Question

Byron deposits 13,500 in an account that earns 2.6% interest compounded monthly. How many years will it take for Byron's money to double if he does not deposit or withdraw any funds during that time?

Answer 1

About 27 years.

Explanation:

Byron deposits $13,500 in an account that earns 2.6% interest compounded monthly, and we want to determine how many years it will take for Byron's money to double.

Recall that compound interest is given by the formula:

`\displaystyle A = P\left( 1 + (r)/(n)\right)^(nt)`

Since our initial deposit is $13,500 at a rate of 2.6% compounded monthly, P = 13500, r = 0.026, and n = 12:

`\displaystyle \begin{aligned} A &= (13500)\left( 1 + ((0.026))/((12))\right)^((12)t) \\ \\ &=13500\left((6013)/(6000)\right)^(12t) \end{aligned}`

For his deposit to double, A must equal $27,000. Hence:

`\displaystyle (27000) &=13500\left((6013)/(6000)\right)^(12t)`

Solve for t:

`\displaystyle \begin{aligned} 27000 &= 13500 \left((6013)/(6000)\right)^(12t) \\ \\ \left((6013)/(6000)\right)^(12t) &= 2 \\ \\ \ln\left((6013)/(6000)\right)^(12t) &= \ln (2)\\ \\ 12t \ln (6013)/(6000) &= \ln 2 \\ \\ t &= (\ln 2)/(12\ln (6013)/(6000)) = 26.6883... \approx 27\end{aligned}`

In conclusion, it will take Byron about 27 years for his deposit to double.