What is the monthly interest rate being charged in this scen

What is the monthly interest rate being charged in this scenario? NO EXCEL and if you use hit and miss method please show all work Given, A college pass out owes $22,500 in loans. He is not able to get a job post his graduation and hence is willing to extend the loan for a time span of 20 years instead of the original 10. Doubling the repayment period would mean that the student pays approximately 29% less in monthly payments. However, the total amount of interest on the loan increases by 121%

Solution

We know that monthly payment = [P x R x (1+R)^N]/[(1+R)^N-1]

P = $22,500. N = 10*12 = 120 months. R = monthly interest rate and the value is unknown.

Thus monthly payment = [22500 x R x (1+R)^120]/[(1+R)^120-1]

Now when n = 20*12 = 240 then monthly payment = [22500 x R x (1+R)^240]/[(1+R)^240-1]

Now the new monthly payment is 29% less than the original monthly payments.

Thus (1-0.29)* [22500 x R x (1+R)^120]/[(1+R)^120-1] = [22500 x R x (1+R)^240]/[(1+R)^240-1]

Or 0.71*[22500 x R x (1+R)^120]/[(1+R)^120-1] = [22500 x R x (1+R)^240]/[(1+R)^240-1]

Solving the above equation mathematcially we get r = 0.00748944

Thus monthly interest rate = 0.00748944*100

= 0.7489% (4 decimal places) or 0.75% (2 decimal places) (this is the monthly rate and not the annual rate).

The answer can be verified by computing total interest in both cases. Interest in case of 10 years or 120 months = 11683.95

Interest in case of 20 years or 240 months = 26041.20

% difference = (26041.2-11683.95)/11683.95 = approximately 122% (this is more than 121% due to rounding off differences).