A property worth $15,000.00 is purchased for 10% down and semi-annual payments of $1750.00 for twelve years. What is the effective rate of interest if interest is compounded semi-annually?
You have $75 to spend on refreshments for a party. Large bags of chips cost $2.50 and sodas cost $0.50. You need to buy exactly five times as many sodas as bags of chips. How many bags of chips and how many sodas can you buy?
To find the effective rate of interest, we first need to calculate the total amount paid over the course of twelve years.
The property was purchased for $15,000.00 and 10% down was paid, which is 10% of $15,000.00 = $1,500.00. Therefore, the loan amount is $15,000.00 - $1,500.00 = $13,500.00.
The term of the loan is twelve years, with semi-annual payments. So, the total number of payments is 12 years * 2 = 24 payments.
Each semi-annual payment is $1,750.00, so the total amount paid over 24 payments is 24 payments * $1,750.00 = $42,000.00.
Now, we can calculate the interest paid by subtracting the loan amount from the total amount paid: $42,000.00 - $13,500.00 = $28,500.00.
To find the effective rate of interest, we can use the formula:
Effective rate of interest = ((1 + (interest rate per period))^number of periods - 1)* 100
In this case, the interest is compounded semi-annually, so the interest rate per period is the annual interest rate divided by 2.
Let's assume the annual interest rate is "r."
So, the interest rate per period is r/2.
We need to find the interest rate for which $13,500.00 grows to $28,500.00 over 24 semi-annual periods.
Using the formula above:
((1 + (r/2))^24 - 1) * 100 = 28,500.00 / 13,500.00
Simplifying the equation, we get:
(1 + (r/2))^24 = 28,500.00 / 13,500.00 + 1
(1 + (r/2))^24 = 3.1111
Now, we can use trial and error or guess-and-check method to find the approximate value of (1 + (r/2)) that raises to the power of 24 and gives us 3.1111.
After solving the equation, we find that (1 + (r/2)) is approximately 1.061.
Now, we subtract 1 from 1.061 to find the interest rate per period:
(r/2) = 1.061 - 1
(r/2) = 0.061
r = 0.061 * 2
r = 0.122 or 12.2%
Therefore, the effective rate of interest, when interest is compounded semi-annually, is 12.2%.