smc-2512-50-CAS solver

Financial Maths, GEN2 2025 VCAA 10

Using profits from his recent film, Declan invests $650 000 into a 10-year annuity.

The annuity earns interest at 6.4% per annum, compounding quarterly.

Declan receives a regular quarterly payment from the annuity.

Halfway through the 10-year annuity, Declan writes a recurrence relation to represent the quarter-to-quarter balance for the remainder of the annuity.

Let $D_n$ be the balance of Declan's annuity $n$ quarters after the halfway point of the annuity.

Complete the recurrence relation below in terms of $D_0, D_{n+1}$ and $D_n$ that can model this balance. (2 marks)

$D_0=\begin{array}{|c|}\hline \rule{0pt}{4ex} \quad \quad \quad \quad \quad & \quad \quad \quad \quad \quad \\ \hline \end{array}, D_{n+1}=1.016 \times D_n+\begin{array}{|c|}\hline \rule{0pt}{4ex} \quad \quad \quad \quad \quad & \quad \quad \quad \quad \quad \\ \hline \end{array}$

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$D_0=376\,159.43, D_{n+1}=1.016 \times D_n-22\,126.27$

Show Worked Solution

$\text{Find annuity quarterly payment.}$

$\text{Solve for \(PMT$ (by CAS):}\)

$N$	$=40$
$I(\%)$	$=6.4$
$PV$	$=-650\,000$
$PMT$	$=\boldsymbol{22\,126.27}$
$FV$	$=0$
$PY$	$=CY=4$

♦♦♦ Mean mark 17%.

$\text{Find annuity balance at halfway:}$

$\text{Solve for \(FV$ (by CAS):}\)

$N$	$=20$
$I(\%)$	$=6.4$
$PV$	$=-650\,000$
$PMT$	$=22\,126.27$
$FV$	$=\boldsymbol{376\,159.428}$
$PY$	$=CY=4$

$\text{Recurrence relation:}$

$D_0=376\,159.43, D_{n+1}=1.016 \times D_n-22\,126.27$

Financial Maths, GEN1 2025 VCAA 24 MC

Dennis invests $800 000 in an annuity.

The annuity earns interest at 4.8% per annum, compounding monthly.

After the interest is credited to the annuity, Dennis receives a monthly payment of $6000.

After some years, the balance of the annuity is $521118.96

For the remainder of the annuity, Dennis receives a monthly payment of $4767.66.

The total number of years for which Dennis receives payments is closest to

Show Answers Only

$A$

Show Worked Solution

$\text{Calculate time until annuity value is}\ \$521\,118.96.$

$\text{Solve for} \ N \ \text {(by CAS):}$

$N$	$=\boldsymbol{84}$
$I(\%)$	$=4.8$
$PV$	$=800\,000$
$PMT$	$=-6000$
$FV$	$=521\,118.96$
$PY$	$=CY=12$

♦ Mean mark 52%.

$\text{After 84 months, calculate time until annuity = 0.}$

$\text{Solve for} \ N \ \text {(by CAS):}$

$N$	$=\boldsymbol{144}$
$I(\%)$	$=4.8$
$PV$	$=521\,118.96$
$PMT$	$=-4767.66$
$FV$	$=0$
$PY$	$=CY=12$

$\therefore \ \text {Length of annuity}=\dfrac{84+144}{12}=19 \text { years.}$

$\Rightarrow A$

Financial Maths, GEN1 2025 VCAA 22 MC

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 $14 000 after three years is closest to

8.4%
14.2%
14.6%
17.2%

Show Answers Only

$A$

Show Worked Solution

$\text{Solve for} \ I \ \text {(by CAS):}$

$N$	$=3 \times 52=156$
$I(\%)$	$=\boldsymbol{8.372…}$
$PV$	$=-4000$
$PMT$	$=-50$
$FV$	$=14\,000$
$PY$	$=CY=52$

$\Rightarrow A$

Financial Maths, GEN1 2024 NHT 23 MC

Todd invested $450 000 in an annuity at the start of 2024.

The interest rate for this annuity is 3.75% per annum compounding monthly.

He will receive regular monthly payments for the 15-year life of the annuity.

In which year will the balance of the annuity first fall below $350 000?

