Compound Interest

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits $4000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $6$ %, compounded semi-annually.

1. Find the amount of interest that Mia will earn over the next $2.5$ years. [3]
   

Ella also deposits AUD into a bank account. Her bank pays a nominal annual $\text{interest}$ rate of $4$ %, compounded monthly. In $2.5$ years, the total amount in Ella's account will be $4000$ AUD.

2. Find the amount that Ella deposits in the bank account. [3]
   

[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of $74\hspace{0.15em}000$ Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of $5.5$ %, compounded monthly.

1. Find the amount of AUD Julia needs to invest to reach $74\hspace{0.15em}000$ AUD after $8$ years. Give your answer correct to the nearest dollar. [3]
   

Julia's parents offer to add $5000$ AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of $3.5$ %, compounded quarterly.

2. Find the number of years it would take Julia to save the $74\hspace{0.15em}000$ AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]
   

[Maximum mark: 6]

On Gary's $50$th birthday, he invests $\$P$ in an account that pays a nominal annual interest rate of $5$ %, compounded monthly.

The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, $\alpha$.

1. Find the value of $\alpha$, giving your answer to four significant figures. [3]
   

Gary makes no further deposits or withdrawals from the account.

2. Find the age Gary will be when the amount of money in his account will be double the amount he invested. [3]
   

[Maximum mark: 13]

Grant wants to save $\$40\hspace{0.15em}000$ over 5 years to help his son pay for his college tuition. He deposits $\$20\hspace{0.15em}000$ into a savings account that has an interest rate of $6\%$ per annum compounded monthly for $5$ years.

1. Show that Grant will not be able to reach his target. [2]
   

2. Find the minimum amount, to the nearest dollar, that Grant would need to deposit initially for him to reach his target. [3]
   

Grant only has $\$20\hspace{0.15em}000$ to invest, so he asks his sister, Caroline, to help him accelerate the saving process. Caroline is happy to help and offers to contribute part of her income each year. Her annual income is $\$37 \hspace{0.15em}500$ per year. She starts by contributing one fifth of her annual income, and then decreases her contributions by half each year until the target is reached. Caroline's contributions do not yield any interest.

3. Show that Grant and Caroline together can reach the target in 5 years. [4]
   

Grant and Caroline agree that Caroline should stop contributing once she contributes enough to complement the deficit of Grant's investment.

4. Find the whole number of years after which Caroline will will stop contributing. [4]
   

[Maximum mark: 15]

Bill takes out a bank loan of $\$100\hspace{0.15em}000$ to buy a premium electric car, at an annual interest rate of $5.49$%. The interest is calculated at the end of each year and added to the amount outstanding.

1. Find the amount of money Bill would owe the bank after $10$ years. Give your answer to the nearest dollar. [3]
   

To pay off the loan, Bill makes quarterly deposits of $\$P$ at the end of every quarter in a savings account, paying a nominal annual interest rate of $3.2$%. He makes his first deposit at the end of the first quarter after taking out the loan.

2. Show that the total value of Bill's savings after $10$ years is $P\bigg[\dfrac{1.008^{40}-1}{1.008-1}\bigg]$. [3]
   

3. Given that Bill's aim is to own the electric car after $10$ years, find the value for $P$ to the nearest dollar. [3]
   

Melinda visits a different bank and makes a single deposit of $\$\hspace{0.05em}Q$, the annual $\text{interest}$ rate being $3.5$%.

4. 1. Melinda wishes to withdraw $\$8000$ at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

      $$\dfrac{8000}{1.035} + \dfrac{8000}{1.035^2} + \dfrac{8000}{1.035^3} + \cdots + \dfrac{8000}{1.035^n}.$$

   2. Hence, or otherwise, find the minimum value of $Q$ that would permit Melinda to withdraw annual amounts of $\$8000$ indefinitely. Give your answer to the nearest dollar. [6]