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QCAA General Mathematics — Loans, investments and annuities 2 Questions & Answers

Q7

2023

QCAA

Paper 2

5 marks

Q7

5 marks

Five years ago, a retiree invested ＄100 000 in a compound interest account earning 3.8% p.a. compounding monthly. They now intend to use the balance of the account to begin a perpetuity that will return 4% p.a. compounding annually and pay them ＄6000 each year.

Provide advice to the retiree about whether their compound interest investment is large enough to finance the perpetuity.

Reveal Answer

Compound interest investment
$A = P(1+i)^n$
$= 100\,000 \left( 1 + \frac{3.8}{12 \times 100} \right)^{5 \times 12}$
$= 120\,888.66$
The balance of the investment account is ＄120 888.66.

Perpetuity
$M = A \times i$
$6000 = A \times 0.04$
$A = \frac{6000}{0.04}$
$= 150\,000$
The present value of the perpetuity needs to be ＄150 000.
$120\,888.66 < 150\,000$

The compound interest investment will not provide enough money to finance the perpetuity.

Marking Criteria

Descriptor	Marks
correctly substitutes into an appropriate rule for compound interest investment	1
determines balance of investment account	1
correctly substitutes into an appropriate rule for perpetuity	1
determines present value of perpetuity	1
determines if the compound interest investment is large enough to finance the perpetuity	1

Q22

2025

VCAA

Paper 1

1 mark

Q22

1 mark

Rita opens a savings account with an initial deposit of ＄4000.

The account earns interest compounding weekly. After the interest is added each week, Rita deposits an additional ＄50 into the account.

Assume there are exactly 52 weeks in one year.

The annual interest rate, compounding weekly, that is required to achieve a balance of ＄14000 after three years is closest to

A

8.4%

B

14.2%

C

14.6%

D

17.2%

Reveal Answer

A

8.4%

Correct Answer

Correct. Setting up the annuity equation $14000 = 4000(1+\frac{r}{52})^{156} + 50\frac{(1+\frac{r}{52})^{156}-1}{\frac{r}{52}}$ or using a financial solver yields an annual rate of $r \approx 8.4\%$ .

B

14.2%

Incorrect. An annual interest rate of $14.2\%$ would result in a final balance of approximately ＄15,800, which exceeds the target of ＄14,000.

C

14.6%

Incorrect. An annual interest rate of $14.6\%$ would result in a final balance of approximately ＄16,100, which is significantly higher than the target of ＄14,000.

D

17.2%

Incorrect. An annual interest rate of $17.2\%$ would result in a final balance of approximately ＄17,700, which is much higher than the target balance of ＄14,000.

Q3

2021

QCAA

Paper 2

5 marks

Q3

5 marks

Jo contributes ＄2500 per quarter to an annuity earning 3.6% p.a. compounding quarterly.

At the end of 4 years, Jo makes a one-off extra contribution of ＄10 000 and continues with the regular quarterly contributions.

Determine the value of the annuity at the end of 6 years, to the nearest dollar.

Reveal Answer

Value of regular contributions
$M = 2500$
$i = \frac{3.6}{400} = 0.009$
$n = 6 \times 4 = 24$

$A = M \left( \frac{(1+i)^n - 1}{i} \right)$
$= 2500 \left( \frac{(1.009)^{24} - 1}{0.009} \right)$
$= 66\,639.94$

Value of extra payment
$P = 10\,000$
$i = \frac{3.6}{400} = 0.009$
$n = 2 \times 4 = 8$

$A = P(1+i)^n$
$= 10\,000(1.009)^8$
$= 10\,743.09$

Total value $= 66\,639.94 + 10\,743.09$
$= 77\,383.03$
$= ＄77\,383$

Marking Criteria

Descriptor	Marks
correctly determines the $i$ and $n$ values	1
substitutes into appropriate annuity rule	1
substitutes into appropriate rule	1
determines sum of two values	1
determines total value, rounded to the nearest dollar	1

Q7

2020

QCAA

Paper 2

7 marks

Q7

7 marks

A couple saved for their retirement by making the same monthly payments for 20 years into an account that earned 4.2% p.a. compounded monthly.