2027
2028
2029
2030
2031

Show Answers Only

$B$

Show Worked Solution

$\text{Find monthly payment (by TVM Solver):}$

$N=15 \times 12 = 180$

$I\%=3.75\%$

$PV=450\,000$

$PMT = \text{SOLVE}\ = -3272.50$

$FV=0$

$P/Y = C/Y = 12$

$\text{Find \(N$ when balance = \$350 000 (by TVM Solver):}\)

$N= \text{SOLVE}\ = 49.61$

$I\%=3.75\%$

$PV=450\,000$

$PMT = -3272.50$

$FV= -350\,000 $

$P/Y = C/Y = 12$

$\text{49.61 months = 4.13 years}\ \ \Rightarrow \ \ \text{4 years after 2024 = 2028}$

$\Rightarrow B$

Financial Maths, GEN2 2024 VCAA 7

Emi decides to invest a $300 000 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

$E_0=300\,000, \quad E_{n+1}=1.003 E_n-2159.41$

Showing recursive calculations, determine the balance of the annuity after two months. Round your answer to the nearest cent. (1 mark)

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For how many years will Emi receive the regular payment? (1 mark)

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Calculate the annual compound interest rate for this annuity. (1 mark)

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If Emi wanted the annuity to act as a perpetuity, what monthly payment, in dollars, would she receive? (1 mark)

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a. $$297\,477.40$

b. $15\ \text{years}$

c. $3.6\%$

d. $$900$

Show Worked Solution

a. $E_0=300\,000$

$E_1=1.003\times 300\,000-2159.41=$298\,740.59$

$E_2=1.003\times 298\,740.59-2159.41=$297\,477.4018\approx $297\,477.40$

♦ Mean mark (a) 49%.

b. $\text{Using CAS:}$

$\text{Number of years}\ =\dfrac{180}{12}=15\ \text{years}$

♦ Mean mark (b) 42%.

c. $\text{Annual interest rate}\ =(1.003-1)\times 12\times 100\% = 3.6\%$

d. $\text{Method 1: Using CAS}$

$\text{Monthly payment}\ =$900$

$\text{Method 2:}$

$\text{Amount must be equal to the amount of monthly interest earned.}$

$\therefore\ \text{Monthly payment}\ =300\,000\times 0.003=$900$

♦ Mean mark (d) 47%.

Financial Maths, GEN1 2024 VCAA 24 MC

André 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, André 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

$189.55
$195.45
$202.35
$246.55

Show Answers Only

$A$

Show Worked Solution

$\text{Step 1: Using CAS}$

$\text{Annual interest rate}\ = (1.002-1) \times 12 = 0.024 = 2.4\% $

$\text{Compounding periods (2 years)}\ = 2 \times 12=24$

$\text{Value of investment after 2 years}\ =$21\,340.18$

♦ Mean mark 46%.

$\text{Step 2: Using CAS}$

$\text{Next 3 years:}\ N=3 \times 12 = 36$

$\text{Minimum value of new payment}\ =$189.55$

$\Rightarrow A$

CORE, FUR2 2021 VCAA 9

Sienna invests $152 431 into an annuity from which she will receive a regular monthly payment of $900 for 25 years. The interest rate for this annuity is 5.1% per annum, compounding monthly.

Let `V` be the balance of the annuity after `n` monthly payments. A recurrence relation written in terms of `V_0 , V_{n + 1}` and `V_n` can model the value of this annuity from month to month.
Showing recursive calculations, determine the value of the annuity after two months.
Round your answer to the nearest cent. (2 marks)

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After two years, the interest rate for this annuity will fail to 4.6%.
To ensure that she will still receive the same number of $900 monthly payments, Sienna will add an extra one-pff amount into the annuity at this time.
Determine the value of this extra amount that Sienna will add.
Round your answer to the nearest cent. (1 mark)

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`$151 \ 925.59`
`$7039.20`

Show Worked Solution

a. `text{Payments are monthly} \ => \ r = 5.1/12 = 0.425text(%) \ text{per month}`

`R`	`= 1 + r/100 = 1.00425`
`V_1`	`= RV_0-text{payment}`
	`= 1.00425 xx 152 \ 431-900`
	`= $152\ 178.83`
`:.V_2`	`= 1.00425 xx 152\ 178.83-900`
	`= $ 151 \ 925.59`

b. `text{Find} \ V_24 \ text{(annuity value after 2 years) by TVM Solver:}`

`N`	`= 24`
`I text{(%)}`	`= 5.1`
`PV`	`= -152 \ 431`
`PMT`	`= 900`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=> FV = 146 \ 073.7405`

`text{Find}\ PV\ text{of annuity needed (by TVM solver):}`

`N`	`= 243 xx 12 = 276`
`Itext{(%)}`	`= 4.6`
`PV`	`= ?`
`PMT`	`= 900`
`FV`	`= 0`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PV = -153\ 112.9399`

`:. \ text{Amount to add}`	`= 153 \ 112.94-146 \ 073.74`
	`= $ 7039.20`

CORE, FUR1 2021 VCAA 22 MC

Joanna deposited $12 000 in an investment account earning interest at the rate of 2.8% per annum, compounding monthly.

