Semiannually compound interest

If you deposit $5000 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:

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

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