Question

To find the interest rate required for an investment of $4,000 to grow to $6,500 in 9 years with interest compounded as follows:

a) Annually (once per year). b) Quarterly (four times per year).
We will use the compound interest formula:
A=P(1+nr)nt
Where:
• A is the future value of the investment.
• P is the principal amount (initial investment), which is $4,000.
• r is the annual interest rate as a decimal (what we want to find).
• n is the number of times the interest is compounded per year.
• t is the number of years, which is 9 years.

Answer 1

To find the required interest rate for an investment to grow from $4,000 to $6,500 in 9 years, use the compound interest formula with different values of 'n' depending on the compounding period (annually n=1, quarterly n=4) and solve for the interest rate 'r'.

Step-by-step explanation:

To find the interest rate required for an investment of $4,000 to grow to $6,500 in 9 years, whether compounded annually or quarterly, we use the compound interest formula. The compound interest formula is A = P(1 + r/n)^(nt), where:

• A is the future value of the investment.
• P is the principal amount, which is $4,000.
• r is the annual interest rate (expressed as a decimal), which we need to find.
• n is the number of times the interest is compounded per year.
• t is the time in years, which is 9 years.

For part a, where interest is compounded annually, n is 1.

For part b, where interest is compounded quarterly, n is 4.

Now, let's solve for r in both scenarios:

Annually Compounded Interest

After plugging in the values into the formula, we have $6,500 = $4,000(1 + r/1)^(1*9). Simplifying the equation, we solve for r and get the required annual interest rate.

Quarterly Compounded Interest

Here, the calculation changes slightly with $6,500 = $4,000(1 + r/4)^(4*9). After simplifying and solving for r, we find the required interest rate for quarterly compounding.