$6900 loan at 12% and eventually repaid $9384 (principal and interest). What was the time period of the loan?
th time was 18
I will assume compound interest:
6900(1.12)^n = 9384
(1.12)^n = .1.36
take log of both sides, and use log rules
nlog1.12 = log1.36
n = log1.36/log1.12 = 2.7
it would take 2 years and appr. 8.5 months
check:
balance after 1 year = 6900(1.12) = 7740.32
balance after 2 years = 7740.32(1.12) = 8669.16
balance after 3 years = 8669.16(1.12) = 9709.46
difference between 2 and 3 years = 1040
and .7 of that is appr 728
8669+728=9397 , close enough
To find the time period of the loan, we can use the formula for calculating compound interest:
A = P(1 + r/n)^(nt)
where:
A = the final amount (principal + interest)
P = the principal amount (loan amount)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the time period (in years)
In this case:
P = $6900
r = 12% = 0.12 (decimal form)
A = $9384
By substituting these values into the formula, we get:
$9384 = $6900(1 + 0.12/n)^(n*t)
We need to solve for t, which represents the time period of the loan. However, since n (the number of times interest is compounded per year) is not provided, we cannot find the exact value of t.
To proceed, we can make an assumption about the compounding period. Let's assume interest is compounded annually (n = 1). Now we can solve for t:
$9384 = $6900(1 + 0.12/1)^(1*t)
$9384 = $6900(1.12)^t
Dividing both sides by $6900:
1.36 = (1.12)^t
Now we can take the logarithm of both sides to solve for t:
log(1.36) = log((1.12)^t)
Using the property of logarithms, we can bring down the exponent:
t * log(1.12) = log(1.36)
Finally, we divide both sides by log(1.12) to get the value of t:
t = log(1.36) / log(1.12)
Using a calculator or computer program, you can find the value of t. The result will give you the time period of the loan in years.