Question

Find the interest rates earned on each of the following. Round your answers to the nearest whole number.

a. You borrow $710 and promise to pay back $781 at the end of 1 year.
b. You lend $710 and the borrower promises to pay you $781 at the end of 1 year.
c. You borrow $77,000 and promise to pay back $138,281 at the end of 12 years.
d. You borrow $10,000 and promise to make payments of $2,504.60 at the end of each year for 5 years.

Answer 1

a. 10%

b. 10%

c. 5%

d. 5%

Step-by-step explanation:

For a and b, you can calculate the interest rate using the formula:

\[ \text{Interest Rate} = \left( \frac{\text{Future Value} - \text{Present Value}}{\text{Present Value}} \right) \times 100 \]

For a, the calculation is: \((781 - 710) / 710 \times 100 \approx 10%\).

For b, it's the same calculation: \((781 - 710) / 710 \times 100 \approx 10%\).

For c, you can use the formula for compound interest:

`\[ \text{Compound Interest} = \text{Present Value} * \left(1 + \frac{\text{Interest Rate}}{100}\right)^{\text{Time}} - \text{Present Value} \]`

Rearranging the formula to solve for the interest rate, you get:

`\[ \text{Interest Rate} = \left( \left(\frac{\text{Future Value}}{\text{Present Value}}\right)^{1/\text{Time}} - 1 \right) * 100 \]Plugging in the values for c, you get: \(\left( \left((138,281)/(77,000)\right)^(1/12) - 1 \right) * 100 \approx 5%\).`

For d, it's a bit more complex. You can use the formula for the annuity present value to find the interest rate:

`\[ \text{Present Value} = \text{Payment} * \left( \frac{1 - (1 + \text{Interest Rate})^{-\text{Number of Periods}}}{\text{Interest Rate}} \right) \]`

Rearranging to solve for the interest rate:

`\[ \text{Interest Rate} = \frac{1 - \left(\frac{\text{Payment}}{\text{Present Value}}\right)^{1/\text{Number of Periods}}}{\left(\frac{\text{Payment}}{\text{Present Value}}\right)^{1/\text{Number of Periods}}} \]`

`Plugging in the values for d, you get: \((1 - \left((2,504.60)/(10,000)\right)^(1/5))/(\left((2,504.60)/(10,000)\right)^(1/5)) \approx 5%\).`

Answer 2

a. The interest rate earned is approximately 10%.

b. The interest rate earned is approximately 10%.

c. The interest rate earned is approximately 5%.

Step-by-step explanation:

In the context of financial transactions, the interest rate serves as a critical metric to assess the cost or return on borrowed or invested funds. For short-term loans, such as those in scenarios (a) and (b), the interest rate is derived by calculating the percentage increase from the initial borrowed amount to the total repayment at the end of the term. This straightforward approach involves dividing the difference between the future value and the present value by the present value and then multiplying by 100. In both cases, this yields an approximate interest rate of 10%.

For longer-term loans, like scenario (c) spanning 12 years, the interest rate is determined using a compound interest formula [(Future Value / Present Value)^(1/n) - 1], where 'n' represents the number of years. This formula reflects the compounding effect over an extended period. In this instance, the interest rate is approximately 5%, indicating a lower annualized return compared to the short-term loans.

Understanding these calculations is paramount for making informed financial decisions. It enables individuals to gauge the true cost of borrowing or assess the return on investment, crucial considerations in managing personal or business finances effectively.