She would like this account to reach a balance of $25 000 after five years.

To achieve this balance, she will make an extra payment into the account each month, immediately after the interest is calculated.

The minimum value of this payment is closest to

$113.85
$174.11
$580.16
$603.22
$615.47

Show Answers Only

`B`

Show Worked Solution

`text{By TVM Solver:}`

`N`	`= 60`
`Itext{(%)}`	`= 2.8`
`PV`	`= -12 000`
`PMT`	`= ?`
`FV`	`= 25 000`
`text(P/Y)`	`= text(C/Y) = 12`

`:. PMT = 174.106 …`

`=> B`

CORE, FUR1 2021 VCAA 18-19 MC

Deepa invests $500 000 in an annuity that provides an annual payment of $44 970.55

Interest is calculated annually.

The first five lines of the amortisation table are shown below.

Part 1

The principal reduction associated with payment number 3 is

$17 962.40
$25 969.37
$27 008.15
$28 088.47
$44 970.55

Part 2

The number of years, in total, for which Deepa will receive the regular payment of `$44\ 970.55` is closest to

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ B`

Show Worked Solution

`text{Part 1}`

`text{Principal reduction}`	`= 449\ 060.08 – 422\ 051.93`
	`= 27\ 008.15`

`=> C`

`text{Part 2}`

♦ Mean mark part (2) 44%.

`text{Interest rate} = {20\ 000}/{500\ 000} = 4text{% p.a.}`

`text{Find}\ N\ text{by TVM solver:}`

`N`	`= ?`
`I(%)`	`= 4`
`PV`	`= -500\ 000`
`PMT`	`= 44\ 970.55`
`FV`	`= 0`
`text(P/Y)`	`= text(C/Y) = 1`

`:. N = 15.000`

`=> B`

Financial Maths, GEN1 2019 NHT 24 MC

Robyn has a current balance of $347 283.45 in her superannuation account.

Robyn’s employer deposits $350 into this account every fortnight.

This account earns interest at the rate of 2.5% per annum, compounding fortnightly.

Robyn will stop work after 15 years and will no longer receive deposits from her employer.

The balance of her superannuation account at this time will be invested in an annuity that will pay interest at the rate of 3.6% per annum, compounding monthly.

After 234 monthly payments there will be no money left in Robyn’s annuity.

The value of Robyn’s monthly payment will be closest to

$3993
$5088
$6664
$8051
$9045

Show Answers Only

`A`

Show Worked Solution

`text(Balance after 15 years (by TVM Solver):)`

`N`	`= 26 xx 15 = 390`
`Itext(%)`	`= 2.5`
`PV`	`= 347\ 283.45`
`PMT`	`= 350`
`FV`	`= ?`
`text(P/Y)`	`= 26`
`text(C/Y)`	`= 26`

`=>\ 670\ 724.87`

`text(Monthly payment (by TVM Solver)):`

`N`	`= 234`
`Itext(%)`	`= 3.6`
`PV`	`= 670\ 724.87`
`PMT`	`= ?`
`FV`	`= 0`
`text(P/Y)`	`= 12`
`text(C/Y)`	`= 12`

`=> PMT = −3993.31`

`=>\ A`

Financial Maths, GEN2 2019 NHT 8

A record producer gave the band $50 000 to write and record an album of songs.

This $50 000 was invested in an annuity that provides a monthly payment to the band.

The annuity pays interest at the rate of 3.12% per annum, compounding monthly.

After six months of writing and recording, the band has $32 667.68 remaining in the annuity.

What is the value, in dollars, of the monthly payment to the band? (1 mark)

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After six months of writing and recording, the band decided that it needs more time to finish the album.

To extend the time that the annuity will last, the band will work for three more months without withdrawing a payment.

After this, the band will receive monthly payments of $3800 for as long as possible.

The annuity will end with one final monthly payment that will be smaller than all of the others.

