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Compound Interest, SMB-030

Michelle decided to invest some of her money at a higher interest rate. She deposited $3000 in an account paying 8.2% per annum, compounding half yearly.

  1. Write down an expression involving the compound interest formula that can be used to find the value of Michelle’s $3000 investment at the end of two years. Find this value correct to the nearest cent.  (2 marks)

  2. How much interest will the $3000 investment earn over a four-year period?

     

    Write your answer correct to the nearest cent.  (2 marks)

Show Answers Only
  1. `$3523.09`
  2. `$1137.40`
Show Worked Solution

a.   `text{Interest rate}\ (r) = \frac{8.2%}{2} = 4.1% = 0.041\ \ \text{(per 6 months)}`

`text(Compounding periods)\ (n) = 2 xx 2 = 4`

`FV` `= PV(1+r)^n`
  `= 3000(1.041)^4`
  `= 3523.093…`
  `= $3523.09\ \ text{(nearest cent)}`

 

b.   `text{Compounding periods}\ (n) = 4 xx 2 = 8`

`FV` `= 3000(1.041)^8`
  `= 4137.396…`

 

`:.\ text(Interest earned)` `= 4137.40-3000`
  `= $1137.40\ \ text{(nearest cent)}`

Compound Interest, SMB-029

The golf club’s social committee has $3400 invested in an account which pays interest at the rate of 4.4% per annum compounding quarterly.

  1. Show that the interest rate per quarter is 1.1%.  (1 mark)

  2. Determine the value of the $3400 investment after three years.

     

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

  3. Calculate the interest the $3400 investment will earn over six years.

     

    Write your answer in dollars correct to the nearest cent.  (2 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `$3876.97`
  3. `$1020.86`
Show Worked Solution

a.   `text{Interest rate}\ = \frac{4.4}{4} = 1.1text{%  (per quarter)}`
 

b.   `text{Compounding periods}\ (n) = 3 xx 4 =12`

`FV` `= PV(1+r)^n`
  `= 3400(1.011)^12`
  `= 3876.973…`
  `= $3876.97`

 

c.   `text{Compounding periods}\ (n) = 6 xx 4 = 24`

`FV` `= 3400(1.011)^24`
  `= 4420.858…`

 
`text(Interest earned over 6 years)`

`= 4420.86-3400`

`= $1020.86\ \ text{(nearest cent)}`

Compound Interest, SMB-027 MC

An amount of $8000 is invested for a period of 4 years.

The interest rate for this investment is 7.2% per annum compounding quarterly.

The interest earned by the investment in the fourth year (in dollars) is given by

  1. `8000 xx 1.018^4 - 8000 xx 1.018^3`
  2. `8000 xx 1.072^4 - 8000 xx 1.072^3`
  3. `8000 xx 1.018^16 - 8000 xx 1.018^12`
  4. `8000 xx 1.072^16 - 8000 xx 1.072^12`
Show Answers Only

 `C`

Show Worked Solution

`text{Interest earned in 4th year}\ = FV\text{(after 4 years)}-FV\text{(after 3 years)}`

`text{Interest rate}\ (r) = \frac{7.2%}{4} = \frac{0.072}{4} = 0.018`

`text{Compounding periods (3 years)}\ = 3 xx 4 = 12`

`text{Compounding periods (4 years)}\ = 4 xx 4 = 16`
 

`FV\text{(after 4 years)}\ = PV(1+r)^n = 8000(1.018)^16`

`FV\text{(after 3 years)}\ = 8000(1.018)^12`

`:.\ text(Interest earned in 4th year) = 8000 xx 1.018^(16)-8000 xx 1.018^(12)`

`=>  C`

Compound Interest, SMB-026 MC

$10 000 is invested for five years. Interest is earned at a rate of 8% per annum, compounding quarterly.

Which one of the following calculations will give the total interest earned, in dollars, by this investment?

  1. `10\ 000 xx 1.02^5-10\ 000`  
  2. `10\ 000 xx 1.02^20-10\ 000`  
  3. `10\ 000 xx 1.08^5 -10\ 000`
  4. `10\ 000 xx 1.08^20-10\ 000`
Show Answers Only

`B`

Show Worked Solution

`text{Interest rate}\ (r) = \frac{8%}{4} =2% = 0.02\ text{(per quarter)}`

`text{Compounding periods}\ (n) =5 xx4=20`

`FV=PV(1+r)^n = 10\ 000(1.02)^20`

`:.\ text(Interest earned)= 10\ 000 xx 1.02^20-10\ 000`

`=>  B`

Compound Interest, SMB-025

$15 000 is invested for 12 months.

