A bank offers an account with a 5% annual interest rate, compounded continuously. Approximately how long would it take for a $5,000 investment to double?

9514 1404 393

  13.9 years

Step-by-step explanation:

The doubling time in years for an account earning interest continuously is given exactly by the formula ...

  t = 69.31472/r . . . . where r is the annual interest rate in percent

We have r = 5, so the doubling time is ...

  t = 69.31472/5 = 13.8629 . . . years

It will take approximately 13.9 years for the investment to double in value.

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Additional comment

The account multiplier for continuous interest is ...

  e^(rt)

That multiplier is 2 when ...

  2 = e^(rt)

  ln(2) = rt . . . take the natural log of both sides

  t = ln(2)/r = 0.6931472/r

Multiplying by 100 so r is in percent, this is ...

  t = 69.31472/r . . . . with r in percent

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This compares with the "rule of 72" that is used to estimate doubling time when interest is paid periodically. That rule would tell you doubling time at 5% is approximately 72/5 = 14.4 years. The value "72" actually depends on interest rate and frequency of compounding, so that rule only gives a crude estimate.

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