Question

Milan needs $2500 for a future project. He can invest $2000 now at an annual rate of 7%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for him to have enough money for his project? Do not round any intermediate computations, and round your answer to the nearest hundredth. If necessary, refer to the list of financial formulas. years​

Answer 1

Answer: To find out how long it will take for Milan to have enough money for his project, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value (amount Milan needs for the project)

P = the principal amount (initial investment) = $2000

r = the annual interest rate (as a decimal) = 7% = 0.07

n = the number of times the interest is compounded per year (quarterly in this case) = 4

t = the number of years

The future value A is the target amount of $2500.

Now, let's solve for t:

$2500 = $2000(1 + 0.07/4)^(4t)

Divide both sides by $2000:

1.25 = (1 + 0.0175)^(4t)

Now, let's take the natural logarithm (ln) of both sides to solve for t:

ln(1.25) = ln((1 + 0.0175)^(4t))

Using the property of logarithms, we can bring the exponent down:

ln(1.25) = 4t * ln(1.0175)

Now, divide both sides by 4 * ln(1.0175) to isolate t:

t = ln(1.25) / (4 * ln(1.0175))

Using a calculator, calculate the value of t:

t ≈ 3.17 (rounded to two decimal places)

So, it will take approximately 3.17 years (or about 3 years and 2 months) for Milan to have enough money for his project.