Respuesta :
We know that the Compound Interest Formula is given as:
[tex]A=P(1+\frac{r}{n})^{n\times t}[/tex]
Where A is the amount,
P is the Principal,
r is the interest rate in decimal (converted from percentage)
n is the number of times interest is compounded per year
t is the number of years.
In our case we will have to find the Amount, A in one year when the principal, P is compounded quarterly at a rate of 6% and compare it with the amount when the same principal is compounded monthly at a rate of 1.25% monthly and see if they are equal or not and explain the possible reason.
Let us begin:
Case 1: 6% Quarterly
We know that there are four quarters in one year. Therefore, our n=4. Therefore, in this case the Amount A will be:
[tex]A=P(1+\frac{0.06}{4})^{4\times 1}=P(1+0.015)^{4}\approx1.061P[/tex]
Case 2 1.25% Monthly
[tex]A=P(1+\frac{0.0125}{12})^{12\times 1}\approxP(1+0.0010417)^{12}\approx1.0126P[/tex]
As we can clearly see the first case gives us 1.061P and the second case gives us 1.0126P.
Therefore, Naomi is incorrect in her reasoning. This can be seen from the fact that the and the are coming out to be different for both the cases and thus effecting the overall results.
Now, to find the correct monthly rate to make the amount the same as the quarterly rate of 1.061P, we will have to proceed as follows:
[tex]P(1+\frac{r}{12})^{12}=1.061P[/tex] (where r is the rate to be found)
[tex]1+\frac{r}{12}=(1.061)^{ \frac{1}{12}}[/tex]
[tex]1+\frac{r}{12}\approx1.00495[/tex]
[tex]\frac{r}{12}=0.00495[/tex]
[tex]r=0.00495\times 12=0.0594[/tex]
Thus, in percentage, r=5.94%.
Therefore, to make the quarterly amount the same as the monthly amount, the monthly rate should be 5.94%.
