Question

Find the time required for an investment of 5000 dollars to grow to 7600 dollars at an interest rate of 7.5 percent per year, compounded quarterly. Your answer is years. You may enter the exact value or round to 2 decimal places.

Answer 1

To find the time required for an investment to grow, we can use the formula for Compound Interest:

A = P * (1 + r/n)^(n*t)

Where:

A = Final amount ($7600)

P = Principal amount ($5000)

r = Annual interest rate (7.5% or 0.075)

n = Number of times interest is compounded per year (quarterly, so n = 4)

t = Time in years (unknown)

We need to solve for t. Rearranging the formula, we have:

(1 + r/n)^(n*t) = A/P

Substituting the given values, we get:

(1 + 0.075/4)^(4*t) = 7600/5000

Simplifying the equation further, we have:

(1 + 0.01875)^(4*t) = 1.52

Taking the natural logarithm (ln) of both sides, we have:

ln(1.01875)^(4*t) = ln(1.52)

(4*t) * ln(1.01875) = ln(1.52)

Now, solving for t:

t = ln(1.52) / (4 * ln(1.01875))

Using a calculator, we find:

t ≈ 3.69

Therefore, the time required for an investment of $5000 to grow to $7600 at an interest rate of 7.5% per year, compounded quarterly, is approximately 3.69 years.