How many SCSA Mathematics Applications questions cover Loans, investments and annuities?

AusGrader has 159 SCSA Mathematics Applications questions on Loans, investments and annuities, all with instant AI grading and detailed marking feedback.

WACE Mathematics Applications — Loans, investments and annuities Questions & Answers

Q17

2022

QCAA

Paper 1

4 marks

Q17

An investment of ＄50 000 that compounds interest monthly is modelled by the recurrence relation

$A_{n+1} = 1.00375A_n$ where $A_0 = 50\,000$ .

Q17a

2 marks

What would be the advertised interest rate per annum, compounding monthly?

Reveal Answer

$r = 1 + \frac{i}{n}$
$1.00375 = 1 + \frac{i}{12}$
$0.00375 = \frac{i}{12}$
$i = 0.045$

Therefore, the annual interest rate is 4.5% p.a. compounding monthly.

Marking Criteria

Descriptor	Marks
Correctly substitutes into an appropriate rule	1
Calculates annual interest rate	1

Q17b

2 marks

How many months would it take for the value of the investment to exceed ＄51 000?

Reveal Answer

$A_0 = 50\ 000$
$A_1 = 50\ 187.50$
$A_2 = 50\ 375.70$
$A_3 = 50\ 564.61$
$A_4 = 50\ 754.23$
$A_5 = 50\ 944.56$
$A_6 = 51\ 135.60$
Therefore, the investment would exceed ＄51 000 at 6 months.

Marking Criteria

Descriptor	Marks
correctly uses an appropriate method	1
determines when the investment would exceed $51 000	1

Q23

2025

VCAA

Paper 1

1 mark

Q23

1 mark

Virat invested ＄5000 into an account that earned interest compounding fortnightly.

The effective annual interest rate for Virat's investment was 4.51%.

Assume that there are exactly 26 fortnights in one year.

After five years, the amount of interest earned by Virat was closest to

A

＄1128

B

＄1234

C

＄1262

D

＄1264

Reveal Answer

A

＄1128

This incorrectly calculates simple interest instead of compound interest ( $5000 \times 0.0451 \times 5 = ＄1127.50$ ).

B

＄1234

Correct Answer

Using the effective annual rate, the total amount after 5 years is $5000(1 + 0.0451)^5 \approx ＄6234.20$ . The interest earned is the total amount minus the principal ( $＄6234.20 - ＄5000 = ＄1234.20$ ).

C

＄1262

This incorrectly treats 4.51% as the nominal annual rate compounded fortnightly, calculating $5000(1 + 0.0451/26)^{130} - 5000 \approx ＄1262$ .

D

＄1264

This incorrectly treats 4.51% as a nominal annual rate compounded daily, which would yield $5000(1 + 0.0451/365)^{1825} - 5000 \approx ＄1264$ .

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

Q12

2023

QCAA

Paper 1

1 mark

Q12

1 mark

A reducing balance loan with an initial balance of ＄6000 is modelled by the recurrence relation
$A_{n+1} = \left(1 + \frac{0.03}{12}\right)A_n - 400$ , where $n$ is the number of months.
The loan balance at the end of two months is closest to

A

＄5100

B

＄5200

C

＄5215

D

＄5230

Reveal Answer

A

＄5100

This value is incorrect. It is significantly lower than the calculated balance and likely results from a calculation error.

B

＄5200

This option incorrectly ignores the interest component. It simply subtracts two repayments from the principal: $6000 - 2(400) = 5200$ .

C

＄5215

This option fails to add interest for the second month. It takes the balance after one month ( $A_1 = 5615$ ) and subtracts the repayment without applying the interest factor: $5615 - 400 = 5215$ .

D

＄5230

Correct Answer

By applying the recurrence relation iteratively: $A_1 = 1.0025(6000) - 400 = 5615$ and $A_2 = 1.0025(5615) - 400 \approx 5229.04$ . This is closest to ＄5230.

Q16

2022

SCSA

Paper 2

7 marks

Q16

After paying a deposit for his new apartment, Declan obtains a bank loan for the remaining amount of ＄112 000 at 3.26% per annum compounded monthly. He can currently afford to repay ＄970 per month at the end of every month.

