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Determine the quarterly tax rate on a deposit of S/.8000 so that in 4 years an interest of S/.6400 is obtained.

 Aug 14, 2023
 #1
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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 \(2.1\%\).

 Aug 14, 2023
 #2
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SpectraSynth: 

 

You made a typo at this step:

 

14,400 / 8,000 ==1.8

 

1.8^(1/16) ==1.0374198 - 1 x 400 ==14.967919% - nominal annual rate compounded quarterly.

 Aug 14, 2023

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