Sara has just purchased a house. She has a $360,000, 30-year mortgage with interest at an annual rate of 5.5% a year, compounded continuously. (a) Assume that Sara pays her mortgage continuously with a fixed amount, then her monthly payment is $___.
(b) If she pays an extra $300 each month, then she can pay off the mortgage in about years and months.
(c) Sara can save about $by paying an extra $300 each month.

a) $2043.14

b) 22.19 years

c) $113,995.6

Step-by-step explanation:

Value of Mortgage ( P ) = $360,000,

Loan Tenure = 30 years

Annual Interest rate = 5.5% = 0.055

monthly interest rate ( m ) = 0.055 / 12 = 0.00458

number of payments ( n ) = 30 * 12 = 360

a) Determine Sara's monthly payment given that Sara pays continuously at a fixed amount

Monthly payments ( M ) = ( P * m ( 1 + m )^n ) / (( 1 + m )^n - 1 )

= ( 360,000 * 0.00458 ( 1.00458)^360 / ((1.00458)^360 - 1 )

= $2043.14

b) when she pays a n extra $300 every month determine time ( n ) when she can pay off the mortgage

i.e. M = 2034.14 + 300 = $2334.14

calculate the value of n from the equation below

M = ( P * m ( 1 + m )^n ) / (( 1 + m )^n - 1 )

n = - In | 1 -(( Pm) /M )) |/ In | 1 + m |

= -In | 1 - (( 360,000 * 0.00458) / 2334.14 )) | / 0.0045

= -In ( 1 - 0.70638436 ) / 0.0045

= 1.225 / 0.0046 = 266.30 months = 22.19 years

c) Determine how much Sara can save

Total amount paid using fixed monthly payment = 2043.14 * 30 * 12 = $735,530.4

Total amount paid by adding an extra $3000 each month

= 2334.14 * 22.19 * 12 = $621,534.80

amount saved = 735,530.4 - 621,534.80 = $113,995.6

b

step-by-step explanation: