount in David’s account after 10 years will be equal to the amount in Sam’s account.
(c) Find the interest David will earn over the 10 years.
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Question 5
Gemma and Kaia started working for different companies on January 1st 2011. Gemma’s starting annual salary was $45 000 , and her annual salary increases 2 % on January 1st each year after 2011.
(a) Find Gemma’s annual salary for the year 2021, to the nearest dollar. Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.
(b) Assuming Kaia’s annual salary can be approximately modelled by the equation S = ax + b , show that Kaia had a higher salary than Gemma in the year 2021, according to the model.
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Question 6
The value of a car is given by the function C = 40 000 (0.91)t, , where t is in years since 1 January 2023 and C is in USD ($).
(a) Write down the annual rate of depreciation of the car.
(b) Find the value of the car on 1 January 2028.
Alvie wants to buy this car. On 1 January 2023, he invested $15 000 in an account that earns 3 % annual interest compounded yearly. He makes no further deposits to, or withdrawals from, the account.
Alvie wishes to buy this car for its value on 1 January 2028. In addition to the money in his account, he will need an extra $M .
(c) Find the value of M.
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Question 7
Darren buys a car for $35 000 . The value of the car decreases by 15 % in the first year.
(a) Find the value of the car at the end of the first year.
After the first year, the value of the car decreases by 11 % in each subsequent year.
(b) Find the value of Darren’s car 10 years after he buys it, giving your answer to the nearest dollar.
When Darren has owned the car for n complete years, the value of the car is less than 10 % of its original value.
(c) Find the least value of n .
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Question 8
Tommaso and Pietro have each been given 1500 euro to save for college. Pietro invests his money in an account that pays a nominal annual interest rate of 2.75 %, compounded half-yearly.
(a) Calculate the amount Pietro will have in his account after 5 years. Give your answer correct to 2 decimal places. Tommaso wants to invest his money in an account such that his investment will increase to 1.5 times the initial amount in 5 years. Assume the account pays a nominal annual interest of r % compounded quarterly.
(b) Determine the value of r .
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Question 9
Juliana plans to invest money for 10 years in an account paying 3.5 % interest, compounded annually. She expects the annual inflation rate to be 2 % per year throughout the 10-year period. Juliana would like her investment to be worth a real value of $4000, compared to current values, at the end of the 10-year period. She is considering two options.
Option 1: Make a one-time investment at the start of the 10-year period.
Option 2: Invest $1000 at the start of the 10-year period and then invest $ x into the account at the end of each year (including the first and last years).
(a) For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.
(b) For option 2, find the minimum value of x that Juliana would need to invest each year. Give your answer to the nearest dollar.
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Question 10
On 1 January 2022, Mina deposited $ 1000 into a bank account with an annual interest rate of 4 % , compounded monthly. At the end of January, and the end of every month after that, she deposits $ 100 into the same account.
(a) Calculate the amount of money in her account at the start of 2024. Give your answer to two decimal places.
(b) Find how many complete months, counted from 1 January 2022, it will take for Mina to have more than $ 5000 in her account.
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Question 1
Two friends Amelia and Bill, each set themselves a target of saving $20 000. They each have $9000 to invest.
(a) Amelia invests her $9000 in an account that offers an interest rate of 7 % per annum compounded annually.
(i) Find the value of Amelia’s investment after 5 years to the nearest hundred dollars.
(ii) Determine the number of years required for Amelia’s investment to reach the target.
(b)
Bill invests his $9000 in an account that offers an interest rate of r % per annum compounded monthly, where r is set to two decimal places.
Find the minimum value of r needed for Bill to reach the target after 10 years.
(c) A third friend Chris also wants to reach the $20 000 target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.
(i) Show that Chris will never reach the target if his initial deposit is $9000.
(ii) Find the amount Chris needs to deposit initially in order to reach the target after 5 years. Give your answer to the nearest dollar.
(a)
(i)
`9000 xx (1 + 7/100)^5; 12622.965... THEN ($)12600`
(ii)
`9000 (1 + 7/100)^x = 20000; THEN 12 (years)`
(b)
attempt to substitute into compound interest formula (condone absence of compounding periods)
`9000 (1 + r / (100 xx 12))^(12 xx 10) = 20000 8.01170...`
r = 8.02 %
(c)
(i)
recognising geometric series (seen anywhere)
r = `4500/9000 = 1/2`
EITHER
considering `S_oo`
`4500/ (1 - 0.5) = 18,000`
correct reasoning that 18000 < 20000
THEN
Therefore, Chris will never reach the target.
