A $8700 investment grows to $11100 in 13 years. If the investment has interest compounded quarterly, determine the interest rate.
A) 9.81% B) 2.38% C) 1.88% D) 19.59%
let the quarterly rate be j
(you will have to multiply j by 4 at the end to get the annual rate)
8700(1+j)^52 = 11100
(1+j)^52 = 1.275862..
take 52nd root, that is
1+j = 1.275862^(1/52) = .....
I matched one of your choices
I got C
To find the interest rate, we can use the formula for compound interest:
A = P * (1 + r/n)^(n*t)
Where:
A = the final amount ($11100 in this case)
P = the initial investment ($8700 in this case)
r = the interest rate (unknown)
n = the number of times interest is compounded per year (quarterly in this case)
t = the number of years (13 in this case)
Plug in the given values into the formula and solve for r:
11100 = 8700 * (1 + r/4)^(4*13)
Divide both sides of the equation by 8700:
11100/8700 = (1 + r/4)^(4*13)
1.2759 = (1 + r/4)^52
Now, we need to isolate the exponential term by taking the logarithm of both sides. Since the exponent is 52, we'll use the logarithm base 52:
log base 52 of 1.2759 = log base 52 of (1 + r/4)^52
Using the logarithmic property, we can bring down the exponent:
52 * log base 52 of 1.2759 = r/4
Simplify the left side:
52 = r/4
Multiply both sides by 4:
4 * 52 = r
r = 208
So the interest rate is 208%. However, since the options given are in percentage form, we need to convert it to a percentage.
Divide r by 100:
r/100 = 208/100 = 2.08
Therefore, the interest rate is 2.08%.
The correct answer is B) 2.38% (the closest option to 2.08%).