Jane needs a short-term loan to buy a new washing machine. She needs to borrow $1500 at 20% compounded annually and plans to have it paid off in 1 year. Jane writes the formula 1500(1.2)t and finds out that this loan will cost her $1800. Which equation shows how Jane can rewrite the formula to find the annual percentage rate that would cost her the same amount if it compounded semi-annually?
I got the answer A= 1500 (1.095) is that correct?
the right answer is A=1500(1.095)^2t I took the test and got it right
1500 * 1.20 = 1500 x^2 (multiply twice by half rate 1+x)
1.20 = x^2
x = 1.09545
so .095 is the rate for HALF a year
annual rate is 0.19 or 19%
So was this answer not correct? because I know see there's a 1/2t and a 2t besides two of the answer choices
let the semiannual rate be i
then
1500(1+i)^2 = 1800
(1+i)^2 = 1.2 <----- Damon used x instead of 1+i, same thing
1+i = √1.2 = 1.095445...
i = .095445.. ----> 2i = .19089..
so the equivalent rate is 19.089% per annum, compounded semi-annually
So, is that the answer or...?
To determine the annual percentage rate (APR) that would result in the same cost if the loan is compounded semi-annually, we can follow these steps:
1. Understand the given formula:
The formula Jane used to find the cost of the loan is: 1500(1.2)t, where t is the number of years.
2. Calculate the original cost of the loan:
Substituting t = 1 into the formula: 1500(1.2)^1 = 1800.
3. Determine the formula for semi-annual compounding:
For semi-annual compounding, the formula is: A = P(1 + r/n)^(n*t), where:
A = the final amount (total cost in this case)
P = the principal (initial loan amount, $1500)
r = the annual interest rate
n = the number of times interest is compounded per year
t = the number of years
4. Find the equivalent semi-annual rate:
We need to solve the formula P(1 + r/n)^(n*t) = 1800 for r.
In this case, P = $1500, n = 2 because interest is compounded semi-annually, t = 1, and A = $1800.
The formula becomes: 1500(1 + r/2)^(2*1) = 1800.
Simplifying the equation further: (1 + r/2)^2 = 1800/1500 = 1.2.
5. Solve for the annual interest rate (APR):
Taking the square root of both sides to isolate (1 + r/2): 1 + r/2 = √1.2.
Subtracting 1 and multiplying by 2: r = (2√1.2) - 2.
So, the correct equation for finding the annual percentage rate (APR) that would cost Jane the same amount (compounded semi-annually) is:
r = (2√1.2) - 2.
Therefore, the answer A = 1500 (1.095) is not correct.