$3.$ Nina and Rafael Ruiz are married and have just been approved for a thirty$-$year, $$150,000$ mortgage, with an APR of $7$ percent. However, they know that they want to make more than the regular payment in order to pay off the loan before thirty years. They have come to you to analyze two options. Option One requires a fixed monthly payment of $$1,200.$ Option Two requires they start with the regular payment but then increase that amount by an effective monthly rate of $0\mathrm{.}35$ percent.

Provide Nina and Rafael answers to the following questions.

a$)$Which payment method will result in a faster payoff?

b$)$What is the difference in total interest payments between the two alternative payment methods? Hint: Estimating the total of payments for Option Two involves using a formula for the partial sum of a geometric series.

c$)$Which repayment method results in higher home equity $($the lower loan balance$)$ after fourteen years?