I am a bit confused about the following problem and I would like to have clarification.
A loan of $12,000$ was made with annual rate of $12\%$ convertible quarterly. Smith plans to make a payments of $750$ at the end of every 6 months until he completely pays off the loan. However, 3 months before the 9th payment Smith refinances the loan to a new rate of $9\%$ convertible monthly and pays $R$ each month. In exactly 30 payments after refinancing, Smith pays off the loan. Calculate $R$.
I understand thus far.
Until the 8th payment, the loan accrues $3\%$ interest every 3 months.
Smith is planning to pay the 9th payment 6 months after he pays the 8th payment, so 3 months before the 9th payment means 3 months after the 8th.
So, the oustanding balance right after the 8th payment, $OB_8$ accrues $3\%$ interest the moment Smith refinances.
The following is what my argument is.
The moment Smith makes his 9th payment, which is the moment he starts his monthly payments is 3 months after the refinancing, so the effective monthly interest $0.75\%$ is acrrued three times. i.e. The outstanding balance right before the 9th payment must be $OB_9=(1.03)(1.0075)^3OB_8$.
The following is my question.
The book that I am working on tells me that the outstanding balance is $(1.03)(1.0075)^2OB_8$.
This makes a rather significant difference in the answer I get and I cannot afford such mistakes in future practice. Can someone tell me what's going on?