Calculate the total number of months that this annuity will last. (2 marks)

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Show Answers Only

`$3000`
`18`

Show Worked Solution

a. `text(By TVM Solver:)`

`N`	`= 6`
`I text(%)`	`=3.12`
`PV`	`=50\ 000`
`PMT`	`= ?`
`FV`	`=-32\ 667.68`
`text(PY)`	`= text(CY) = 12`

`=> PMT = -3000.00`

`:. \ text(Monthly payment) = $3000`

b. `text{Value after 3 more months (by TVM Solver):}`

`N`	`= 3`
`I text(%)`	`=3.12`
`PV`	`=32\ 667.68`
`PMT`	`= 0`
`FV`	`=?`
`text(PY)`	`= text(CY) = 12`

`=> FV = -32\ 923.15`

`text(Find) \ \ N \ text(when) \ FV= 0 \ text{(by TVM Solver):}`

`N`	`= ?`
`I text(%)`	`=3.12`
`PV`	`=-32\ 923.15`
`PMT`	`= 3800`
`FV`	`=0`
`text(PY)`	`= text(CY) = 12`

`=> N = 8.77`

`:. \ text(Total months of annuity)`	`= 6 + 3 + 9`
	`= 18`

Financial Maths, GEN2 2019 NHT 7

Tisha plays drums in the same band as Marlon.

She would like to buy a new drum kit and has saved $2500.

Tisha could invest this money in an account that pays interest compounding monthly.

The balance of this investment after `n` months, `T_n` could be determined using the recurrence relation below

`T_0 = 2500, \ \ \ \ T_(n+1) = 1.0036 xx T_n`

Calculate the total interest that would be earned by Tisha's investment in the first five months.

Round your answer to the nearest cent. (2 marks)

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Tisha could invest the $2500 in a different account that pays interest at the rate of 4.08% per annum, compounding monthly. She would make a payment of $150 into this account every month.

Let `V_n` be the value of Tisha's investment after `n` months.

Write down a recurrence relation, in terms of `V_0`, `V_n` and `V_(n + 1)`, that would model the change in the value of this investment. (1 mark)

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Tisha would like to have a balance of $4500, to the nearest dollar, after 12 months.

What annual interest rate would Tisha require?

Round your answer to two decimal places. (1 mark)

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Show Answers Only

`$45.33`
`V_0 = 2500, \ V_(n+1) = 1.0034 xx V_n + 150`
`5.87%`

Show Worked Solution

a. `T_1 = 1.0036 xx 2500 = 2509`

`T_2 = 1.0036 xx 2509 = 2518.0324`

`vdots`

`T_5 = 2545.33`

`:. \ text(Total interest) `	`= 2545.33-2500`
	`= $45.33`

b. `text(Monthly interest) = (4.08)/(12) = 0.34%`

`:. \ V_0 = 2500, \ V_(n+1) = 1.0034 xx V_n + 150`

c. `text(By TVM Solver:)`

`N`	`= 12`
`I text(%)`	`=?`
`PV`	`=-2500`
`PMT`	`=-150`
`FV`	`=4500`
`text(PY)`	`= text(CY)=12`

`=> I = 5.87%`

Financial Maths, GEN2 2019 NHT 8

Phil invests $200 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after `n` months, `A_n`, can be modelled by the recurrence relation

`A_0 = 200\ 000, qquad A_(n + 1) = 1.0035\ A_n - 3700`

What monthly payment does Phil receive? (1 mark)

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Show that the annual percentage compound interest rate for this annuity is 4.2%. (1 mark)

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At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.

What is the balance of the annuity when it first falls below the monthly payment amount?

Round your answer to the nearest cent. (1 mark)

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If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.

What monthly payment could Phil have received from this perpetuity? (1 mark)

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Show Answers Only

`$3700`
`text(Proof)\ text{(See Worked Solutions)}`
`$92.15`
`$700`

Show Worked Solution

a. `$3700`

b.	`text(Monthly rate)`	`= 0.0035 = 0.35%`
	`text(Annual rate)`	`= 12 xx 0.35 = 4.2%`

c. `text(Find)\ N\ text(when)\ FV = 0\ \ text{(by TVM solver)}:`

`N`	`= ?`
`I(%)`	`= 4.2`
`PV`	`= 200\ 000`
`PMT`	`= 3700`
`FV`	`= 0`
`text(P/Y)`	`= 12`
`text(C/Y)`	`= 12`

`=> N = 60.024951`

`text(Find)\ \ FV\ \ text(when)\ \ N = 60.024951\ \ text{(by TVM solver):}`

`=>FV = $92.15`

d. `text(Perpetuity) => text(monthly payment) = text(monthly interest)`

`:.\ text(Perpetuity payment)`	`= 200\ 000 xx 4.2/(12 xx 100)`
	`= $700`

CORE, FUR2 2018 VCAA 6

Julie has retired from work and has received a superannuation payment of $492 800.

