# Question #46f24

May 18, 2015

The original year and value are almost irrelevant to the answer. Percentage is about proportion, which will only be determined by the factor ${1.009}^{\left(\frac{2}{3}\right) x}$.

Notice that

${1.006}^{3} = {\left(1.0 + 0.006\right)}^{3}$

$= 1.0 + \left(3 \times 0.006\right) + \left(3 \times {0.006}^{2}\right) + {0.006}^{3}$

$= 1.0 + 0.018 + 0.000108 + 0.000000216$

$= 1.018108216$

And that

${1.009}^{2} = {\left(1.0 + 0.009\right)}^{2}$

$= 1.0 + 0.018 + 0.000081$

$= 1.018081$

So ${1.009}^{2} \cong {1.006}^{3}$ and ${1.009}^{\frac{2}{3}} \cong 1.006$.

${1.009}^{\left(\frac{2}{3}\right) x} = {\left({1.009}^{\frac{2}{3}}\right)}^{x} \cong {1.006}^{x}$

Multiplying by $1.006$ is the same as adding $0.6$%