Q16a

3 marks

Calculate how much he would owe after the 40th repayment.

Reveal Answer

Using the financial app with
N = 40, I = 3.26, PV = -112 000, PMT = 970, P/Y = C/Y = 12
FV = 83 910.19

He would owe ＄83 910.19 after the 40th repayment

Marking Criteria

Descriptor	Marks
states at least 4 correct entries	1
states all correct entries	1
states correct answer	1

Q16b

4 marks

Declan decided to deposit a one-off extra amount of ＄1600, after the 16th repayment. Calculate the new amount he would owe after the 40th repayment.

Reveal Answer

Step 1. Using the financial app
N = 16, I = 3.26, PV = –112 000, PMT = 970, P/Y = C/Y = 12
FV = ＄101 128.46
New PV = ＄101 128.46 − 1600 = ＄99 528.46

Step 2. Using financial app
N = 24(40 − 16), I = 3.26, PV = −99 528.46, PMT = 970, P/Y = C/Y = 12
FV = ＄82 202.54

New amount owing after the 40th repayment is ＄82 202.54

Marking Criteria

Descriptor	Marks
states all correct entries in step 1	1
correctly subtracts 1600 from FV in step 1 to give new PV	1
states all correct entries in step 2	1
correctly determines new FV	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

Q6

2024

VCAA

Paper 2

2 marks

Q6

Emi invested profits of ＄10000 into a savings account that earns interest compounding fortnightly, for one year.

The effective interest rate, rounded to two decimal places, is 5.07%.

Assume that there are exactly 26 fortnights in a year.

Q6a

1 mark

What is the nominal percentage rate of interest for the account?

Round your answer to two decimal places.

Reveal Answer

4.95%

Marking Criteria

Descriptor	Marks
States the correct nominal percentage rate of interest (4.95%)	1

Q6b

1 mark

Explain why the nominal interest rate appears lower than the effective interest rate.

Reveal Answer

It does not take into account the fortnightly compounding.

Marking Criteria

Descriptor	Marks
Explains that the nominal interest rate does not take into account the effect of compounding (e.g., fortnightly compounding)	1

Q21

2023

VCAA

Paper 1

1 mark

Q21

1 mark

Use the following information to answer the question.

For taxation purposes, Audrey depreciates the value of her ＄3000 computer over a four-year period. At the end of the four years, the value of the computer is ＄600.

If Audrey uses reducing balance depreciation, the depreciation rate, per annum is closest to

A

10%

B

15%

C

20%

D

25%

E

33%

Reveal Answer

A

10%

Incorrect. A 10% reducing balance rate would leave a final value of $3000(1 - 0.10)^4 = ＄1968.30$ , which is much higher than the actual final value of ＄600.

B

15%

Incorrect. Applying a 15% reducing balance rate gives a final value of $3000(1 - 0.15)^4 = ＄1566.02$ , which does not match the target of ＄600.

C

20%

Incorrect. This represents the straight-line depreciation rate, where the computer loses ＄600 per year (20% of ＄3000), rather than the reducing balance rate.

D

25%

Incorrect. A 25% reducing balance rate results in a final value of $3000(1 - 0.25)^4 = ＄949.22$ , which is still higher than ＄600.

E

33%

Correct Answer

Correct. Using the reducing balance formula $V_n = V_0(1 - r)^n$ , we set up the equation $600 = 3000(1 - r)^4$ . Solving for $r$ yields $r = 1 - \sqrt[4]{0.2} \approx 0.331$ , which is closest to 33%.

Q25

2022

QCAA

Paper 1

5 marks

Q25

A couple borrow money to complete home renovations. Their bank has loaned the amount at 2.4% p.a. compounding monthly with repayments of ＄993.14 each month for 15 years.

Q25a

3 marks

Determine the amount of money borrowed.

Reveal Answer

$i = \frac{2.4}{1200} = 0.002$
$n = 15 \times 12 = 180$
$M = 993.14$

$A = M \left( \frac{1 - (1+i)^{-n}}{i} \right)$
$= 993.14 \left( \frac{1 - (1+i)^{-180}}{i} \right)$
$= 150\ 000.29$

They borrowed ＄150 000.