She has two options for investing her money.

Option 1

Julie could invest the $492 800 in a perpetuity. She would then receive $887.04 each fortnight for the rest of her life.

At what annual percentage rate is interest earned by this perpetuity? (1 mark)

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Option 2

Julie could invest the $492 800 in an annuity, instead of a perpetuity.

The annuity earns interest at the rate of 4.32% per annum, compounding monthly.

The balance of Julie’s annuity at the end of the first year of investment would be $480 242.25

1. What monthly payment, in dollars, would Julie receive? (1 mark)
  
  --- 1 WORK AREA LINES (style=lined) ---
2. How much interest would Julie’s annuity earn in the second year of investment?
3. Round your answer to the nearest cent. (1 mark)
  
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Show Answers Only

`4.68 text(%)`
1. `$2800`
2. `$20\ 488.89`

Show Worked Solution

a.	`text(Annual interest)`	`= 26 xx 887.04`
		`= $23\ 063.04`

♦ Mean mark 41%.

`:.\ text(Annual percentage rate)`	`= (23\ 063.04)/(492\ 800)`
	`= 4.68 text(%)`

b.i. `text(Find the monthly payment by TVM Solver:)`

♦ Mean mark 48%.

`N`	`= 12`
`I(%)`	`= 4.32`
`PV`	`= -492\ 800`
`PMT`	`= ?`
`FV`	`= 480\ 242.25`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PMT = $2800.00`

♦♦♦ Mean mark 16%.

b.ii.	`text(Year 2 start balance)`	`= $480\ 242.25`
	`text(Year 2 end balance)`	`= $467\ 131.14`
	`text(Balance reduction)`	`= 480\ 242.25-467\ 131.14`
		`= 13\ 111.11`

`text(Year 2 total payment) = 12 xx 2800 = 33\ 600`

`:.\ text(Interest)`	`= 33\ 600-13\ 111.11`
	`= $20\ 488.89`

CORE, FUR1 2018 VCAA 24 MC

Mariska plans to retire from work 10 years from now.

Her retirement goal is to have a balance of $600 000 in an annuity investment at that time.

The present value of this annuity investment is $265 298.48, on which she earns interest at the rate of 3.24% per annum, compounding monthly.

To make this investment grow faster, Mariska will add a $1000 payment at the end of every month.

Two years from now, she expects the interest rate of this investment to fall to 3.20% per annum, compounding monthly. It is expected to remain at this rate until Mariska retires.

When the interest rate drops, she must increase her monthly payment if she is to reach her retirement goal.

The value of this new monthly payment will be closest to

$1234
$1250
$1649
$1839
$1854

Show Answers Only

`E`

Show Worked Solution

`text(Find)\ FV\ text{after 2 years (by TVM solver):}`

♦ Mean mark 45%.

`N`	`= 24`
`I(%)`	`= 3.24`
`PV`	`= −265\ 298.48`
`PMT`	`= −1000`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=>FV = $307\ 794.50\ text{(nearest cent)}`

`text(Find new monthly payment:)`

`N`	`= 8 xx 12 = 96`
`I(%)`	`= 3.20`
`PV`	`= –307\ 794.50`
`PMT`	`= ?`
`FV`	`= 600\ 000`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PMT = −1854.05`

`=> E`

CORE, FUR2 2017 VCAA 7

Alex sold his mechanics’ business for $360 000 and invested this amount in a perpetuity.

The perpetuity earns interest at the rate of 5.2% per annum.

Interest is calculated and paid monthly.

What monthly payment will Alex receive from this investment? (1 mark)

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Later, Alex converts the perpetuity to an annuity investment.

This annuity investment earns interest at the rate of 3.8% per annum, compounding monthly.

For the first four years Alex makes a further payment each month of $500 to his investment.

This monthly payment is made immediately after the interest is added.

After four years of these regular monthly payments, Alex increases the monthly payment.

This new monthly payment gives Alex a balance of $500 000 in his annuity after a further two years.

What is the value of Alex’s new monthly payment?

