Question

You expect to receive 12,000 at graduation in 3 years. You plan on investing this money at 13 percent until you have 75,000. How many years will it be until this occurs?

1) 17.99 years
2) 11.99 years
3) 21.90 years
4) 14.99 years
5) 25 years

Answer 1

To find out how many years it will take for a $12,000 investment to grow to $75,000 at a 13% annual interest rate, we use the future value compound interest formula and solve for n, the number of years. The calculation yields an answer of approximately 14.99 years.

Step-by-step explanation:

The question concerns determining the number of years needed for an investment to grow from $12,000 to $75,000 with a compound annual growth rate (CAGR) of 13%. This is a question of using the future value formula for compound interest: FV = PV * (1 + r)^n, where FV is the future value, PV is the present value, r is the interest rate, and n is the number of periods.

First, we will substitute the known values into the formula: $75,000 = $12,000 * (1 + 0.13)^n. To find n, we need to solve for n, which entails doing some algebraic rearrangement and ultimately using a logarithmic operation.

The algebraic steps would look something like this:

1. Divide both sides of the equation by $12,000 to isolate the growth factor on one side: ($75,000 / $12,000) = (1 + 0.13)^n.
2. Calculate the left side of the equation to get the growth factor: 6.25 = (1 + 0.13)^n.
3. Now, apply logarithms to both sides of the equation to solve for n: log(6.25) = n * log(1.13).
4. Finally, divide both sides by log(1.13) to solve for n: n = log(6.25) / log(1.13).

Using a calculator, you can find that n roughly equals 14.99 years. Therefore, it will take approximately 14.99 years for the initial $12,000 investment to grow to $75,000 at a 13% annual interest rate.