Compound Interest

[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 $1$st of January $2022$, Grace invests $\$P$ in an account that pays a nominal annual interest rate of $6$ %, compounded quarterly.

The amount of money in Grace'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]
   

Grace makes no further deposits or withdrawals from the account.

2. Find the year in which the amount of money in Grace's account will become triple the amount she 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: 14]

Alex and Julie each have a goal of saving $\$30\hspace{0.15em}000$ to put towards a house deposit. They each have $\$16\hspace{0.15em}000$ to invest.

1. Alex chooses his local bank and invests his $\$16\hspace{0.15em}000$ in a savings account that offers an interest rate of $5\%$ per annum compounded annually.

   1. Find the value of Alex's investment after $7$ years, to the nearest hundred dollars.

   2. Alex reaches his goal after n years, where n is an integer. Determine the value of n. [4]
      

2. Julie chooses a different bank and invests her $\$16\hspace{0.15em}000$ in a savings account that offers an interest rate of $r\%$ per annum compounded monthly, where $r$ is set to two decimal places.

   Find the minimum value of $r$ needed for Julie to reach her goal after $10$ years. [3]
   

3. Xavier also wants to reach a savings goal of $\$30 \hspace{0.15em}000$. He doesn't trust his local bank so he decides to put his money into a safety deposit box where it does not earn any interest. His system is to add more money into the safety deposit box each year. Each year he will add one third of the amount he added in the previous year.

   1. Show that Xavier will never reach his goal if his initial deposit into the safety deposit box is $\$16\hspace{0.15em}000$.

   2. Find the amount Xavier needs to initially deposit in order to reach his goal after $7$ years. Give your answer to the nearest dollar. [7]