(ii)
recognising geometric sum
`(u_1 (1 - 0.5^5)) / 0.5 = 20000`
10322.58...
($)10323
Question 2
In a local weekly lottery, tickets cost $2 each. In the first week of the lottery, a player will receive $D for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving $10 is 0.03. The grand prize in the first week of the lottery is $1000.
(a) Find the value of c.
(b) Determine whether this lottery is a fair game in the first week. Justify your answer.
If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be $2000 in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.
(c) Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of n for the value of the grand prize in the nth week of the lottery.
The wth week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the wth week, his expected profit is $p
(d) Find the value of p .
a)
considering that sum of probabilities is 1
`0.85 + c + 0.03 + 0.002 + 0.0001 = 1; 0.1179`
(b)
valid attempt to find
`E(D) = (0 xx 0.85) + (2 xx 0.1179) + (10 xx 0.03) + (50 xx 0.002) + (1000 xx 0.0001); E(D) = 0.7358`
No, not a fair game
for a fair game, would be $2
(c)
recognition of GP with r = 2
`1000 xx 2^(n-1)`
(d)
recognizing
correct expression for wth week (or wth week)
`(0 xx 0.85) + (2 xx 0.1179) + (10 xx 0.03) + (50 xx 0.002) + (1000 xx 2^(n-1) xx 0.0001)`
correct inequality (accept equation
`0.6358 + (1000 xx 2^(n-1) xx 0.0001) > 2`
EITHER
n - 1 > 3.76998
THEN
w = 5
expected profit per ticket = their E(D) - 2
= 0.2358
Question 3
In this question, give all answers correct to 2 decimal places.
Raul and Rosy want to buy a new house and they need a loan of 170 000 Australian dollars (AUD) from a bank. The loan is for 30 years and the annual interest rate for the loan is 3.8 %, compounded monthly. They will pay the loan in fixed monthly instalments at the end of each month.
(a) Find the amount they will pay the bank each month.
(b)
(i) Find the amount Raul and Rosy will still owe the bank at the end of the first 10 years.
(ii) Using your answers to parts (a) and (b) (i), calculate how much interest they will have paid in total during the first 10 years.
`(a) N = 360; I% = 3.8; PV = (+-)170000; FV = 0; P/Y = 12; C/Y = 12; PMT = 792.13`
(b)
(i) `N = 120; I% = 3.8; PV = (+-)170000; PMT = (+-)792.13; P/Y = 12; C/Y = 12; FV = 133019.94`
(ii) amount of money paid: 120 x 792.13 = 95055.60
loan paid off: 170000 - 133019.94 = 36,980.06
interest paid: 95055.60 - 36980.06 = 58,075.54
Question 4
Sam invests $ 1700 in a savings account that pays a nominal annual rate of interest of 2.74 %, compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.
(a) Find the amount that Sam will have in his account after 10 years. David also invests $ 1700 in a savings account that pays an annual rate of interest of r % , compounded yearly. David makes no further payments or withdrawals from this account.
(b) Find the value of r required so that the amount in David’s account after 10 years will be equal to the amount in Sam’s account.
(c) Find the interest David will earn over the 10 years.
(a)
`1700 (1 + 0.0274 / 2)^(2 xx 10)`
= 2231.71
(b)
`1700 (1 + r / 100)^10 = 2231.71...`
THEN
r = 2.75876
r = 2.76
(c)
531.71
Question 5
Gemma and Kaia started working for different companies on January 1st 2011. Gemma’s starting annual salary was $45 000 , and her annual salary increases 2 % on January 1st each year after 2011.
(a) Find Gemma’s annual salary for the year 2021, to the nearest dollar. Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.
(b) Assuming Kaia’s annual salary can be approximately modelled by the equation S = ax + b , show that Kaia had a higher salary than Gemma in the year 2021, according to the model.
(a)
using geometric sequence with r = 1.02
correct expression or listing terms correctly 45000 x 1.0210
(b)
finds `a = 1096.89... and b = - 2160753.8...` (accept b = - 2.16 x 106)
Kaia's salary in 2021 is 56063.21 (accept 56817.09 from b = - 2.16 x 106)
Kaia had a higher salary than Gemma in 2021
Question 6
The value of a car is given by the function C = 40 000 (0.91)t, , where t is in years since 1 January 2023 and C is in USD ($).
(a) Write down the annual rate of depreciation of the car.