At the age of 65, the couple retired and used all their savings to purchase a perpetuity with an interest rate of 5.76% p.a. compounded monthly, paying ＄3600 each month.

How much did they save each month to prepare for their retirement?

Reveal Answer

Perpetuity — find the size of the savings
$M = 3600$
$i = \frac{0.0576}{12} = 0.0048$
$A = ?$

$A = \frac{M}{i}$
$= \frac{3600}{0.0048}$
$= 750\,000$

Use the total savings to find the size of the monthly payment
$A = 750\,000$
$M = ?$
$i = \frac{0.042}{12} = 0.0035$
$n = 20 \times 12 = 240$

$A = M \left(\frac{(1+i)^n-1}{i}\right)$

$750\,000 = M \times 375.13...$

$M = 1999.281$

The monthly savings were ＄1999.29.

Marking Criteria

Descriptor	Marks
Correctly determines the $i$ value	1
Correctly recalls the perpetuity rule	1
Determines purchase price of perpetuity	1
Correctly determines the $i$ and $n$ values	1
Correctly selects the appropriate annuity rule	1
Determines payment	1
Shows logical organisation, communicating key steps	1

Q24

2023

VCAA

Paper 1

1 mark

Q24

1 mark

The following recurrence relation models the value, $P_n$ , of a perpetuity after $n$ time periods.

$P_0 = a, \quad P_{n+1} = RP_n - d$

The value of $R$ can be found by calculating

A

$a + d$

B

$\frac{a + d}{a}$

C

$\frac{a + d}{d}$

D

$1 + \frac{a + d}{a}$

E

$1 + \frac{a + d}{d}$

Reveal Answer

A

$a + d$

This represents the value of $Ra$ , not $R$ . It fails to divide by the initial value $a$ when solving the equation.

B

$\frac{a + d}{a}$

Correct Answer

In a perpetuity, the balance remains constant over time, so $P_{n+1} = P_n = a$ . Substituting this into the recurrence relation gives $a = Ra - d$ , which algebraically solves to $R = \frac{a + d}{a}$ .

C

$\frac{a + d}{d}$

This incorrectly divides by the payment amount $d$ instead of the principal amount $a$ when solving for $R$ .

D

$1 + \frac{a + d}{a}$

This expression is equivalent to $1 + R$ , which does not correctly solve the perpetuity equation $a = Ra - d$ for $R$ .

E

$1 + \frac{a + d}{d}$

This expression is mathematically incorrect and does not follow from solving the constant balance equation $a = Ra - d$ .

Q21

2025

VCAA

Paper 1

1 mark

Q21

1 mark

Donald invests ＄250000 into a perpetuity at an interest rate of 5% per annum.

Donald receives a payment at the end of each year.

When will the sum of all annual payments to Donald first exceed ＄250000?

A

at the end of year 13

B

at the end of year 17

C

at the end of year 21

D

at the end of year 25

Reveal Answer

A

at the end of year 13

Incorrect. The annual payment is $0.05 \times 250,000 = 12,500$ . After 13 years, the sum of payments is $13 \times 12,500 = 162,500$ , which is less than ＄250,000.

B

at the end of year 17

Incorrect. The annual payment is ＄12,500. After 17 years, the sum of payments is $17 \times 12,500 = 212,500$ , which has not yet reached ＄250,000.

C

at the end of year 21

Correct Answer

Correct. The annual payment is $0.05 \times 250,000 = 12,500$ . After 20 years, the total received is exactly ＄250,000 ( $20 \times 12,500$ ), so the sum first exceeds ＄250,000 at the end of year 21.

D

at the end of year 25

Incorrect. Although the sum of payments after 25 years ( $25 \times 12,500 = 312,500$ ) exceeds ＄250,000, year 25 is not the first time this occurs.

Q21

2024

QCAA

Paper 1

3 marks

Q21

A perpetuity earns interest quarterly at 5.2% p.a. and pays ＄975 each quarter.

