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Sagot :
To determine the best interest rate we need to calculate how much he'll pay in total for the loan. We can do that for all the options and choose the one where he'll pay less.
For the first option we have:
[tex]\begin{gathered} M=C\cdot(1+\frac{r}{4})^{4\cdot t} \\ M=5000\cdot(1+\frac{0.05}{4})^{4\cdot5} \\ M=5000\cdot(1+0.0125)^{20}=6410.19 \end{gathered}[/tex]On the first loan he'll pay a total of $6410.19
For the second option:
[tex]\begin{gathered} M=C\cdot(1+r) \\ M=5000\cdot(1+0.05)=5000\cdot(1.05)=5250 \end{gathered}[/tex]On the second option he'll pay a total of $5250
For the third option we have:
[tex]\begin{gathered} M=C\cdot(1+\frac{r}{12})^{12\cdot5} \\ M=5000\cdot(1+\frac{0.05}{12})^{60} \\ M=5000\cdot(1+0.0042)^{60}=6429.6 \end{gathered}[/tex]On the third option he'll pay a total of $6429.6.
For the fourth option we have:
[tex]\begin{gathered} M=C\cdot(1+\frac{r}{365})^{365\cdot r} \\ M=5000\cdot(1+\frac{0.05}{365})^{365\cdot5} \\ M=5000\cdot(1+0.000137)^{1825}=6420.18 \end{gathered}[/tex]On the last option he'd pay a total of $6420.18
The best option is 5% simple interest, because he'd pay less.
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