What is the accumulated value at the end of two years? > Suppose we make a $\$100$ deposit at $t=0$ . A year later, you make a withdrawal of $\$50$. If we have an annual simple interest rate of $10\%$, what is the accumulated value at $t=2$? Well, obviously, at the end of a year we have $\$100\times1.10 = \$110 $ and if we withdraw $\$50$, then we have $\$60$ and finally at $t=2$ we have $\$60\times1.10$ which gives $\$66$. However, I don't understand why the solution key gives $\$65.46$. Am I missing something?

The only answer I can see is that the interest is **simple**. In other words, no interest is earned on the part of the balance that represents accumulated interest.

When you withdrew 50, the principal was 100 and the interest 10. Suppose the withdrawal was **allocated pro rata**. Then you withdrew 45.45 of principal, leaving 54.55 and you withdrew 4.55 of interest, leaving 5.45. Over the final year, the principal earned interest of 5.46, giving you a balance at the end of year 2 of 65.46