Formula Sheet Simple interest PrT 1 =7 100 Compound interest A = PR" , where R = 1+- r 100 The annuities formula - reducing balance loans Q(Rn - 1) r A = PRn R-1 , where R = 1+ 100 The annuities formula - investments A = PRn + Q (Rn - 1) r R -1 , where R = 1 + 100 Interest-only loans (& Perpetuities) Payment (or Q) = Pr 100 Effective Interest Rate Teffective 100 -1)x 10Question 1 Nate is uprading his means of transportation and wishes to purchase a new electric scooter. The scooter he wants has a cash price of $1700. Nate could enter into an agreement to buy one of these scooters from a supplier where he pays a deposit of $425 and 18 monthly payments of $90. A flat rate of interest applies. a. What would be the best way to describe Nate's agreement: reducing balance loan, interest-only loan, hire purchase, cash payment? b. What is the total cost of the scooter under this agreement? C. How much money would Nate actually borrow under this agreement and how much interest would he pay? d i Write down the simple interest formula and change the formula so that the rate r is the subject. ii Find the annual flat rate of interest that applies to Nate's agreement. (Give your answer as a percentage to the nearest whole number). iii Suppose that Nate could negotiate a discount with the supplier so that his payments are reduced to $80 per month for 18 months. Determine the annual flat rate of interest (as a percentage to the nearest whole number) under these new conditions.Question 2 Nate has received a gift of $1500 from a generous aunt. However, this is not enough to pay cash for the scooter. To pay the cash price he needs a further $200 and he hopes to obtain this amount by investing the $1500 gift in two different ways; simple interest and compound interest. a Suppose Nate is able to invest his $1500 at a flat rate of 6% p.a. Write down the simple interest formula and change the formula so that the time period 7 is the subject. ii Hence find the investment time required so that he earns interest to the value of at least $200. Give your answer as a whole number of months. b Suppose Nate is able to invest his $1500 at a rate of 10% p.a., compounded quarterly. Interest is added to his account at the end of each quarter. He hopes to earn interest of $200. The compound interest formula is A = PR". Determine the values of A, P and R for this situation. Under these different conditions, find the investment time required so that he earns interest to the value of at least $200. Give your answer as a whole number of quarters.Question 3 A different electric scooter retail outlet offers an alternative payment plan for the type of scooter Nate wants to own. The purchase price is $2100, not $1700. Nate would be able to take the scooter home without paying a deposit on the understanding that monthly payments over 2 years would be required. Interest would be charged at a rate of 12% p.a. compounded monthly. The monthly repayment is calculated from the annuities formula: A, = PR" _ O(R" -1) R-1 a State the values of P, R and n. By substituting into the formula, show that the monthly repayment has to be $98.85 Show your working. C Use the formula to find the amount still owing after one year of repayments. Show your working. Give your answer to the nearest cent. d If Nate decided to accept this plan, how much interest would he end up paying over the term of the loan? Give your answer to the nearest cent.Question 4 (round all answers to 2 decimal places) Mortgages can be taken out at a fixed interest rate over a period of time, which means that the interest rate cannot be changed during the time period agreed upon. It is common for interest rates to be variable, which means that the bank can adjust them up or down at any time. At the time that a loan is taken out, fixed interest rates are usually higher than variable interest rates. Suppose that Mike and Eliza borrow $620 000 from the Commonwealth Bank at a fixed interest rate of 5.3% p.a., compounded monthly, for 10 years. How much would they still owe at the end of this time if they make monthly repayments of $4000? b Consider the following situation. Another couple, Dan and Olivia borrow the same amount ($620 000) as Mike and Eliza from the Commonwealth Bank. They borrow at the same time and make the same repayments of $4000. However, Dan and Olivia choose the variable interest rate of 4% p.a., compounded monthly. Suppose that after 4 years the interest rate increases to 5% p.a., compounded monthly, where it remains for the next 6 years. How much would Dan and Olivia still owe after 4 years? How much would Dan and Olivia still owe after 10 years? C Compare Mike and Eliza's situation after 10 years with that of Dan and Olivia. Which couple would be better off? Explain. d Another person, Cici, intends to borrow $500 000 from ANZ Bank at 5.5% p.a. compounded monthly as an interest-only loan to buy a two-bedroom apartment as an investment. She hopes eventually to make a capital gain (i.e. money arising from an increase in value of the apartment over time). What would Cici's monthly repayment be? ii How much would Cici still owe after 10 years?Question 5 Cici's mother, Rai, is about to retire and is wondering what to do with her superannuateon money. She has $900 000 to invest, and is considering the following options: Option A Option B Reducing Balance Annuity Perpetuity Interest: 5% per annum Interest: 3.8% per annum (compounding monthly) (compounding weekly) a. How many monthly payments would Rai receive if she invests in the annuity for 25 years? Calculate the payment she would receive per month if she invests in the reducing balance annuity for 25 years? Give your answer to 2 decimal places. b. If she invests $900 000 in the perpetuity, how much will Rai receive per week? Give your answer to 2 decimal places. How much interest would Rai earn over 25 years, if she chooses the perpetuity