Metropolis currently has $1,150,000 in cash. How long would it take them to accumulate $2,000,000 in cash? Assume an interest rate of 5 percent.
To determine how long it would take Metropolis to accumulate $2,000,000 in cash with an interest rate of 5 percent, we need to use the concept of compound interest.
Compound interest is calculated using the formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial cash)
r = the interest rate (in decimal form)
n = the number of compounding periods per year
t = the number of years
In this case, we know that Metropolis currently has $1,150,000 and wants to accumulate $2,000,000. The interest rate is 5 percent, which can be expressed as 0.05. We will assume the interest is compounded annually, so n = 1.
Now, let's plug in the given information into the formula:
$2,000,000 = $1,150,000(1 + 0.05/1)^(1*t)
Simplifying the equation:
2/1.15 = 1.05^t
Next, we need to isolate t by taking the logarithm of both sides:
log(2/1.15) = log(1.05^t)
Using the logarithm property log(a^b) = b*log(a):
log(2/1.15) = t*log(1.05)
Finally, we can solve for t by dividing both sides by log(1.05):
t = log(2/1.15) / log(1.05)
Using a calculator, we can evaluate this expression to find the value of t:
t ≈ 5.8 years
Therefore, it would take Metropolis approximately 5.8 years to accumulate $2,000,000 in cash with an interest rate of 5 percent.