Q21a

1 mark

Determine the quarterly interest rate.

Reveal Answer

Quarterly interest rate, $i = \frac{5.2}{100 \times 4}$
$= 0.013$

Marking Criteria

Descriptor	Marks
correctly determines the quarterly interest rate	1

Q21b

2 marks

Calculate the value of the perpetuity.

Reveal Answer

$A = \frac{M}{i}$

$A = \frac{975}{0.013}$

$A = 75\,000$

The value of the perpetuity is ＄75 000.

Marking Criteria

Descriptor	Marks
substitutes into appropriate rule	1
calculates value of perpetuity	1

Q11

2023

QCAA

Paper 1

1 mark

Q11

1 mark

An annuity with an initial zero balance has ＄500 deposited at the end of every month. The annuity earns 4.8% p.a. interest, compounding monthly. At the end of the fourth month, the balance is closest to

A

＄2002

B

＄2008

C

＄2012

D

＄2014

Reveal Answer

A

＄2002

This option is incorrect because it significantly underestimates the interest earned. The total principal deposited is ＄2000, and the compound interest over four months amounts to approximately ＄12, not ＄2.

B

＄2008

This option is incorrect. While it accounts for some interest, it is lower than the actual future value calculated using the monthly compounding rate of $0.4\%$ .

C

＄2012

Correct Answer

This is the correct answer. Using the monthly interest rate $i = \frac{4.8\%}{12} = 0.004$ and the future value of an annuity formula $FV = 500 \times \frac{(1.004)^4 - 1}{0.004}$ , the balance is approximately ＄2012.03.

D

＄2014

This option is incorrect as it overestimates the final balance. The correct calculation yields a value closer to ＄2012.

Q2

2022

QCAA

Paper 2

5 marks

Q2

5 marks

Tam deposits a fixed amount at the end of each month into an account paying 8.6% p.a. compounding monthly. From an initial zero balance, she accumulates ＄51 343.85 in four years.

A financial planner has advised Tam that she would have been at least ＄3000 better off if she had instead deposited half of the fixed amount at the end of each fortnight into an account paying 7.9% p.a. compounding fortnightly.

Evaluate the reasonableness of this advice.

Reveal Answer

Monthly amount
$A = M \left( \frac{(1+i)^n - 1}{i} \right)$
$51\,343.85 = M \left( \frac{(1+\frac{0.086}{12})^{48} - 1}{\frac{0.086}{12}} \right)$
$51\,343.85 = M \times 57.0487$

$\therefore M = 900$

Fortnightly annuity balance
$A = M \left( \frac{(1+i)^n - 1}{i} \right)$
$A = 450 \left( \frac{(1+\frac{0.079}{26})^{104} - 1}{\frac{0.079}{26}} \right)$
$= 54\,941.61$

$\text{Diff} = 54\,941.61 - 51\,343.85$
$= 3597.76$

The advice that she would have been at least ＄3000 better off is reasonable as ＄3597.76 > ＄3000.

Marking Criteria

Descriptor	Marks
correctly substitutes parameters into the appropriate annuity rule	1
correctly determines the monthly amount	1
determines value of fortnightly annuity	1
determines difference in annuity balances	1
compares values to evaluate the reasonableness of the advice	1

Q24

2024

VCAA

Paper 1

1 mark

Q24

1 mark

Andr0e invested ＄18 000 in an account for five years, with interest compounding monthly.

He adds an extra payment into the account each month immediately after the interest is calculated.

For the first two years, the balance of the account, in dollars, after $n$ months, $A_n$ , can be modelled by the recurrence relation

$A_0 = 18\,000, \quad A_{n+1} = 1.002 A_n + 100$

After two years, Andr0e decides he would like the account to reach a balance of ＄30 000 at the end of the five years.

He must increase the value of the monthly extra payment to achieve this.

