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I've been told that for extract brewing it's best to simply calculate the original gravity instead of measuring it. Is this correct? What are the pros/cons either way? What about with extract+grains or partial mashes?

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If you boil the entire volume of your extract batches, go ahead and measure it. But most extract brewers do a partial boil and add top up water afterward. In that case, it's REALLY REALLY REALLY hard to get the extract and water mixed thoroughly enough to get an accurate reading. The extract is heavier due to the sugar in it and sinks to the bottom of the fermenter. I speak from experience...this was the basis of the first question I ever posted to the internet nearly 14 years ago. In these cases, it's more accurate to calculate the OG based on the ingredients and the final volume of the beer. For instance, you can assume that liquid extract contributes about 36 ppg (points/pound/gal.) and dry extract about 45. So 5 lb. of dry extract in 5 gal. would give you a gravity of 1.045. For steeping grains you can assume about 35-50% efficiency and add that in, and for partial mashes about 65-75% efficiency.

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+1. For an example of how measuring your OG when doing a partial-boil extract recipe can go awry, check out this question I posted about a year ago: homebrew.stackexchange.com/questions/1650/… –  JackSmith Feb 1 '11 at 15:12
    
This explains a lot.. –  corymathews Feb 1 '11 at 22:26
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