(b) Find the value of the car on 1 January 2028.
Alvie wants to buy this car. On 1 January 2023, he invested $15 000 in an account that earns 3 % annual interest compounded yearly. He makes no further deposits to, or withdrawals from, the account.
Alvie wishes to buy this car for its value on 1 January 2028. In addition to the money in his account, he will need an extra $M .
(c) Find the value of M.
(a) 9%
(accept 0.09)
(b) t = 5
(seen anywhere) 24961.28... 25000 (dollars)
(c)
`15000 (1 + 3/100)^5 (= 17389.11...)`
THEN subtracting their value from their answer to part (b)
7572.17 ...
7570 (dollars)
Question 7
Darren buys a car for $35 000 . The value of the car decreases by 15 % in the first year.
(a) Find the value of the car at the end of the first year.
After the first year, the value of the car decreases by 11 % in each subsequent year.
(b) Find the value of Darren’s car 10 years after he buys it, giving your answer to the nearest dollar.
When Darren has owned the car for n complete years, the value of the car is less than 10 % of its original value.
(c) Find the least value of n .
(a)
recognition that a 15% loss leaves 85% OR finding 15% and subtracting from original 0.85 x 35000
= ($) 29750
(b)
29750 x 0.899
THEN value (FV) = $ 10423
(c)
attempt to solve the inequality (or equation)
`29750 xx 0.89^(n-1) < 3500`
n = 20
Question 8
Tommaso and Pietro have each been given 1500 euro to save for college. Pietro invests his money in an account that pays a nominal annual interest rate of 2.75 %, compounded half-yearly.
(a) Calculate the amount Pietro will have in his account after 5 years. Give your answer correct to 2 decimal places. Tommaso wants to invest his money in an account such that his investment will increase to 1.5 times the initial amount in 5 years. Assume the account pays a nominal annual interest of r % compounded quarterly.
(b) Determine the value of r .
(a)
`1500 (1 + 2.75 / (2 xx 100))^(2 xx 5) = 1719.49 euro`
(b)
`(1 + r / (4 xx 100))^(4 xx 5) = 1.5; r = 8.19 (8.19206...)`
Question 9
Juliana plans to invest money for 10 years in an account paying 3.5 % interest, compounded annually. She expects the annual inflation rate to be 2 % per year throughout the 10-year period. Juliana would like her investment to be worth a real value of $4000, compared to current values, at the end of the 10-year period. She is considering two options.
Option 1: Make a one-time investment at the start of the 10-year period.
Option 2: Invest $1000 at the start of the 10-year period and then invest $ x into the account at the end of each year (including the first and last years).
(a) For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.
(b) For option 2, find the minimum value of x that Juliana would need to invest each year. Give your answer to the nearest dollar.
(a)
METHOD – (with FV = 4000)
EITHER
`N = 10; I = 1.5; FV = 4000; P/Y = 1; C/Y = 1`
(b)
METHOD (Finding the future value of the investment using PV from part (a))
`N = 10; I = 3.5; PV = 3446.66...`
(FV=) 4861.87
so payment required (from TVM) will be $294 or $295
Question 10
On 1 January 2022, Mina deposited $ 1000 into a bank account with an annual interest rate of 4 % , compounded monthly. At the end of January, and the end of every month after that, she deposits $ 100 into the same account.
(a) Calculate the amount of money in her account at the start of 2024. Give your answer to two decimal places.
(b) Find how many complete months, counted from 1 January 2022, it will take for Mina to have more than $ 5000 in her account.
(a)
`N = 24; I = 4; PV = +-1000; PMT = +-100; P/Y = 12; C/Y = 12; FV = ($)3577.43`
(b)
N = 36.5 (36.4689...)
N = 37 (months)
Question 1
Two friends Amelia and Bill, each set themselves a target of saving $20 000. They each have $9000 to invest.
(a) Amelia invests her $9000 in an account that offers an interest rate of 7 % per annum compounded annually.
(i) Find the value of Amelia’s investment after 5 years to the nearest hundred dollars.
(ii) Determine the number of years required for Amelia’s investment to reach the target.
(b)
Bill invests his $9000 in an account that offers an interest rate of r % per annum compounded monthly, where r is set to two decimal places.
Find the minimum value of r needed for Bill to reach the target after 10 years.
(c) A third friend Chris also wants to reach the $20 000 target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.
(i) Show that Chris will never reach the target if his initial deposit is $9000.
(ii) Find the amount Chris needs to deposit initially in order to reach the target after 5 years. Give your answer to the nearest dollar.
