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Financial Maths, GEN1 2022 VCAA 24 MC

On 1 January 2020, Dion invested $10 500 into an investment account paying compound interest of 0.52% quarterly.

At the end of each quarter, after the interest was credited, Dion added an additional amount of money.

Let \(D_n\) represent the additional amount, in dollars, added at the end of quarter \(n\).

This additional amount per quarter is modelled by the recurrence relation

\(D_1=C,\ \ \ D_{n+1}=D_n\)

The balance of Dion's investment account on 1 January 2022 was $12 700.95

The value of \(C\) is

  1. $71.69
  2. $215.55
  3. $260.22
  4. $270.15
  5. $275.12
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Annual interest rate}\ = 0.52  \times 4 = 2.08\%\)

\(D_n = D_{n+1}\ \text{indicates the additional payment is constant}\) 

\(\text{By TVM Solver:}\)

\(N\) \(=4 \times 2 = 8\)  
\(I\%\) \(=2.08\)  
\(PV\) \(=-10\ 500\)  
\(PMT\) \(=?\)  
\(FV\) \(=12\ 700.95\)
  
\(\text{P/Y}\) \(=4\)  
\(\text{C/Y}\) \(=4\)  

 
\(PMT = -215.55\)

\(\Rightarrow B\)


♦♦ Mean mark 32%.

CORE, FUR2 2020 VCAA 9

Samuel opens a savings account.

Let `B_n` be the balance of this savings account, in dollars, `n` months after it was opened.

The month-to-month value of `B_n` can be determined using the recurrence relation shown below.

`B_0 = 5000, qquad B_(n+1) = 1.003B_n`

  1. Write down the value of `B_4`, the balance of the savings account after four months.
  2. Round your answer to the nearest cent.   (1 mark)

  3. Calculate the monthly interest rate percentage for Samuel’s savings account.   (1 mark)

  4. After one year, the balance of Samuel’s savings account, to the nearest dollar, is $5183.

     

    If Samuel had deposited an additional $50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year?

     

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

Show Answers Only
  1. `$5060.27`
  2. `0.3 text(%)`
  3. `$610`
Show Worked Solution
a.   `B_1` `= 1.003 (5000)`
  `B_2` `= 1.003^2 (5000)`

`vdots`

`:. B_4` `= 1.003^4 (5000)`
  `= $5060.27`

 

b.  `text(Monthly interest rate)`

`= (1.003-1) xx 100`

`= 0.3%`

 

c.   `text(Extra)\ =\ text(value of annuity after 12 months)`

`text(By TVM solver:)`

`N` `= 12`
`I(%)` `= 3.6`
`PV` `= 0`
`PMT` `= 50`
`FV` `= ?`
`text(PY)` `= text(CY) = 12`

 
`FV = 609.84`

`:.\ text(Extra money) = $610`

CORE, FUR1 2020 VCAA 24 MC

Manu invests $3000 in an account that pays interest compounding monthly.

The balance of his investment after `n` months, `B_n` , can be determined using the recurrence relation

`B_0 = 3000, qquad B_(n+1) = 1.0048 xx B_n`

The total interest earned by Manu’s investment after the first five months is closest to

  1. $57.60
  2. $58.02
  3. $72.00
  4. $72.69
  5. $87.44
Show Answers Only

`D`

Show Worked Solution
`text(Total interest)` `= 3000 xx 1.0048^5 – 3000`
  `~~ $72.69`

`=>  D`

CORE, FUR1 2020 VCAA 22 MC

An asset is purchased for $2480.

The value of this asset after `n` time periods, `V_n` , can be determined using the rule

`V_n = 2480 + 45n`

A recurrence relation that also models the value of this asset after `n` time periods is

  1. `V_0 = 2480, qquad V_(n + 1) = V_n + 45n`
  2. `V_n = 2480, qquad V_(n + 1) = V_n + 45n`
  3. `V_0 = 2480, qquad V_(n + 1) = V_n + 45`
  4. `V_1 = 2480, qquad V_(n + 1) = V_n + 45`
  5. `V_n = 2480, qquad V_(n + 1) = V_n + 45`
Show Answers Only

`C`

Show Worked Solution

`V_0 = 2480`

`V_(n+1)` `= 2480 + 45(n+1)`
  `= 2480 + 45n + 45`
  `= V_n + 45`

 
`=>  C`

CORE, FUR1 2019 VCAA 18 MC

The value of a compound interest investment, in dollars, after `n` years, `V_n`, can be modelled by the recurrence relation shown below.

`V_0 = 100\ 000, qquad V_(n + 1) = 1.01 V_n`

The interest rate, per annum, for this investment is

  1.     `0.01 text(%)`
  2.     `0.101 text(%)`
  3.     `1 text(%)`
  4.     `1.01 text(%)`
  5. `101 text(%)`
Show Answers Only

`C`

Show Worked Solution

`V_(n + 1) = 1.01 V_n`

`text(Interest) = 1%`

`=>  C`

CORE, FUR2 2018 VCAA 4

 

Julie deposits some money into a savings account that will pay compound interest every month.

