# Compound Interest - Example - Saving Goal

The time to reach a value G from an asset of value P with a compound interest r is $$t = \frac{\ln \left( \frac{G}{P} \right)}{\ln \left( 1+ r \right)}$$

## Example

You have placed \$1000 in your savings account. The rate, 1%, is compounded annually. Your goal is to wait until your account contains \$1500. You are solving the following equation for t \begin{align*} 1000 \left( 1 + 0.01 \right)^t &= 1500 \\ \left( 1 + 0.01 \right)^t &= \frac{1500}{1000} \\ 1.01^t &= 1.5 \\ \ln \left( 1.01^t \right) = \ln \left( 1.5 \right) \\ t \ln \left( 1.01 \right) = \ln \left( 1.5 \right) \\ t = \frac{\ln \left( 1.5 \right)}{\ln \left( 1.01 \right)} \\ t \approx 40.75 \\ \end{align*} So it will take at least 41 years to reach \\$1500

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