Question

Hannah would like to make an investment that will turn 7000 dollars into 37000 dollars in 6 years. what quarterly rate of interest, compounded four times per year, must she receive to reach her goal? give your answer correct to at least 3 decimal places.

Answer 1

To determine the quarterly interest rate needed to grow an investment from $7,000 to $37,000 in 6 years with quarterly compounding, use the compound interest formula. After plugging in the known values and solving for the annual rate r, express the rate as a decimal to get the required quarterly rate to the nearest thousandth.

Step-by-step explanation:

To calculate the quarterly interest rate required to grow an investment of $7,000 to $37,000 in 6 years with compound interest, we can use the compound interest formula: A = P(1 + r/n)^(NT), where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (decimal).
• n is the number of times that interest is compounded per year.
• t is the time the money is invested for, in years.

In Hannah's case, A is $37,000, P is $7,000, n is 4 (since the interest is compounded quarterly), and t is 6 years.

Step 1: Plug the known values into the formula:

37000 = 7000(1 + r/4)^(4*6)

Step 2: Solve the equation for r, the annual interest rate:

37000 / 7000 = (1 + r/4)^(24)

5.2857 = (1 + r/4)^24

Step 3: Take the 24th root of both sides:

(5.2857)^(1/24) = 1 + r/4

Step 4: Subtract 1 from both sides:

r/4 = (5.2857)^(1/24) - 1

Step 5: Multiply by 4 to find the annual rate:

r = 4[(5.2857)^(1/24) - 1]

After performing the calculations, you will get the quarterly interest rate, which you should express as a decimal.