The minimum value of the new payment for the last three years is closest to

A

＄189.55

B

＄195.45

C

＄202.35

D

＄246.55

Reveal Answer

A

＄189.55

Correct Answer

First, calculate the balance after 24 months: $A_{24} = 18000(1.002)^{24} + 100 \frac{1.002^{24}-1}{0.002} \approx ＄21336.77$ . Then, solve for the new payment ( $PMT$ ) over the remaining 36 months: $30000 = 21336.77(1.002)^{36} + PMT \frac{1.002^{36}-1}{0.002}$ , yielding $PMT \approx ＄189.78$ , making this the closest option.

B

＄195.45

This option is incorrect and likely results from miscalculating the accumulated balance after the first two years or using an incorrect number of compounding periods for the second phase.

C

＄202.35

This option is incorrect. A common mistake is calculating the future value of the principal and the annuity separately without compounding the intermediate balance correctly over the final three years.

D

＄246.55

This option is incorrect and likely stems from using the original ＄18,000 principal instead of the accumulated two-year balance as the starting amount for the final three years.

Q14

2021

SCSA

Paper 2

11 marks

Q14

Patrick has retired and invested his lump sum superannuation payout of ＄717 850 at a rate of 5.7% per annum compounded monthly. He begins the investment strategy from 1 January.

Q14a

Patrick will receive ＄4500 at the end of each month for general living expenses and will also receive a further ＄4000 at the end of each year for an annual holiday.

Q14a (i)

1 mark

Identify this type of investment account.

Reveal Answer

Annuity

Marking Criteria

Descriptor	Marks
states correct answer	1

Q14a (ii)

4 marks

Determine the balance in the account at the end of the first year.

Reveal Answer

N = 12, I = 5.7, PV = $-717\ 850$ , PMT = 4500, P/Y = 12, C/Y = 12
FV = 704 420.20

Balance at end of year 1 = $704\ 420.20 - 4000 = ＄700\ 420.20$

Marking Criteria

Descriptor	Marks
uses at least 4 correct values for N, I, PV, PMT, P/Y, C/Y	1
uses all correct values for N, I, PV, PMT, P/Y, C/Y	1
determines correct value for FV	1
determines correct end of year balance	1

Q14a (iii)

3 marks

Determine the balance in the account at the end of the second year.

Reveal Answer

N = 12, I = 5.7, PV = $-700\ 420.20$ , PMT = 4500, P/Y = 12, C/Y = 12
FV = 685 970.53

Balance at end of year 2 = $685\ 970.53 - 4000 = ＄681\ 970.53$

Marking Criteria

Descriptor	Marks
uses correct value for PV	1
determines correct FV	1
determines correct end of year 2 balance	1

Q14b

3 marks

When Patrick retired, he also considered the option of setting up a perpetuity with his superannuation payout still at 5.7% per annum compounded monthly. Calculate the quarterly payments Patrick would have received with this perpetuity in place.

Reveal Answer

N = 2 (can be any value), I = 5.7, PV = $-717\ 850$ , FV = 717 850, P/Y = 4, C/Y = 12

Quarterly payments = ＄10 278.03

Marking Criteria

Descriptor	Marks
uses at least 4 correct values for N, I, PV, FV, P/Y, C/Y	1
uses all correct values for N, I, PV, FV, P/Y, C/Y	1
states correct quarterly payments	1

Q22

2025

QCAA

Paper 1

4 marks

Q22

A ＄50 000 perpetuity earning fortnightly interest at 4.94% p.a. provides a regular fortnightly payment.

Q22a

2 marks

Calculate the fortnightly payment.

Reveal Answer

Fortnightly payment = $\frac{0.0494}{26} \times 50 000 =＄95$

Marking Criteria

Descriptor	Marks
correctly provides mathematical reasoning or working to support the answer	1
calculates fortnightly payment	1

Q22b

2 marks

Calculate the perpetuity's effective annual rate of interest as a percentage.

Reveal Answer

\begin{align*} i_{\text{effective}} &= \left(1 + \frac{i}{n}\right)^n - 1\\ &= \left(1 + \frac{0.0494}{26}\right)^{26} - 1\\ &\approx 0.05059 \end{align*}

The effective annual rate of interest is 5.06% p.a.

