Question

You expect to receive $41,000 at graduation in two years. You plan on investing it at 9.25 percent until you have $176,000. Required: How long will you wait from now? (Enter rounded answer as directed, but do not use rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Answer 1

To find out how long it will take for $41,000 to grow to $176,000 at a 9.25% annual interest rate, we use the compound interest formula. After calculating, we determine that it will take approximately 15.53 years from the time of investment, which would be 17.53 years from now including the two years until graduation.

Step-by-step explanation:

The student is asking how long it will take for an investment of $41,000 to grow to $176,000 at an annual interest rate of 9.25%. This is a problem involving compound interest, where interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

To solve for the duration, we use the future value formula for compound interest:

FV = PV * (1 + r)^t

Where:

• FV is the future value of the investment
• PV is the present value of the investment
• r is the annual interest rate (in decimal form)
• t is the time (in years)

Plugging in the values: $176,000 = $41,000 * (1 + 0.0925)^t

First, divide both sides by $41,000: 4.29268 = (1 + 0.0925)^t

Now, take the natural logarithm of both sides: ln(4.29268) = t * ln(1.0925)

Then, divide by ln(1.0925): t = ln(4.29268) / ln(1.0925)

Calculating this gives us: t ≈ 15.53 years

But we must also consider the two years until graduation, therefore the total time from now is approximately:

15.53 + 2 = 17.53 years