Ralph's Machine Shop purchased a computer to use in tuning engines. To finance the purchase, the company borrowed $20,000 at 7% compounded semi-annually. To repay the loan, equal monthly payments are made over two years, with the first payment due one year after the date of the loan. What is the size of each monthly payment?
Ralph's Machine Shop purchased a computer to use in tuning engines. To finance the purchase, the company borrowed $20,000 at 7% compounded semi-annually. To repay the loan, equal monthly payments are made over two years, with the first payment due one year after the date of the loan. What is the size of each monthly payment?
To determine the size of each monthly payment, we can use the formula for the present value of an annuity:
PVA = PMT × [(1 - (1 + r)^(-n)) / r]
Where:
PVA = Present Value of the Annuity
PMT = Payment amount per period
r = Interest rate per period
n = Total number of periods
First, we need to calculate the values for the variables:
PVA = $20,000
r = 7% compounded semi-annually, so the effective interest rate per period is (1 + 7%/2)^(2/12) - 1 = 0.582%
n = 2 years, which equals 24 months (since payments are made monthly)
Now we can substitute these values into the formula:
$20,000 = PMT × [(1 - (1 + 0.582%)^(-24)) / 0.582%]
Next, we can simplify the equation to solve for PMT:
$20,000 = PMT × [(1 - (1.00582)^(-24)) / 0.00582]
Now we can calculate the value within square brackets:
(1 - (1.00582)^(-24)) / 0.00582 ≈ 15.3155
Finally, we can determine the monthly payment amount (PMT):
$20,000 = PMT × 15.3155
PMT ≈ $1,305.57
Therefore, the size of each monthly payment is approximately $1,305.57.
To find the size of each monthly payment, we need to use the formula for calculating the monthly payment on a loan. This formula is called the amortization formula.
The formula for calculating the monthly payment on a loan is:
P = (r * A) / (1 - (1+r)^(-n))
Where:
P = Monthly payment
r = Monthly interest rate
A = Loan amount
n = Total number of payments
Let's calculate each of these values step by step:
1. Calculate the monthly interest rate (r):
Since the loan is compounded semi-annually, we need to divide the annual interest rate by 2 and then convert it to a decimal.
r = (7% / 2) / 100 = 0.035
2. Calculate the total number of monthly payments (n):
Given that the loan needs to be repaid over two years, with monthly payments, there will be:
n = 2 years * 12 months = 24 months
3. Calculate the loan amount (A):
The loan amount is given as $20,000.
Now, let's substitute these values into the formula and calculate the monthly payment (P):
P = (0.035 * 20000) / (1 - (1+0.035)^(-24))
Using a calculator or a spreadsheet, we can simplify and calculate the value of P.
P ≈ $946.03
Hence, each monthly payment will be approximately $946.03.
To repay the loan over two years, there will be a total of 24 monthly payments (12 payments per year * 2 years).
The interest is compounded semi-annually, so the interest rate for each period will be 7%/2 = <<7/2=3.5>>3.5%.
To calculate the size of each monthly payment, we can use the monthly payment formula for a loan:
M = P * (r(1+r)^n) / ((1+r)^n - 1)
Where:
M = Monthly payment
P = Principal loan amount
r = Monthly interest rate
n = Number of payments
Substituting the given values into the formula, we get:
M = $20,000 * (0.035(1+0.035)^24) / ((1+0.035)^24 - 1)
M = $20,000 * 0.035 * 1.419826 / (1.419826 - 1)
M = $20,000 * 0.0496672291 / 0.419826
M = $2,668.67
Therefore, the size of each monthly payment is $2,668.67.
To find the size of each monthly payment, we need to calculate the loan repayment amount.
The loan is for $20,000 at an annual interest rate of 7% compounded semi-annually. This means that after one year, the loan will accrue interest and the principal will grow.
First, let's calculate the interest earned on the loan after one year:
Annual interest rate = 7%
Semi-annual interest rate = 7% / 2 = 3.5%
Interest earned after one year = $20,000 * 3.5% = $20,000 * 0.035 = $700
Now, let's calculate the remaining loan balance after one year:
Principal loan amount = $20,000
Remaining balance after one year = Principal loan amount + Interest earned after one year
Remaining balance after one year = $20,000 + $700 = $20,700
The loan needs to be repaid over 2 years, which is equivalent to 24 months.
To calculate the size of each monthly payment, we divide the remaining balance by the number of months:
Monthly payment = Remaining balance / Number of months
Monthly payment = $20,700 / 24
Monthly payment ≈ $862.50
Therefore, the size of each monthly payment is approximately $862.50.