Final answer:
The time to pay off half of a loan can be calculated using an amortization formula which requires knowing the loan amount, interest rate, and monthly payment amount. The calculation would reveal how long it takes with different payment amounts, under the condition that the payments are sufficient to cover interest and reduce the principal.
Step-by-step explanation:
The question pertains to the calculation of the time it takes to pay off half of a loan while considering the conditions such as interest rate, payment frequency, and installment amounts. In the scenario provided, we can use the formula for an amortizing loan to find how long it will take to pay off half the loan. This involves understanding concepts like the present value of an annuity and the effects of compounding interest.
For the specific case of a $20,000 car loan with a 6% annual interest rate converted monthly and a monthly payment of $500, you would first calculate the monthly interest rate, then set up and solve the loan amortization schedule equation to find the number of months it takes to reach the point where half the loan is paid off. Similar steps would follow to determine the timeline when paying $100 monthly; however, it's important to note that the loan may never be paid off if the payments do not cover the interest accruing each month.