Question If a $$ \$ 1,555 $$ loan is paid off with a payment of 5,800 after 20 years, what is the interest rate compounded monthly? Round your answer to one decimal place.

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UpStudy AI · Accepted Answer

We start with the equation for compound interest:

$$
A = P(1 + i)^n
$$

where  
$$ A = 5800 $$ (the final amount),  
$$ P = 1555 $$ (the principal),  
$$ n = 20 \times 12 = 240 $$ months, and  
$$ i $$ is the monthly interest rate.

Substituting the given values, we have:

$$
5800 = 1555(1+i)^{240}
$$

Solve for $$(1+i)^{240}$$:

$$
(1+i)^{240} = \frac{5800}{1555}
$$

Calculate the ratio:

$$
\frac{5800}{1555} \approx 3.7328
$$

Now, solve for $$1+i$$ by taking the 240th root:

$$
1+i = \left(3.7328\right)^{\frac{1}{240}}
$$

Taking the natural logarithm of both sides:

$$
\ln(1+i) = \frac{1}{240}\ln(3.7328)
$$

Using a calculator, we find:

$$
\ln(3.7328) \approx 1.3170
$$

So:

$$
\ln(1+i) \approx \frac{1.3170}{240} \approx 0.0054875
$$

Exponentiating both sides gives:

$$
1+i \approx e^{0.0054875} \approx 1.0055
$$

Thus:

$$
i \approx 1.0055 - 1 = 0.0055
$$

Since $$i$$ is the monthly interest rate, the annual nominal interest rate compounded monthly is:

$$
r = 12 \times 0.0055 \approx 0.066 \text{ or } 6.6\%
$$

Therefore, the interest rate is approximately $$6.6\%$$ compounded monthly.
The interest rate is approximately 6.6% compounded monthly.

Question
If a loan is paid off with a payment of 5,800 after 20 years, what is the interest rate compounded monthly? Round
your answer to one decimal place.

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Question If a loan is paid off with a payment of 5,800 after 20 years, what is the interest rate compounded monthly? Round your answer to one decimal place.