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# Tax question

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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