Semiannually compound interest

If you deposit $ 5,000 into an account paying 8.25% annual interest compounded semiannually, how long until there is $9350 in the account?

Correct answer:

t =  96 m

Step-by-step explanation:

$$\begin{gathered} p=8.25\%=\frac{8.25}{100}=0.0825 \\ i=1+p/2=1+\frac{33}{400}/2=\frac{833}{800}\doteq 1.0413 \\ 5000\cdot i^n=9350 \\ {i}^n=9350/5000 \\ n\log i=\log 9350/5000 \\ n=\frac{\log (9350/5000)}{\log i}=\frac{\log (9350/5000)}{\log 1.0413}\doteq 15.4851 \\ {n}_1=\lceil n\rceil =\lceil 15.4851\rceil =16 \\ t=12/2\cdot n_1=12/2\cdot 16=96\text{m} \end{gathered}$$

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