For the first six months the interest rate is 6.1% per annum compounding monthly.

After six months the interest rate increases to 6.25% per annum compounding monthly.

Calculate the total interest earned by this investment over 12 months, giving your answer to the nearest dollar.   (4 marks)

Show Answers Only

`$953`

Show Worked Solution

`text{In the first 6 months:}`

`text{Interest rate}\ (r) = \frac{0.061}{12}`

`FV(\text{after 6 months})` `=PV(1+r)^n`  
  `=15\ 000(1 + \frac{0.061}{12})^6`   
  `=$15\ 463.35…`  

 
`text{In the second 6 months:}`

`text{Interest rate}\ (r) = \frac{0.0625}{12}`

`FV(\text{after 2nd 6 months})` `=15\ 463.35(1 + \frac{0.0625}{12})^6`  
  `=$15\ 952.91…`   

 

`:.\ text(Interest)` `= 15\ 952.91-15\ 000`
  `= $953\ \ \text{(nearest dollar)}`

Compound Interest, SMB-021

Callum invests $20 000 in a term deposit account that adds 3.8% interest annually, calculated on the account balance at the end of each year.

Calculate the interest paid in the third year of Callum's investment.   (3 marks)

Show Answers Only

`$818.86`

Show Worked Solution

`r=3.8% = 0.038`

`text(Using)\ \ FV=PV(1+r)^n :`

`text(Value after 2 years)` `= 20\ 000(1.038)^2`
  `= $21\ 548.88`
 `text(Value after 3 years)` `= 20\ 000(1.038)^3`
  `= $22\ 367.74`

 

 `:.\ text(Interest paid in 3rd year)` `= 22\ 367.74-21\ 548.88`
  `= $818.86`

Compound Interest, SMB-020

Tim invests $3000 in a term deposit account that adds 6.5% interest annually, calculated on the account balance at the end of each year.

The interest paid in the fourth year is

  1. `$221.16`
  2. `$235.55`
  3. `$3623.85`
  4. `$3859.40`
Show Answers Only

`B`

Show Worked Solution

`r=6.5% = 0.065`

`text(Using)\ \ FV=PV(1+r)^n :`

`text(Value after 3rd year)` `= 3000(1.065)^3`
  `= $3623.85`
 `text(Value after 4th year)` `= 3000(1.065)^4`
  `= $3859.40`

 

 `:.\ text(Interest paid in 4th year)` `= 3859.40-3623.85`
  `= $235.55`

`=>  B`

Compound Interest, SMB-017

Ekamjot invests $13 000 for two years.

Interest is calculated at the rate of 7.2% per annum, compounding quarterly.

How much interest does Ekamjot earn from this investment, giving your answer to the nearest cent?   (2 marks)

Show Answers Only

`$1994.28`

Show Worked Solution

`text{Annual interest rate}\ = 7.2% = 0.072`

`text(Quarterly interest rate)\ (r) = \frac{0.072}{4} = 0.018`

`text(Compounding periods)\ (n) = 2 xx 4 =8`

`FV(text{after 2 years})` `= PV(1+r)^n`
  `=13\ 000(1 + 0.018)^8`
  `=$14\ 994.278….`
  `=$14\ 994.28`

 
`text{Interest earned}\ =14\ 994.28-13\ 000=$1994.28`

Compound Interest, SMB-016 MC

Mary invests $1200 for two years.

Interest is calculated at the rate of 3.35% per annum, compounding monthly.

The amount of interest she earns in two years is closest to

  1. `$6.71`
  2. `$80.40`
  3. `$81.75`
  4. `$83.03`
Show Answers Only

`D`

Show Worked Solution

`text{Annual interest rate}\ = 3.35% = 0.0335`

`text(Monthly interest rate)\ (r) = \frac{0.0335}{12}`

`text(Compounding periods)\ (n) = 2 xx 12 =24`

`FV(text{after 2 years})` `= PV(1+r)^n`
  `=1200(1 + \frac{0.0335}{12})^24`
  `=$1283.03…`

 
`text{Interest earned}\ =1283.03-1200=$83.03`

`=> D`