Round your answer to the nearest cent. (2 marks)

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Show Answers Only

`$1560`
`$805.65 (text(nearest cent))`

Show Worked Solution

a.	`text(Monthly payment)`	`= 360\ 000 xx 0.052/12`
		`= $1560`

♦ Mean mark part (a) 50%.

b. `text(By TVM Solver,)`

`text(Find balance after 4 years:)`

`N`	`= 4 xx 12 = 48`
`I(%)`	`= 3.8`
`PV`	`=-360\ 000`
`PMT`	`=-500`
`FV`	`= ?`
`text(PY)`	`=\ text(CY) = 12`

`=> FV`	`= 444\ 872.9445`

`:.\ text(Balance is $444 872.9445)`

♦♦ Mean mark 29%.
MARKER’S COMMENT: A common error was entering the $500 payment as a positive value. Know why this is incorrect!

`text(Find)\ \ PMT\ \ text(when)\ \ FV = 500\ 000\ \ text(and)\ \ N = 24:`

`N`	`= 24`
`I(%)`	`= 3.8`
`PV`	`=-444\ 872.9445`
`PMT`	`= ?`
`FV`	`= 500\ 000`
`text(PY)`	`= text(CY) = 12`

`=> PMT`	`=-805.6505…`

`:. text(Alex’s new monthly payment = $805.65 (nearest cent))`

CORE, FUR1 2016 VCAA 24 MC

Mai invests in an annuity that earns interest at the rate of 5.2% per annum compounding monthly.

Monthly payments are received from the annuity.

The balance of the annuity will be $130 784.93 after five years.

The balance of the annuity will be $66 992.27 after 10 years.

The monthly payment that Mai receives from the annuity is closest to

$1270
$1400
$1500
$2480
$3460

Show Answers Only

`C`

Show Worked Solution

`text(By TVM solver, starting at 5 years,)`

♦♦ Mean mark 30%.
MARKER’S COMMENT: Many students incorrectly chose `E`, by allocating the same sign to both the PV and FV.

`N`	`= 5 xx 12=60`
`I(text(%))`	`= 5.2`
`PV`	`= −130\ 784.93`
`PMT`	`= ?`
`FV`	`= 66\ 992.27`
`text(P/Y)`	`= text(C/Y) = 12`

`:. PMT`	`= 1500.00…`

`=> C`

CORE, FUR1 2016 VCAA 23 MC

Sarah invests $5000 in a savings account that pays interest at the rate of 3.9% per annum compounding quarterly. At the end of each quarter, immediately after the interest has been paid, she adds $200 to her investment.

After two years, the value of her investment will be closest to

$5805
$6600
$7004
$7059
$9285

Show Answers Only

`D`

Show Worked Solution

`text(By TVM Solver, after 2 years)`

`N`	`= 2 xx 4 = 8`
`I(text(%))`	`= 3.9`
`PV`	`= −5000`
`PMT`	`= −200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 4`

`:. FV`	`= 7059.25`

`=> D`

CORE*, FUR2 2006 VCAA 3

The company prepares for this expenditure by establishing three different investments.

$7000 is invested at a simple interest rate of 6.25% per annum for eight years.
Determine the total value of this investment at the end of eight years. (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
$10 000 is invested at an interest rate of 6% per annum compounding quarterly for eight years.
Determine the total value of this investment at the end of eight years.
Write your answer correct to the nearest dollar. (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
$500 is deposited into an account with an interest rate of 6.5% per annum compounding monthly.
Deposits of $200 are made to this account on the last day of each month after interest has been paid.
Determine the total value of this investment at the end of eight years.
Write your answer correct to the nearest dollar. (1 mark)
--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`$10\ 500`
`$16\ 103`
`$25\ 935`

Show Worked Solution

a.	`I`	`= (PrT)/100`
		`= (7000 xx 6.25 xx 8)/100`
		`= $3500`

`:.\ text(Total value of investment)`

`= 7000 + 3500`

`= $10\ 500`

b. `text(Compounding periods) = 8 xx 4 = 32`

`text(Interest rate)`	`= (text(6%))/4`
	`= 1.5text(% per quarter)`

`:.\ text(Total value of investment)`

`= PR^n`

`= 10\ 000(1.015)^32`

`= 16\ 103.24…`

`= $16\ 103\ \ text{(nearest $)}`

c. `text(By TVM Solver,)`

`N`	`= 8 xx 12 = 96`
`I(text(%))`	`= 6.5`
`PV`	`= 500`
`PMT`	`= 200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=> FV = −25\ 935.30…`

`:.\ text(Total value of investment is $25 935.)`

CORE*, FUR2 2012 VCAA 4

Arthur invested $80 000 in a perpetuity that returns $1260 per quarter. Interest is calculated quarterly.

