Your friend tells you he has a very simple trick for taking one$-$third of the time it takes to repay your mortgage: Use your holiday bonus to make an extra payment on January $1$ of each year $($that is$,$ pay your monthly payment due on that day twice$).$ Assume that the mortgage has an original balance of $$100,000,$ has an original term of $30$ years, and has an APR of $12\mathrm{.}0\%.$

a$.$ If you take out your mortgage on January $1($so that your first payment is due on February $1),$ and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage?

b$.$ If you take out your mortgage on July $1($so that the first payment is on August $1),$ and you make the extra payment each January, in how many months will you pay off your mortgage?

c$.$ How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan?

Note: Make sure to round all intermediate calculations to at least $6$ decimal places.

a$.$ If you take out your mortgage on January $1($so that your first payment is due on February $1),$ and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage?

The loan payment is $$,($Round to the nearest cent.$)$

The approximate number of years to pay off the loan is years. $($Round to two decimal places.$)$

Because the mortgage will take about years to pay off this way$-$which $,$ close to $\frac{2}{3}$ of its life of $30$

years$-$your friend is

$($Round to the nearest integer and select from the drop$-$down menus.$)$

b$.$ If you take out your mortgage on July $1($so that the first payment is on August $1),$ and you make the extra payment each January, in how many months will you pay off your mortgage?

The approximate number of years is years. $($Round to two decimal places.$)$