Sara buys a washer and a dryer for $2112.She pays $500 and borrows the remaining amount. A year and a half later she pays off the loan, which compounded semi-annually, was Sara being charged.
To find out how much Sara was charged for the loan, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the total amount paid
P = the principal amount (initial loan amount)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, Sara borrowed the remaining amount after paying $500, so the principal amount (P) is $2112 - $500 = $1612.
The interest is compounded semi-annually, which means it is compounded twice a year (n = 2). The loan term is a year and a half, so t = 1.5 years.
To calculate the annual interest rate (r), we can rearrange the formula:
A = P(1 + r/n)^(nt)
A/P = (1 + r/n)^(nt)
(1 + r/n)^(nt) = A/P
1 + r/n = (A/P)^(1/nt)
r/n = [(A/P)^(1/nt)] - 1
r = n * [(A/P)^(1/nt)] - 1
Substituting the given values:
r = 2 * [(2112/1612)^(1/(2*1.5))] - 1
Using this formula, we can calculate the annual interest rate (r).
To find out how much interest Sara was charged on the loan, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (loan balance + interest)
P = the principal amount borrowed ($2112 - $500 = $1612)
r = annual interest rate (unknown)
n = number of times interest is compounded per year (semi-annually, so n = 2)
t = time in years (1.5 years)
We can rearrange the formula to solve for the annual interest rate (r):
r = ([(A / P)^(1/(n*t))] - 1) * n
Using the given information, we can plug these values into the formula and solve for r:
A = $2112 (since Sara pays off the entire loan)
P = $1612
n = 2
t = 1.5
r = ([(2112 / 1612)^(1 / (2 * 1.5))] - 1) * 2
Calculating this expression, we find that r is approximately 0.1478, or 14.78% (rounded to 2 decimal places).
Therefore, Sara was charged an annual interest rate of 14.78% on her loan.
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