I have been asked the following problem by a student of mine and there is a specific method that he requested.
A mortgage of $\$450,000$ is loaned for a monthly payment for $30$ years with nominal annual interest rate of $6\%$. This loaned is planned to be paid off by a monthly payment of $\$2697.98$. If one decides to pay $\$3500$ each month, how many years earlier will he/she finish paying the debt?
Fortunately I have been studying financial math these days, so I understand that
However, in order to solve the number of years that it will take to pay off the loan with monthly payment of $\$3500$, I am thinking that the number of months $n$ needed can be calculated by
$$n \approx 206.44$$
or about 17 years (therefore the answer is about 13 years earlier), which is a fairly complex calculation for a student who just started precalc.
The student knows the straight forward compound interest, and he claims that he has never seen or heard the word annuity. Is there a way to solve this problem without using the concept of annuity but compound interest?