Determine the quarterly tax rate on a deposit of S/.8000 so that in 4 years an interest of S/.6400 is obtained.
+121 To determine the quarterly tax rate, we need to use the formula for compound interest:
\[A = P \times \left(1 + \frac{r}{n}\right)^{nt},\]
where:
- \(A\) is the amount of money accumulated after \(n\) years, including interest.
- \(P\) is the principal amount (initial deposit).
- \(r\) is the annual interest rate (decimal form).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested for.
In this case, \(P = 8000\), \(A = 8000 + 6400 = 14400\) (since the interest is \(6400\)), \(n = 4\) (since it's quarterly compounded), and \(t = 4\) years.
We need to solve for \(r\), the annual interest rate. Rearranging the formula gives:
\[r = n \times \left[\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right].\]
Substitute the known values:
\[r = 4 \times \left[\left(\frac{14400}{8000}\right)^{\frac{1}{4 \times 4}} - 1\right].\]
Calculate the expression within the square brackets first:
\[\left(\frac{14400}{8000}\right)^{\frac{1}{16}} = 1.2^{\frac{1}{16}}.\]
Now calculate \(1.2^{0.0625}\) (since \(1/16 = 0.0625\)):
\[1.2^{0.0625} \approx 1.00676874544.\]
Finally, plug this back into the formula:
\[r = 4 \times (1.00676874544 - 1) \approx 0.02107498178.\]
So, the quarterly tax rate (annual interest rate) is approximately \(0.021\) or \(2.1\%\).
