If you borrow $15,000 from your dad for college and you agree to pay him $25,000 back at 8% compound interest per year. How many years do you think you have to do this? In other words, given the amounts and the interest rate, what is the number of years?
so
25,000 = 15,000(1.08)^n
5/3 = 1.08^n
log(5/3) = n log 1.08
n = log(5/3) / log 1.08 = appr 6.6 years
Holy Waa!!
Your dad is a loan shark.... call the cops!
To determine the number of years required to pay back the loan amount, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $25,000)
P = the principal amount (the loan amount, $15,000)
r = the annual interest rate (8% or 0.08)
n = the number of times interest is compounded per year (assuming it is compounded annually, n = 1)
t = the number of years
Substituting the given values into the formula, we can solve for t:
25,000 = 15,000(1 + 0.08/1)^(1*t)
Simplifying the equation:
25,000 = 15,000(1.08)^t
Dividing both sides of the equation by 15,000:
1.6667 = 1.08^t
Taking the natural logarithm of both sides:
ln(1.6667) = ln(1.08^t)
Using the property of logarithms:
t * ln(1.08) = ln(1.6667)
Dividing both sides by ln(1.08):
t = ln(1.6667) / ln(1.08)
Using a calculator, we find:
t ≈ 4.18
Therefore, it will take approximately 4.18 years to pay back the loan amount.
To calculate the number of years needed to pay back the loan, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the total amount to be paid back ($25,000 in this case)
P = the initial loan amount ($15,000 in this case)
r = the interest rate per compounding period (8% or 0.08 as a decimal)
n = the number of compounding periods per year (since it is not specified, we will assume it is compounded annually, so n = 1)
t = the time in years (which we want to find)
Now we can rewrite the formula as:
25,000 = 15,000(1 + 0.08/1)^(1t)
Next, let's simplify the equation:
25,000/15,000 = (1.08)^t
1.67 = 1.08^t
To find the value of t (the number of years), we need to take the logarithm of both sides of the equation. Since the base of the logarithm is not specified, we'll use the natural logarithm (base e):
ln(1.67) = ln(1.08^t)
Now, we can use logarithmic rules to simplify the equation further:
ln(1.67) = t * ln(1.08)
Using a calculator, we can determine the natural logarithm of 1.67 and 1.08:
t = ln(1.67) / ln(1.08)
By evaluating this expression, we can find the approximate value of t, which represents the number of years needed to pay back the loan at compound interest.