Question

. a local finance company quotes an interest rate of 18.4 percent on one-year loans. so, if you borrow $20,000, the interest for the year will be $3,680. because you must repay a total of $23,680 in one year, the finance company requires you to pay $23,680/12, or $1,973.33, per month over the next 12 months. is the interest rate on this loan 18.4 percent? what rate would legally have to be quoted? what is the effective annual rate?

Answer 1

The interest rate on the loan is 18.4%. The legally required rate to be quoted is the annual percentage rate (APR). The effective annual rate is a measure that accounts for compounding and represents the true cost of borrowing.

Step-by-step explanation:

To calculate the interest rate on a loan, you can use the formula:

Interest = Principal imes Rate imes Time

From the given information, the principal is $20,000 and the interest for one year is $3,680. Using these values, we can calculate the rate:

$3,680 = $20,000 imes Rate imes 1

Solving for Rate, we get:

Rate = $3,680 / ($20,000 imes 1) = 0.184 = extbf{18.4%}

Therefore, the interest rate on this loan is indeed 18.4%. However, legally, the interest rate would need to be quoted as the extbf{annual percentage rate (APR)}. The effective annual rate is a measure that takes into account compounding and reflects the true cost of borrowing over a year.

Answer 2

The interest rate on this loan is not 18.4 percent. To calculate the true interest rate, we need to use the formula for compound interest. The interest rate is approximately 17.92 percent and rounded to the nearest 0.125 percent, it is 18 percent. The effective annual rate (EAR) is approximately 19.56 percent.

Step-by-step explanation:

The interest rate on this loan is not 18.4 percent.

To calculate the true interest rate, we need to use the formula for compound interest:

`\[ A = P \left(1 + (r)/(n)\right)^(nt) \]`

Where:

A symbolizes the final amount after time t

P symbolizes the principal amount borrowed

r symbolizes the annual interest rate (as a decimal)

n symbolizes the number of times interest is compounded per year

t symbolizes the time in years

In this case, we know that A = $23,680, P = $20,000, t = 1 year, and n = 12 (monthly compounding).

Using the formula, we can solve for r (the interest rate):

`\[ 23,680 = 20,000 \left(1 + (r)/(12)\right)^(12) \]`

Simplifying this equation, we find that the interest rate is approximately 17.92 percent.

To legally quote the interest rate, the finance company would have to round this rate to the nearest 0.125 percent, which would be 18 percent.

The effective annual rate (EAR) takes into account the effects of compounding. To calculate the EAR, we can use the formula:

`\[ \text{EAR} = \left(1 + (r)/(n)\right)^n - 1 \]`

This formula calculates the actual annual interest rate when interest is compounded more frequently than annually.

Using the values from the original loan, the EAR would be approximately 19.56 percent.