Marking Criteria

Descriptor	Marks
correctly provides mathematical reasoning or working to support the answer	1
calculates effective interest rate as a percentage	1

Q7

2024

VCAA

Paper 2

4 marks

Q7

Emi decides to invest a ＄300000 inheritance into an annuity.

Let $E_n$ be the balance of Emi's annuity after $n$ months.

A recurrence relation that can model the value of this balance from month to month is

$E_0 = 300000, \quad E_{n+1} = 1.003E_n - 2159.41$

Q7a

1 mark

Showing recursive calculations, determine the balance of the annuity after two months.
Round your answer to the nearest cent.

Reveal Answer

$E_0 = 300\,000$

$E_1 = 1.003 \times 300\,000.00 - 2159.41 = 298\,740.59$

$E_2 = 1.003 \times 298\,740.59 - 2159.41 = 297\,477.401 = ＄297\,477.40$

Marking Criteria

Descriptor	Marks
Shows recursive calculations and determines the correct balance after two months, rounded to the nearest cent (＄297,477.40)	1

Q7b

1 mark

For how many years will Emi receive the regular payment?

Reveal Answer

15 years

Marking Criteria

Descriptor	Marks
Correctly determines the number of years Emi will receive the regular payment (15)	1

Q7c

1 mark

Calculate the annual compound interest rate for this annuity.

Reveal Answer

3.6%

Marking Criteria

Descriptor	Marks
Correctly calculates the annual compound interest rate (3.6%)	1

Q7d

1 mark

If Emi wanted the annuity to act as a perpetuity, what monthly payment, in dollars, would she receive?

Reveal Answer

＄900

Marking Criteria

Descriptor	Marks
Correctly determines the monthly payment for a perpetuity (＄900)	1

Q12

2025

SCSA

Paper 2

13 marks

Q12

A retiring mathematics teacher donates ＄4000 to the school where she has worked for many years to pay for a prize to be awarded to a student at the school's annual prize-giving ceremony.

The school principal sets up an annuity with this money, receiving an interest rate of 0.3% compounded monthly and using ＄250 at the end of each year to purchase the prize.

Q12c

The school principal is considering changing this investment to a perpetuity after ten years so there will always be money available to award this prize. The financial institution at that time will offer them an annual interest rate of 4.2% compounded monthly.

The school principal states that the new minimum value of the annual prize should be ＄130.

Q12a (i)

1 mark

Calculate the nominal annual interest rate.

Reveal Answer

$0.3 \times 12 = 3.6\%$

Marking Criteria

Descriptor	Marks
states correct rate	1

Q12a (ii)

2 marks

Determine a recursive rule to model the balance of the annuity at the end of each year.

Reveal Answer

Effective annual rate of interest = 3.66% (CAS 2 d.p.)
$T_{n+1} = 1.0366T_n - 250, \quad T_0 = 4000$

Marking Criteria

Descriptor	Marks
determines effective annual rate of interest	1
states correct rule	1

Q12b (i)

1 mark

Determine how much money will be left in the annuity after five years.

Reveal Answer

$T_5 = 3442.669$

i.e. ＄3442.67

Marking Criteria

Descriptor	Marks
correctly determines the amount left after 5 years	1

Q12b (ii)

2 marks

Determine the number of years the school will be able to award this prize using this annuity.

Reveal Answer

$T_{24} = 123.30$ , $T_{25} = -122.18$
Therefore the school will be able to award the prize for 24 years.

Marking Criteria

Descriptor	Marks
correctly calculates $24^{\text{th}}$ and $25^{\text{th}}$ terms	1
correctly concludes it is 24 years	1

Q12c

4 marks

Show that the yearly perpetuity amount received by the school will be insufficient to purchase the annual prize.

Reveal Answer

\begin{align*} T_{10} &= 2775.60\\ 2775.60 &= 2775.60 \left(1 + \frac{0.042}{12}\right)^{12} - x\\ x &= 118.85 \end{align*}

Therefore yearly amount = ＄118.85
Therefore there is not enough money for the yearly prize (less than ＄130).

