how long will it take for $700 to double if it is invested at 10% annual interest compounded 3 times a year…

Explanation:

Step1: Use compound - interest formula

The compound - interest formula is (A = P\left(1+\frac{r}{n}\right)^{nt}). Here, (P=$700), (A = 2P=$1400), (r = 0.1), (n = 3). Substitute into the formula: (1400=700\left(1 +\frac{0.1}{3}\right)^{3t}). Divide both sides by (700): (2=\left(1+\frac{0.1}{3}\right)^{3t}). Take the natural logarithm of both sides: (\ln(2)=3t\ln\left(1+\frac{0.1}{3}\right)). Solve for (t): (t=\frac{\ln(2)}{3\ln\left(1+\frac{0.1}{3}\right)}). Calculate (\ln\left(1+\frac{0.1}{3}\right)=\ln\left(\frac{3 + 0.1}{3}\right)=\ln\left(\frac{3.1}{3}\right)\approx\ln(1.0333)\approx0.0328). (t=\frac{\ln(2)}{3\times0.0328}), since (\ln(2)\approx0.6931), (t=\frac{0.6931}{0.0984}\approx7.044).

Step2: Use continuous - compounding formula

The continuous - compounding formula is (A = Pe^{rt}). Substitute (A = 1400), (P = 700), (r=0.1). (1400 = 700e^{0.1t}). Divide both sides by (700): (2=e^{0.1t}). Take the natural logarithm of both sides: (\ln(2)=0.1t). Solve for (t): (t=\frac{\ln(2)}{0.1}). Since (\ln(2)\approx0.6931), (t = 6.931).

Answer:

It will take (7.044) years. Compounded continuously, it would only take (6.931) years.