Question
Suppose that a loan of \( \$ 5000 \) is given at an interest rate of \( 17 \% \) compounded each year. Assume that no payments are made on the loan. Follow the instructions below. Do not do any rounding. (a) Find the amount owed at the end of 1 year. \( \$ \square \) (b) Find the amount owed at the end of 2 years. \( \$ \square \)
Ask by Davey Bernard. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Amount owed at the end of 1 year:** \$5850
**Amount owed at the end of 2 years:** \$6844.5
Solution
Let's calculate the amount owed at the end of each year using the compound interest formula:
\[
A = P \times (1 + r)^n
\]
where:
- \( A \) is the amount owed,
- \( P \) is the principal amount (\$5,000),
- \( r \) is the annual interest rate (17% or 0.17),
- \( n \) is the number of years.
### (a) Amount Owed at the End of 1 Year
For \( n = 1 \):
\[
A = 5000 \times (1 + 0.17)^1 = 5000 \times 1.17 = 5850
\]
**Amount owed at the end of 1 year:** \(\$5850\)
### (b) Amount Owed at the End of 2 Years
For \( n = 2 \):
\[
A = 5000 \times (1 + 0.17)^2 = 5000 \times 1.3689 = 6844.5
\]
**Amount owed at the end of 2 years:** \(\$6844.5\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To calculate the amount owed when interest is compounded annually, you can use the formula: \[ A = P(1 + r)^n \] where \( A \) is the amount owed, \( P \) is the principal amount (initial loan), \( r \) is the annual interest rate (in decimal), and \( n \) is the number of years. (a) For 1 year: \[ A = 5000(1 + 0.17)^1 = 5000(1.17) = 5850 \] So, the amount owed at the end of 1 year is \( \$ 5850 \). (b) For 2 years: \[ A = 5000(1 + 0.17)^2 = 5000(1.17)^2 = 5000(1.3689) = 6844.5 \] Therefore, the amount owed at the end of 2 years is \( \$ 6844.5 \).
