Equivalent Interest Rates

Question

Suppose that the APR on a certain product $x$ whose dollar value is $x_1$ is 5%. Now suppose we subtract some amount $x_2$ from $x_1$ where $x_2 < x_1$. Call this new amount $y$. How does one find the new percentage that generates the same dollar value in interest for $y$ as was generated from $x$?

Answer 1

Roughly speaking, you need the same interest from $y$ as you were getting, which is $x_1 \cdot 0.05$, so you need an interest rate of $0.05 \cdot \frac {x_1}y$ When you use APR, that comes with a lot of legal baggage specifying exactly how you calculate it, so the answer might be different. This says that if you reduce the loan balance by one third, you need the interest rate to go up to $7.5 \%$ to compensate.