Question 2
In a local weekly lottery, tickets cost $2 each. In the first week of the lottery, a player will receive $D for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving $10 is 0.03. The grand prize in the first week of the lottery is $1000.
(a) Find the value of c.
(b) Determine whether this lottery is a fair game in the first week. Justify your answer.
If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be $2000 in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.
(c) Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of n for the value of the grand prize in the nth week of the lottery.
The wth week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the wth week, his expected profit is $p
(d) Find the value of p .
Question 3
In this question, give all answers correct to 2 decimal places.
Raul and Rosy want to buy a new house and they need a loan of 170 000 Australian dollars (AUD) from a bank. The loan is for 30 years and the annual interest rate for the loan is 3.8 %, compounded monthly. They will pay the loan in fixed monthly instalments at the end of each month.
(a) Find the amount they will pay the bank each month.
(b)
(i) Find the amount Raul and Rosy will still owe the bank at the end of the first 10 years.
(ii) Using your answers to parts (a) and (b) (i), calculate how much interest they will have paid in total during the first 10 years.
Question 4
Sam invests $ 1700 in a savings account that pays a nominal annual rate of interest of 2.74 %, compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.
(a) Find the amount that Sam will have in his account after 10 years. David also invests $ 1700 in a savings account that pays an annual rate of interest of r % , compounded yearly. David makes no further payments or withdrawals from this account.
(b) Find the value of r required so that the amount in David’s account after 10 years will be equal to the amount in Sam’s account.
(c) Find the interest David will earn over the 10 years.
Question 5
Gemma and Kaia started working for different companies on January 1st 2011. Gemma’s starting annual salary was $45 000 , and her annual salary increases 2 % on January 1st each year after 2011.
(a) Find Gemma’s annual salary for the year 2021, to the nearest dollar. Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.
(b) Assuming Kaia’s annual salary can be approximately modelled by the equation S = ax + b , show that Kaia had a higher salary than Gemma in the year 2021, according to the model.
Question 6
The value of a car is given by the function C = 40 000 (0.91)t, , where t is in years since 1 January 2023 and C is in USD ($).
(a) Write down the annual rate of depreciation of the car.
(b) Find the value of the car on 1 January 2028.
Alvie wants to buy this car. On 1 January 2023, he invested $15 000 in an account that earns 3 % annual interest compounded yearly. He makes no further deposits to, or withdrawals from, the account.
Alvie wishes to buy this car for its value on 1 January 2028. In addition to the money in his account, he will need an extra $M .
(c) Find the value of M.
Question 7
Darren buys a car for $35 000 . The value of the car decreases by 15 % in the first year.
(a) Find the value of the car at the end of the first year.
After the first year, the value of the car decreases by 11 % in each subsequent year.
(b) Find the value of Darren’s car 10 years after he buys it, giving your answer to the nearest dollar.
When Darren has owned the car for n complete years, the value of the car is less than 10 % of its original value.
(c) Find the least value of n .
Question 8
Tommaso and Pietro have each been given 1500 euro to save for college. Pietro invests his money in an account that pays a nominal annual interest rate of 2.75 %, compounded half-yearly.
(a) Calculate the amount Pietro will have in his account after 5 years. Give your answer correct to 2 decimal places. Tommaso wants to invest his money in an account such that his investment will increase to 1.5 times the initial amount in 5 years. Assume the account pays a nominal annual interest of r % compounded quarterly.
(b) Determine the value of r .
Question 9
Juliana plans to invest money for 10 years in an account paying 3.5 % interest, compounded annually. She expects the annual inflation rate to be 2 % per year throughout the 10-year period. Juliana would like her investment to be worth a real value of $4000, compared to current values, at the end of the 10-year period. She is considering two options.
Option 1: Make a one-time investment at the start of the 10-year period.
Option 2: Invest $1000 at the start of the 10-year period and then invest $ x into the account at the end of each year (including the first and last years).
(a) For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.
(b) For option 2, find the minimum value of x that Juliana would need to invest each year. Give your answer to the nearest dollar.
Question 10
On 1 January 2022, Mina deposited $ 1000 into a bank account with an annual interest rate of 4 % , compounded monthly. At the end of January, and the end of every month after that, she deposits $ 100 into the same account.
(a) Calculate the amount of money in her account at the start of 2024. Give your answer to two decimal places.
(b) Find how many complete months, counted from 1 January 2022, it will take for Mina to have more than $ 5000 in her account.
