# How do I change the base year of real GDP using the GDP deflator and nominal GDP?

I'll use the US as an example. I have three data series

1. nominal GDP $(Y)$
2. real GDP in 2005 USD $(\bar{Y})$
3. the GDP deflator $(d)$, with 2005 as the base year, so $d_{2005} = 100$

I want to change the base year to 2000. Are these calculations accurate? I use the notation $\$_{t}$for USD in year$t$prices to help myself keep the units straight. My goal is$\bar{Y}_{t} \ \$_{2000}$. \begin{align} \frac{d_{2000}}{d_{t}} \cdot Y_{t} \ \$_t &= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000} \ \$_{2005}} \cdot \frac{\bar{Y}_{t} \ \$_{2005}}{Y_{t} \ \$_{t}} \cdot Y_{t} \ \$_t \\ &= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000}} \cdot \bar{Y}_{t} \\ &= \bar{Y}_{t} \ \$_{2000} \cdot \frac{Y_{2000}}{\bar{Y}_{2000}} \end{align}

I think those are all the correct unit cancellations, but now I'm stuck with the unitless quantity $\frac{Y_{2000}}{\bar{Y}_{2000}}$, so I don't know how to complete the conversion.

Am I doing this right?