Marking Criteria

Descriptor	Marks
correctly determines value of annuity after 10 years	1
sets up correct equation	1
calculates correct yearly payment	1
states correct conclusion about amount of money for the yearly prize	1

Q12d

3 marks

Determine the largest number of years the school principal can maintain the annuity before changing to a perpetuity and receive enough to cover the annual prize of ＄130.

Reveal Answer

$T_9 = 2918.78$ , $T_8 = 3056.89$

uses finance app
I = 4.2, PV = $-2918.78$ , N = any positive value, FV = 2918.78, P/Y = 1, C/Y = 12,
gives PMT = 124.98, i.e. ＄124.98, which is not enough

uses finance app
I = 4.2, PV = $-3056.89$ , N = any positive value, FV = 3056.89, P/Y = 1, C/Y = 12,
gives PMT = 130.89, i.e. ＄130.89, which is enough

Therefore the school principal can maintain the annuity for eight years.

Marking Criteria

Descriptor	Marks
correctly determines value of annuity after nine and eight years	1
calculates correct yearly payments (PMT)	1
states that eight years is the largest number of years the school principal can maintain the annuity	1

Q10

2024

SCSA

Paper 2

13 marks

Q10

Matt is saving up to purchase a new boat. He deposits ＄14 500 into a savings account which is compounded monthly. The account pays an annual interest rate of 4.8% and he also deposits ＄300 into the account at the end of each month.

Q10c

After four years, Matt makes a one-off deposit of ＄2500 into the savings account. His goal is to have a total of ＄50 000 by the end of the fifth year.

Q10a (i)

1 mark

Calculate the monthly interest rate.

Reveal Answer

4.8 ÷ 12 = 0.4%

Marking Criteria

Descriptor	Marks
calculates correct rate	1

Q10a (ii)

2 marks

Determine a recursive rule to model the balance of the savings account at the end of each month.

Reveal Answer

$T_{n+1} = 1.004T_n + 300, \quad T_0 = 14 500$

Marking Criteria

Descriptor	Marks
states correct rule	1
states correct initial value	1

Q10b

2 marks

After how many months will the balance of Matt's account first exceed ＄20 000?

Reveal Answer

$T_{14} = 19 644.42$
$T_{15} = 20 023.00$
Therefore after 15 months

Marking Criteria

Descriptor	Marks
correctly calculates 14th and 15th terms	1
correctly concludes it is 15 months	1

Q10c

5 marks

Determine the equal monthly deposits during the fifth year he will need to make to reach this amount.

Reveal Answer

$T_{48} = 33 402.99$

33 402.99 + 2500 = 35 902.99

N = 12, I = 4.8, PV = –35 902.99, FV = 50 000, P/Y = 12, C/Y = 12
PMT = –1005.52

Therefore, deposits of ＄1005.52 per month

Marking Criteria

Descriptor	Marks
correctly calculates balance after 4 years	1
adds 2500 to balance after 4 years	1
states correct PV	1
correctly states the remaining parameters	1
determines correct monthly deposit	1

Q10d

3 marks

Matt purchases his new boat, which costs him ＄47 500. He decides to take the remaining money and re-invest it in one of the following high-interest savings accounts.

Option 1: 5.52% per annum, compounded six-monthly.

Option 2: 5.5% per annum, compounded quarterly.

Determine which option Matt should choose, by calculating the effective annual rates of interest.

Reveal Answer

Option 1: $i_e = \left(1 + \frac{0.0552}{2}\right)^2 - 1 = 0.05596 \times 100 = 5.60$

Option 2: $i_e = \left(1 + \frac{0.055}{4}\right)^4 - 1 = 0.0561 \times 100 = 5.61$

Therefore option 2 is the better choice as it has a higher effective interest rate.

Marking Criteria

Descriptor	Marks
correctly calculates effective interest rate for option 1	1
correctly calculates effective interest rate for option 2	1
correctly states option 2 is the better choice	1

QCAA General Mathematics Loans, investments and annuities 2

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