Calculate the annual interest rate of Arthur’s investment. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
After Arthur has received 20 quarterly payments, how much money remains invested in the perpetuity? (1 mark)

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Arthur’s wife, Martha, invested a sum of money at an interest rate of 9.4% per annum, compounding quarterly.

She will be paid $1260 per quarter from her investment.

After ten years, the balance of Martha’s investment will have reduced to $7000.

Determine the initial sum of money Martha invested.

Write your answer, correct to the nearest dollar. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`6.3text(%)`
`$80\ 000`
`$35\ 208`

Show Worked Solution

a. `text(Let)\ \ r =\ text(annual interest rate)`

♦♦ Mean mark of all parts (combined) was 26%.

`80\ 000 xx r/(4 xx 100)`	`= 1260`
`:. r`	`= (1260 xx 400)/(80\ 000)`
	`= 6.3text(%)`

b. `$80\ 000`

`text{(The principal invested in a perpetuity}`

`text{remains unchanged.)}`

c. `text(Find)\ PV\ text(using TVM Solver:)`

`N`	`= 4 xx 10 = 40`
`I(text(%))`	`= 9.4`
`PV`	`= ?`
`PMT`	`= 1260`
`FV`	`= 7000`
`text(P/Y)`	`= text(C/Y) = 4`

`=> PV =-35\ 208.002…`

`:.\ text{Martha initially invested $35 208 (nearest $)}`

CORE*, FUR2 2013 VCAA 2

Hugo won $5000 in a road race and invested this sum at an interest rate of 4.8% per annum compounding monthly.

What is the value of Hugo’s investment after 12 months?
Write your answer in dollars, correct to the nearest cent. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---
1. Suppose instead that at the end of each month Hugo added $200 to his initial investment of $5000.
2. Find the value of this investment immediately after the 12th monthly payment of $200 is made.
3. Write your answer in dollars, correct to the nearest cent. (1 mark)
  --- 1 WORK AREA LINES (style=lined) ---
4. Assume Hugo follows the investment that is described in part b.i.
5. Determine the total interest he would earn over the 12-month period.
6. Write your answer in dollars, correct to the nearest cent. (1 mark)
  --- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`$5245.35`
1. `$7698.86`
2. `$298.86`

Show Worked Solution

a. `text(Monthly interest rate)`

♦ Mean mark for all parts combined was 39%.

`= 4.8/12`

`= 0.4text(%)`

`:.\ text(Value after 12 months)`

`= 5000(1 + 0.4/100)^12`

`= 5000(1.004)^12`

`= 5245.351…`

`= $5245.35\ \ text{(nearest cent)}`

b.i. `text(Using TVM Solver,)`

`N`	`= 12`
`I(text(%))`	`= 4.8`
`PV`	`=-5000`
`PMT`	`=-200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`FV = 7698.8614…`

`:. text(Value of interest) = $7698.86`

b.ii. `text(Total interest)`

`=\ text{Final value − (original + payments)}`

`= 7698.86-(5000 + 12 xx 200)`

`= 7698.86-7400`

`= $298.86`

CORE*, FUR2 2014 VCAA 3

The cricket club had invested $45 550 in an account for four years.

After four years of compounding interest, the value of the investment was $60 000.

How much interest was earned during the four years of this investment? (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Interest on the account had been calculated and paid quarterly.

What was the annual rate of interest for this investment?
Write your answer, correct to one decimal place. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

The $60 000 was re-invested in another account for 12 months.

The new account paid interest at the rate of 7.2% per annum, compounding monthly.

At the end of each month, the cricket club added an additional $885 to the investment.

1. The equation below can be used to determine the account balance at the end of the first month, immediately after the $885 was added.
2. Complete the equation by filling in the boxes. (1 mark)
  
  --- 0 WORK AREA LINES (style=lined) ---
3. What was the account balance at the end of 12 months?
4. Write your answer, correct to the nearest dollar. (1 mark)
  
  --- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`14\ 450`
`text(6.9%)`
1. `text(account balance)\ = 60\ 000 xx (1 + (7.2)/(12 xx 100)) + 885`
2. `75\ 443`

Show Worked Solution

a. `text(Interest earned)`

`= text(End value − original investment)`

`= 60\ 000-45\ 550`

`= $74\ 450`

♦ Mean mark of all parts (combined) was 40%.

