find the interest rate to the nearest hundredth of a percent that will produce $2500, if $2000 is left at interest compounded semiannually for 4.5 year
To find the interest rate that will produce $2500, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after the interest has compounded
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
In this case, we have:
P = $2000
A = $2500
t = 4.5 years
n = 2 (since interest is compounded semiannually)
We need to solve for 'r'. Let's rearrange the formula and substitute the given values:
A = P(1 + r/n)^(nt)
$2500 = $2000(1 + r/2)^(2 * 4.5)
First, divide both sides of the equation by $2000:
$2500/$2000 = (1 + r/2)^(2 * 4.5)
1.25 = (1 + r/2)^9
Next, take the ninth root of both sides of the equation:
(1.25)^(1/9) = 1 + r/2
Subtract 1 from both sides:
(1.25)^(1/9) - 1 = r/2
Multiply both sides by 2:
2 * ((1.25)^(1/9) - 1) = r
Now we have the value for 'r', but we need to round it to the nearest hundredth of a percent. Multiply the obtained value by 100 and round to the nearest hundredth:
r = 2 * ((1.25)^(1/9) - 1) ≈ 1.59%
Therefore, the interest rate to the nearest hundredth of a percent that will produce $2500 is approximately 1.59%.