Paul has just been hired for a new job. To travel to his new job, he needed a new car. Reading the newspaper, he noticed an ad for a $2024$ Kia Sportage. It was just the car he wanted.

The ad quoted both a cash purchase price of $$33,650\mathrm{.}00$ and a monthly lease payment options. Since he did not have enough money to pay for the car, he would have to borrow the money from the bank, paying interest of $5\mathrm{.}3\%$ compounded annually on the loan.

The lease option showed payments of $$339$ a month for $48$ months with a $3,300\mathrm{.}00$ down payment or equivalent trade. Freight and air tax were included. Paul did not currently have a vehicle to offer as a trade$-$in$.$ If the vehicle was leased, after $48$ months it could be purchased for $$20,242\mathrm{.}00($residual value$).$ The lease was based on a finance interest rate of $5\mathrm{.}3\%.$ During the term of the lease, kilometers were limited to $81,600$km with a charge of $$0\mathrm{.}15$ per kilometer for excess kilometers. The cost included freight, but excluded taxes, registration, license, and dealer administration charges. Paul was particularly impressed with the $$seven$-$years or $115,000-$kilometre$$ warranty on the engine and transmission. They also offered $24-$hour roadside assistance.

Paul must decide whether to buy or lease this car. He lives in a province with a combined PST and GST tax rate of $13\%.$ He realizes that the costs of license and insurance must be paid, but he will ignore these in his calculations.

Questions

If Paul buys the car, what is the total purchase price, including taxes? Assume tax is charged on any additional costs.

Since Paul has no down payment, he must finance the car if he purchases it$.$

Is it cheaper to borrow money from the bank or to lease?

The dealer is offering a vehicle loan rate of $6\%$ compounded annually. Is it better to buy or to lease the car at this rate?

Find the rate of interest at which the cost of buying is the same as the cost of leasing. Calculate the answer to two decimals places.

Suppose Paul has a $$2000\mathrm{.}00$ down payment for this car.

What is the purchase price of the car if he pays cash for it$?$ Assume the down payment is subtracted from the price of the car including tax.

If the monthly lease payment is $$279\mathrm{.}00,$ is it cheaper to lease or buy the car if Paul can get the special dealer rate of $5\%?$

Is there an advantage to having a $$2000\mathrm{.}00$ down payment if you want to lease this car? Why or why not?