# The Johnson Company purchased

The Johnson Company purchased a building for \$200,000 at aninterest rate of 5% for 20 years. If they decided to pay \$,2400additional a year toward the borrowed principle, while everythingelse remained constant, how many years from today can they pay offthe mortgage

Loan Amount P = \$200000

Interest Rate = 5%

Number of payment periods = n = 20 years

Let the required annual payments made be X

Hence, the sum of present value of annual payments must be equalto the value of the loan amount

=> X/(1+r) + X/(1+r)2 +….+ X/(1+r)N =P

=> X[1- (1+r)-N]/r = P

=> X = rP(1+r)N/[(1+r)N-1]

Hence, Annual Payments=  rP(1+r)N/[(1+r)N-1]

= 200000*( 0.05)*(1+ 0.05)20/((1+0.05)20-1) = \$16048.52

However, actual payment made = A = 16048.52 + 2000 =18048.52

Let the number of years in which mortgage will be paid off bem

Hence, present value of all the future payments should be equalto the loan amount

Using the formula above,

A[1- (1+r)-m]/r = P

=> 18048.52[1- (1+0.05)-m]/0.05 = 200000

=> [1- (1+0.05)-m] = 200000*0.05/18048.52

=> (1+0.05)-m = 1 – 200000*0.05/18048.52

=> (1+0.05)-m = 0.4459

Take log on both sides

=> m = – ln (0.4459) / ln (1+0.05) = 16.55

Hence, the loan will be paid off in 16.55 years

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