Compound Interest Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
How long does it take for an investment to double at 8%8\% annual interest compounded monthly?

Solution

  1. 1
    Set A=2PA = 2P in A=P(1+r/n)ntA = P(1 + r/n)^{nt}: 2=(1+0.08/12)12t=(1.006‾)12t2 = (1 + 0.08/12)^{12t} = (1.00\overline{6})^{12t}.
  2. 2
    Take the natural log: ln⁡2=12t⋅ln⁡(1.006‾)\ln 2 = 12t \cdot \ln(1.00\overline{6}).
  3. 3
    Solve: t=ln⁡212ln⁡(1.006‾)=0.693112×0.006645≈0.69310.07974≈8.69t = \frac{\ln 2}{12 \ln(1.00\overline{6})} = \frac{0.6931}{12 \times 0.006645} \approx \frac{0.6931}{0.07974} \approx 8.69 years.

Answer

t≈8.69 yearst \approx 8.69 \text{ years}
To find the time to double, set A=2PA = 2P and use logarithms to solve for tt. Note the Rule of 72 gives a quick estimate: 72/8=972/8 = 9 years.

About Compound Interest

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after tt years, and A=PertA = Pe^{rt} gives the continuously compounded amount.

Learn more about Compound Interest →

More Compound Interest Examples

Example 1 easy

You invest [formula]5{,}000[formula]6%$ annual interest compounded quarterly. Find the amount after

Example 3 medium

Find the amount when [formula]2{,}000[formula]5%$ compounded continuously for 4 years.

Example 4 easy

You deposit [formula]1{,}200[formula]4%$ compounded annually. What is the balance after 5 years?

Example 5 hard

What annual interest rate, compounded monthly, is needed to grow [formula]3{,}000[formula][formula]

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