Final answer:
To find out how long it will take for Simone's money to double, we can use the formula for compound interest and solve for the number of years. In this case, it will take approximately 9.94 years for Simone's $6,000 investment to double.
Step-by-step explanation:
To determine how many years it will take for the money to double, we can use the formula for compound interest. In this case, Simone invests $6,000 at an interest rate of 7% compounded quarterly. The formula to calculate the future value of the investment is:
FV = PV * (1 + r/n)^(n*t)
Where:
- FV is the future value
- PV is the present value (initial investment)
- r is the interest rate (as a decimal)
- n is the number of compounding periods per year
- t is the number of years
In this case, we want to find the value of t that makes FV equal to twice the initial investment (2 * $6,000). So we plug in the values into the formula and solve for t:
2 * $6,000 = $6,000 * (1 + 0.07/4)^(4*t)
Dividing both sides by $6,000, we get:
2 = (1 + 0.07/4)^(4*t)
Raising both sides to the power of (4*t), we get:
(1 + 0.07/4)^(4*t) = 2
Taking the natural logarithm on both sides to isolate the exponent, we get:
ln[(1 + 0.07/4)^(4*t)] = ln(2)
Using logarithmic properties, we can bring the exponent down:
4*t * ln(1 + 0.07/4) = ln(2)
Finally, we solve for t:
t = ln(2) / (4 * ln(1 + 0.07/4))
Using a calculator, we find approximately t = 9.94 years.