Marking Criteria

Descriptor	Marks
Correctly determines the $i, n$ and $M$ values	1
Substitutes into the appropriate annuity formula	1
Determines amount of money borrowed, including units	1

Q25b

2 marks

Write a recurrence relation for the amount owing after $n$ months.

Reveal Answer

$A_{n+1} = rA_n - R$
$A_{n+1} = \left( 1 + \frac{2.4}{1200} \right) A_n - 993.14$

$A_{n+1} = 1.002A_n - 993.14$

Marking Criteria

Descriptor	Marks
Correctly selects the appropriate formula	1
Determines recurrence relation	1

Q17

2021

QCAA

Paper 1

4 marks

Q17

4 marks

Determine the monthly repayment on a reducing balance loan of ＄720 000 at 4.8% p.a. over 25 years.
Give your answer to the nearest dollar.

Reveal Answer

$A = 720\,000$

$M = ?$
$i = \frac{0.048}{12} = 0.004$
$n = 25 \times 12 = 300$

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

$A = M\left(\frac{1-(1+0.004)^{-300}}{0.004}\right)$

$720\,000 = M \times 174.520 ...$

$M = \frac{720\,000}{174.520 ...}$

$M = 4125.578 ...$

The monthly repayment will be ＄4126 each month for 25 years.

Marking Criteria

Descriptor	Marks
correctly determines the $i$ and $n$ values	1
substitutes into appropriate annuity rule	1
determines monthly repayment	1
states solution to the nearest dollar	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

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

Q19

2023

QCAA

Paper 1

4 marks

Q19

4 marks

Ngarra compares two investment options and decides option A will provide the better return.

Option A: 5.60% p.a. compounding monthly
Option B: 5.62% p.a. compounding quarterly

Use the effective annual rate of interest formula to evaluate the reasonableness of Ngarra’s decision.

Reveal Answer

Option A: $i = 0.056, n = 12$
$i_{\text{effective}} = \left( 1 + \frac{i}{n} \right)^n - 1$
$= \left( 1 + \frac{0.056}{12} \right)^{12} - 1$
$\approx 0.05745...$

Option B: $i = 0.0562, n = 4$
$i_{\text{effective}} = \left( 1 + \frac{i}{n} \right)^n - 1$
$= \left( 1 + \frac{0.0562}{4} \right)^4 - 1$
$\approx 0.05739...$

$0.05745 > 0.05739$

Ngarra's decision is reasonable because option A has a higher effective interest rate.

Marking Criteria

Descriptor	Marks
Correctly substitutes into appropriate rule for either option	1
Calculates effective interest rate for option A	1
Calculates effective interest rate for option B	1
Provides a statement of reasonableness linked to effective interest rate	1

Q19

2025

VCAA

Paper 1

1 mark

Q19

1 mark

The table below compares the value of an asset at various times using the flat rate and reducing balance methods of depreciation.

	Flat rate (＄)	Reducing balance (＄)
Original value	60000.00	60000.00
Value after 1 year	56000.00	55200.00
Value after 2 years	52000.00	50784.00
Value after 3 years	48000.00	46721.28

After how many years will the value using flat rate depreciation first be lower than the value using reducing balance depreciation?

A

5

B

6

C

7

D

8

Reveal Answer

A

5

At year 5, the flat rate value (＄40,000) is still greater than the reducing balance value (＄39,544.89).

B

6

Correct Answer

At year 6, the flat rate value (＄36,000) drops below the reducing balance value (＄36,381.30) for the first time.

C

7

Although the flat rate value is lower than the reducing balance value at year 7, year 6 is the first time this occurs.

D

8

Although the flat rate value is lower than the reducing balance value at year 8, year 6 is the first time this occurs.

SCSA Mathematics Applications Loans, investments and annuities

Frequently Asked Questions

Ready to practise SCSA Mathematics Applications?

SCSA Mathematics Applications Loans, investments and annuities

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Sample Answer

Frequently Asked Questions

Ready to practise SCSA Mathematics Applications?