b. `text(By TVM Solver:)`

`N`	`= 4 xx 4 = 16`
`I(text(%))`	`= ?`
`PV`	`=-45\ 550`
`PMT`	`= 0`
`FV`	`= 60\ 000`
`text(P/Y)`	`= text(C/Y) = 4`

`I(text(%)) = 6.948…`

`:.\ text(Annual interest rate = 6.9%)`

c.i. `text(account balance)`

`= 60\ 000 xx (1 + 7.2/(12 xx 100)) + 885`

c.ii. `text(By TVM Solver:)`

`N`	`= 12`
`I(text(%))`	`= 7.2`
`PV`	`=-60\ 000`
`PMT`	`=-885`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`FV = 75\ 443.014…`

`:. text(Balance after 12 months) = $75\ 443`

CORE*, FUR2 2015 VCAA 4

As their business grows, Jane and Michael decide to invest some of their earnings.

They each choose a different investment strategy.

Jane opens an account with Red Bank, with an initial deposit of $4000.

Interest is calculated at a rate of 3.6% per annum, compounding monthly.

Determine the amount in Jane’s account at the end of six months.

Write your answer correct to the nearest cent. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Michael decides to open an account with Blue Bank, with an initial deposit of $2000.

At the end of each quarter, he adds an additional $200 to his account.

Interest is compounded at the end of each quarter.

The equation below can be used to determine the balance of Michael’s account at the end of the first quarter.

account balance = 2000 × (1 + 0.008) + 200

Show that the annual compounding rate of interest is 3.2%. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Determine the amount in Michael’s account, after the $200 has been added, at the end of five years.

Write your answer correct to the nearest cent. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`$4072.54`
`3.2text(%)`
`$6664.63`

Show Worked Solution

a. `text(Amount in account after 6 months)`

`= 4000 xx (1 + 3.6/(12 xx 100))^6`

`=4072.542…`

`=$4072.54\ \ text{(nearest cent)}`

b. `text(Annual compounding rate)`

`=0.008 xx text(100%) xx 4`

`= 3.2text(%)`

c. `text(By TVM Solver, after 5 years)`

`N`	`=5 xx 4 = 20`
`I (%)`	`=3.2`
`PV`	`=-2000`
`PMT`	`=200`
`FV`	`=?`
`text(P/Y)`	`= text(C/Y) = 4`

`FV=6664.629…`

`:.\ text(The amount in Michael’s account is $6664.63)`

CORE*, FUR1 2005 VCAA 7 MC

Gregor invests $10 000 and earns interest at a rate of 6% per annum compounding quarterly.

Every quarter, after interest has been added, he withdraws $500.

At the end of four years, after interest has been added and he has made the $500 withdrawal, the value of the remaining investment will be closest to

A. $3720

B. $4220

C. $5440

D. $21 660

E. $22 160

Show Answers Only

`A`

Show Worked Solution

`text(By TVM Solver:)`

`N`	`= 16`
`I text(%)`	`=6`
`PV`	`=-10\ 000`
`PMT`	`= 500`
`FV`	`=?`
`text(PY)`	`= text(C/Y) = 4`

`:. FV = 3723.67`

`=> A`

\(N\)	\(=40\)
\(I(\%)\)	\(=6.4\)
\(PV\)	\(=-650\,000\)
\(PMT\)	\(=\boldsymbol{22\,126.27}\)
\(FV\)	\(=0\)
\(PY\)	\(=CY=4\)

\(N\)	\(=20\)
\(I(\%)\)	\(=6.4\)
\(PV\)	\(=-650\,000\)
\(PMT\)	\(=22\,126.27\)
\(FV\)	\(=\boldsymbol{376\,159.428}\)
\(PY\)	\(=CY=4\)

\(N\)	\(=\boldsymbol{84}\)
\(I(\%)\)	\(=4.8\)
\(PV\)	\(=800\,000\)
\(PMT\)	\(=-6000\)
\(FV\)	\(=521\,118.96\)
\(PY\)	\(=CY=12\)

\(N\)	\(=\boldsymbol{144}\)
\(I(\%)\)	\(=4.8\)
\(PV\)	\(=521\,118.96\)
\(PMT\)	\(=-4767.66\)
\(FV\)	\(=0\)
\(PY\)	\(=CY=12\)

\(N\)	\(=3 \times 52=156\)
\(I(\%)\)	\(=\boldsymbol{8.372…}\)
\(PV\)	\(=-4000\)
\(PMT\)	\(=-50\)
\(FV\)	\(=14\,000\)
\(PY\)	\(=CY=52\)