# 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 series1. 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?

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