Boucher Steel Corporation borrowed $80,000 for 10 years to purchase a machine. Calculate the nominal interest rate compounded semi-annually if $1250 was paid at the beginning of every month. Express as a percent to 2 decimal places. Eg. 3.44% [a]%
Solution:
To calculate the nominal interest rate compounded semi-annually, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the loan
P = the principal amount borrowed
r = the nominal interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal amount borrowed (P) is $80,000, the nominal interest rate (r) is what we need to find, the number of times interest is compounded per year (n) is 2 (semi-annually), and the number of years (t) is 10.
We also know that $1250 is paid at the beginning of every month. Since there are 12 months in a year, the total amount paid per year is $1250 * 12 = $15,000.
Now, let's substitute the values into the formula and solve for r:
A = P(1 + r/n)^(nt)
$80,000 = $15,000(1 + r/2)^(2*10)
To simplify the equation, let's divide both sides by $15,000:
$80,000/$15,000 = (1 + r/2)^(2*10)
5.3333 = (1 + r/2)^20
Now, let's take the 20th root of both sides to isolate (1 + r/2):
(1 + r/2) = √5.3333
Finally, let's solve for r:
r/2 = √5.3333 - 1
r = 2(√5.3333 - 1)
To express the nominal interest rate as a percent to 2 decimal places, we multiply r by 100:
r = 2(√5.3333 - 1)%