The balance of Julie’s account, in dollars, after `n` months, `V_n` , can be modelled by the recurrence relation shown below.

`V_0 = 12\ 000, qquad V_(n + 1) = 1.0062 V_n` 

  1. How many dollars does Julie initially invest?   (1 mark)
  2. Recursion can be used to calculate the balance of the account after one month.
    1. Write down a calculation to show that the balance in the account after one month, `V_1`, is  $12 074.40.   (1 mark)
    2. After how many months will the balance of Julie’s account first exceed $12 300?   (1 mark)
  3. A rule of the form  `V_n = a xx b^n`  can be used to determine the balance of Julie's account after `n` months.
    1. Complete this rule for Julie’s investment after `n` months by writing the appropriate numbers in the boxes provided below.   (1 mark)

    2. Balance = 
       
      		  × 
       
      		  `n`
    3. What would be the value of  `n`  if Julie wanted to determine the value of her investment after three years?   (1 mark)

Show Answers Only

  1. `$12\ 000`
    1. `text(Proof)\ \ text{(See Worked Solutions)}`
    2. `4\ text(months)`
    1. `text(balance) = 12\ 000 xx 1.0062^n`
    2. `36`

Show Worked Solution

a.   `$12\ 000`
 

b.i.   `V_1` `= 1.0062 xx V_0`
    `= 1.0062 xx 12000`
    `= $12\ 074.40\ text(… as required.)`

 

b.ii.   `V_2` `= 1.0062 xx 12\ 074.40 = 12\ 149.26`
  `V_3` `= 1.0062 xx 12\ 149.26 = 12\ 224.59`
  `V_4` `= 1.0062 xx 12\ 224.59 = 12\ 300.38`

 
`:.\ text(After 4 months)`

 
c.i.  `text(balance) = 12\ 000 xx 1.0062^n`

 
c.ii.  `n = 12 xx 3 = 36`

CORE, FUR2 2017 VCAA 6

Alex sends a bill to his customers after repairs are completed.

If a customer does not pay the bill by the due date, interest is charged.

Alex charges interest after the due date at the rate of 1.5% per month on the amount of an unpaid bill.

The interest on this amount will compound monthly.

  1. Alex sent Marcus a bill of $200 for repairs to his car.

     

    Marcus paid the full amount one month after the due date.

     

    How much did Marcus pay?   (1 mark)

Alex sent Lily a bill of $428 for repairs to her car.

Lily did not pay the bill by the due date.

Let `A_n` be the amount of this bill `n` months after the due date.

  1. Write down a recurrence relation, in terms of `A_0`, `A_(n + 1)` and `A_n`, that models the amount of the bill.   (2 marks)

  2. Lily paid the full amount of her bill four months after the due date.

     

    How much interest was Lily charged?

     

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

Show Answers Only
  1. `$203`
  2. `A_o = 428,qquadA_(n + 1) = 1.015A_n`
  3. `$26.26\ \ (text(nearest cent))`
Show Worked Solution
a.    `text(Amount paid)` `= 200 + 200 xx 1.5text(%)`
    `= 1.015 xx 200`
    `= $203`

♦ Mean mark part (b) 47%.
MARKER’S COMMENT: A recurrence relation has the initial value written first. Know why  `A_n=428 xx 1.015^n`  is incorrect.

 

b.   `A_o = 428,qquadA_(n + 1) = 1.015A_n`

 

c.    `text(Total paid)\ (A_4)` `= 1.015^4 xx 428`
    `= $454.26`

♦♦ Mean mark part (c) 29%.

`:.\ text(Total Interest)` `= 454.26-428`
  `= $26.26\ \ (text(nearest cent))`

CORE, FUR1 SM-Bank 2 MC

Derek invests $48 000 in an investment that guarantees an annual interest rate of 4.8%, compounded monthly.

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

Which recurrence relation below models the investment?

  1. `V_0 = 48\ 000,qquadV_(n + 1) = 1.048V_n`
  2. `V_0 = 48\ 000,qquadV_(n + 1) = 1.0048V_n`
  3. `V_0 = 48\ 000,qquadV_(n + 1) = 1.48V_n`
  4. `V_0 = 48\ 000,qquadV_(n + 1) = 1.004V_n`
  5. `V_0 = 48\ 000,qquadV_(n + 1) = 1.04V_n`
Show Answers Only

`D`

Show Worked Solution

`text(Monthly interest rate)`

`= (4.8%)/12`

`= 0.4%`

`= 0.004`
 

`:. V_(n + 1) = 1.004V_n`

`=> D`