Question

Find the time required for an investment of $5000 to grow to $8000 at an interest rate of 7.5% per year, compounded quarterly.

Answer 1

To find the time required for an investment to grow to a certain amount with compound interest, we can use the formula: A = p(1 + r/n)^(nt). Using the provided values ($5000 principal, $8000 final amount, 7.5% annual interest rate compounded quarterly), we can calculate that it would take approximately 5.8 years for the investment to grow to $8000.

Step-by-step explanation:

To find the time required for an investment to grow to a certain amount with compound interest, we can use the formula:

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

Where:

• A is the final amount
• p is the principal investment
• r is the annual interest rate (as a decimal)
• n is the number of times the interest is compounded per year
• t is the number of years

In this case, we have:

• p = $5000
• A = $8000
• r = 0.075 (7.5% as a decimal)
• n = 4 (quarterly compounding)

Substituting these values into the formula:

$8000 = $5000(1 + 0.075/4)^(4t)

Dividing both sides by $5000:

8/5 = (1 + 0.075/4)^(4t)

Taking the natural logarithm of both sides to solve for t:

ln(8/5) = 4t * ln(1 + 0.075/4)

Dividing both sides by 4 * ln(1 + 0.075/4):

t = ln(8/5) / (4 * ln(1 + 0.075/4))

Using a calculator, we can find that t ≈ 5.8 years.

Answer 2

Answer: 6.3 years

Step-by-step explanation:

To find the time in years, we will use the Compound interest formula:

F = P( 1 + i/m)^mn

Where F = future value of investment ($8000); P = Amount invested ($5000); I

i = interest rate (7.5%); m = number of times money is compounded in a year (m = 4 for quarterly) and n = time of investment in years

Substituting;

8000 = 5000( 1 + 0.075/4)^4n

Divide both side by 5000 and simplify the bracket on the right hand side;

8000/5000 = (1.01875)^4n

1.6 = (1.01875)^4n

Since n is the power, to solve for it we can introduce the natural logarithm ( ln);

ln (1.6) = ln (1.01875)^4n

The power can betaken down according to the Laws of logarithms;

ln (1.6) = 4n x ln(1.01875)

To get n, divide both sides by 4ln (1.01875);

ln (1.6)/ 4ln(1.01875) = n

Therefore; n = 6.3 years