. Mr. Kane recently borrowed $15,000 from his Aunt Jemima, and he has promised to pay his aunt $5,000 per year at 15%. How long will it take Mr. Kane to pay off the entire loan from his aunt?
2. You just borrowed $1,000 from Mr. Loan Shark. Mr. Shark requires you to pay $150 per week for the next 10 weeks. What is the effective annual interest rate on this loan?
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To answer these questions, we need to use two financial formulas: the future value of an ordinary annuity and the formula for calculating the effective annual interest rate.
1. For Mr. Kane's loan:
To determine how long it will take Mr. Kane to pay off the entire loan from his Aunt Jemima, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r) ^ n - 1] / r
Where:
FV = Future value (total loan amount)
P = Periodic payment ($5,000)
r = Interest rate per period (15% or 0.15)
n = Number of periods (unknown)
We know P = $5,000 and FV = $15,000. Substituting these values into the formula, we get:
$15,000 = $5,000 * [(1 + 0.15) ^ n - 1] / 0.15
To solve for n, we can isolate it on one side of the equation:
[(1 + 0.15) ^ n - 1] / 0.15 = 3
[(1.15) ^ n - 1] / 0.15 = 3
(1.15) ^ n - 1 = 3 * 0.15
(1.15) ^ n - 1 = 0.45
(1.15) ^ n = 0.45 + 1
(1.15) ^ n = 1.45
To solve for n, we need to take the logarithm of both sides:
n * log(1.15) = log(1.45)
n = log(1.45) / log(1.15)
Using a calculator, we find that n is approximately 3.3972. Since n represents the number of years, we can round up to the nearest whole number, making it 4. Therefore, it will take Mr. Kane approximately 4 years to pay off the entire loan from Aunt Jemima.
2. For your loan from Mr. Loan Shark:
To calculate the effective annual interest rate on this loan, we can use the formula:
EAR = (1 + periodic interest rate) ^ number of periods - 1
Where:
EAR = Effective annual interest rate (what we want to find)
Periodic interest rate = Interest paid per period / Amount borrowed (150 / 1000 = 0.15)
Number of periods = Total loan duration in weeks (10)
Substituting these values into the formula, we get:
EAR = (1 + 0.15) ^ 10 - 1
Calculating this expression, we find:
EAR = 4.046 - 1
EAR = 3.046
So, the effective annual interest rate for the loan from Mr. Loan Shark is approximately 304.6%.