Find the periodic payment R required to amortize a loan of P dollars over t years with interest earned at the rate of r %/year compounded m times a year. P=30,000
r = 8
t = 12
m =1
To find the periodic payment required to amortize a loan, we can use the formula for calculating the payment amount of an amortizing loan, which is given by:
R = (P * r * (1 + r/m)^(m*t)) / ((1 + r/m)^(m*t) - 1)
Let's substitute the given values into the formula:
P = $30,000
r = 8% (converted to decimal form by dividing by 100, so r = 0.08)
t = 12 years
m = 1
Now we can calculate the periodic payment R:
R = (30,000 * 0.08 * (1 + 0.08/1)^(1*12)) / ((1 + 0.08/1)^(1*12) - 1)
First, let's simplify the exponent terms:
R = (30,000 * 0.08 * (1 + 0.08)^(12)) / ((1 + 0.08)^(12) - 1)
Next, evaluate the exponent terms:
R = (30,000 * 0.08 * (1.08)^(12)) / ((1.08)^(12) - 1)
Using a calculator, calculate (1.08)^12 ≈ 1.96824.
R = (30,000 * 0.08 * 1.96824) / (1.96824 - 1)
Now, subtract 1 from 1.96824:
R = (30,000 * 0.08 * 1.96824) / 0.96824
Finally, let's calculate the value for R:
R ≈ 6,128.36
Therefore, the periodic payment required to amortize the loan of $30,000 over 12 years with an annual interest rate of 8% compounded annually would be approximately $6,128.36.
To find the periodic payment required to amortize the loan, we can use the formula for calculating the monthly payment on a loan:
R = (P * (r/100) * ((1 + (r/100)/m)^(m*t))) / (((1 + (r/100)/m)^(m*t)) - 1)
Plugging in the given values:
P = $30,000
r = 8%
t = 12 years
m = 1 (interest compounded once a year)
R = (30,000 * (8/100) * ((1 + (8/100)/1)^(1*12))) / (((1 + (8/100)/1)^(1*12)) - 1)
Simplifying the calculations step-by-step:
R = (30,000 * 0.08 * ((1 + (0.08)/1)^(12))) / (((1 + (0.08)/1)^(12)) - 1)
R = (30,000 * 0.08 * ((1 + 0.08)^(12))) / (((1 + 0.08)^(12)) - 1)
R = (30,000 * 0.08 * (1.08^(12))) / ((1.08^(12)) - 1)
R = (30,000 * 0.08 * 1.593848) / (1.593848 - 1)
R = (38,172.72) / (0.593848)
R ≈ $64,383.61
Therefore, the periodic payment required to amortize the loan is approximately $64,383.61.