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Beer Math with Brett Shegogue

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As I mentioned in my Recipe Setup article, it is nice to utilize brewing software so you don't have to make complex calculations. While this is true, some of us would like to know the formulas behind those calculations. Knowledge of your formulas is useful for making adjustments on-the-fly, or to answer questions that friends may have about your beer. I will go over a few different commonly used formulas and discuss how you can use them in your brewing.

Do you even math bro? Young brewers doing beer math. Source.
For ease of use, when working with gravity in the following equations, ignore the 1 and multiply by 1000. For brewing purposes that leaves us working with round numbers from 0-120 or so. For example a 1.050 gravity would be 50 for the purpose of these calculations.
ABV
A question many homebrewers receive after handing their most recent brew to a friend is "What's the alcohol content?" Whether your friend is trying to be a responsible drinker, or just trying to get a buzz quickly, what they really want to know is the ABV percentage. This is a fairly simple equation and comes from taking the difference between your Final Gravity (FG) and Original Gravity (OG) and multiplying by the constant of .131*.

Lets use an example of a 1.050 OG pale ale with a FG of 1.012. (remember we are converting the gravities to easier to use values, i.e. 50 and 12)

Quick and simple. Brewing software uses this calculation in association with an estimated attenuation percentage (from the yeast you use) to predict your ABV. So what if you want to make this prediction yourself? Well lets work backwards to derive that value.
Attenuation and Predicting FG
Calculating a predicted final gravity is another simple formula, but the trick is determining your attenuation. Most yeast strains have a percentage range for attenuation which can be found from the yeast manufacturer. Many of us are familiar with the standard american "chico" ale yeast. Take a quick look at White Labs WLP001 or Wyeast 1056, and you will see the estimated attenuation for this strain is anywhere from 73% - 80%. Fairly large range, right? Well, when you think about it, the attenuation really depends on a few different things:
  1. Type and amount of fermentable sugars: Using table sugar which is usually 100% fermentable can adjust the overall attenuation percentage.
  2. Pitch rate: too little yeast and you may not attenuate fully
  3. Environment: Temperature, oxygen, vessel size and shape, and alcohol content (certain yeasts can't keep going once a certain alcohol content is reached) all affect the attenuation.
But, I digress... The formula for attenuation is the difference between the OG and FG, divided by the OG.

It will help to see how attenuation is calculated so we can manipulate it to solve for a predicted gravity based on our yeast's attenuation. In our example of the 1.050 OG beer that finishes at 1.012 FG we can see that our attenuation on this beer was 76%

Therefore when you are calculating a recipe and you want to figure out the Final Gravity you would need to use an estimated attenuation (from the yeast manufacturer above) to solve for FG. That formula breaks down to this:

Lets say you are planning on mashing really low to produce a highly fermentable wort with a OG of 1.050 and you are expecting to hit the upper range for this yeast's attenuation which is 80%. What would your final gravity be?

Gravity Points
So we covered ABV, Attenuation and FG. While these are all important formulas, the most useful formula I offer today is regarding gravity points. Understanding gravity points and how to manipulate them can help you make mid-brew adjustments.
When you plug your recipe into brewing software, it will often give you a starting gravity, which is the gravity of your wort at the beginning of the boil. These gravity readings are based off your volume of wort at the start of the boil. Multiply the starting, pre-boil, gravity by the volume to get your total gravity points of the wort:

For this batch, your software predicts a pre-boil SG of 1.040, or 40 points per gallon. You will be starting with a pre-boil volume of 6.25 gallons. This means you have a total of 250 gravity points.

If we assume your boil off rate is 1.25 gallons per hour, we can solve for your final volume FG using our formula.

As you may have guessed, our five gallon batch came out to 50 points per gallon for a gravity of 1.050.
As mentioned, this would become very useful, so lets to do some troubleshooting of potential issues that may arise on a brew day. Say you take a reading at the start of the boil and you have overshot your estimated pre-boil gravity of 1.040 and are instead sitting at 1.045. The first thing you should check is your volume. It is possible you didn't end up with your full 6.25 gallons. It is also possible that you had better extract efficiency than you expected. We will assume you had better extract efficiency (makes you feel good, right?). Using our total gravity points formula, we see your predicted to have a FG of 56.25 points per gallon:

So you are forced with a decision; do you roll with the brew and take the higher gravity? Or, do you want to hit your FG dead-on? Lets say you really wanted to hit that 1.050. Use the formula to see what volume we would need to hit 1.050.

As you see, to hit your original gravity of 1.050, you would need to end up with 5.625 gallons at the end of the boil. You would need to add .625 gallons of water to your kettle during the boil to hit this number (assuming you also achieve your 1.25 gallon boil off rate).

Putting it Together

Okay, you made it through the algebra lesson in one piece! But no math lesson would be complete without a bit of homework. I present to you the following word problems, let us know the answers in the comments below. If you get them correct, treat yourself to a homebrew.
  1. Austin has crafted what he thinks is the perfect recipe for a pale ale. A balanced malt backbone of munich malt and american 2-row are complemented by the centennial and cascade hops, which come through for a nice citrus punch. Austin measures his Original Gravity of 1.058 and is using WLP002 English ale yeast. He mashed at a really low temperature and is predicting an attenuation of 70%, the upper range for this yeast. If he hits this attenuation, what will Austin's Final Gravity and estimated ABV of this pale ale be?
  2. Brett is testing out his new induction burner and decided to brew a Belgian Tripel. If he has a pre-boil volume of 7 gallons and a measure gravity of 1.070 wort, and after his 90 minute boil has a post-boil original gravity of 1.089, how much wort is left in his brew kettle?
  3. Brett used a pound and a half of highly fermentable belgian candi sugar in his tripel. The FG was measured at a fairly dry 1.008. What was the yeasts attenuation percentage?
* - There is some debate on how this constant is derived and it usually comes from using other values including the ratio of alcohol weight to water. The values of 1.05 are multiplied to the OG and .79 to the FG. This 1.05:.79 ratio is equivalent to .132911 but I assume the multipliers have been rounded. Anyways, .131 is close enough for me!
 
Great article. One of the things I like most about brewing IS the math. I have always done all the calculations myself, and still do so despite the fact that I eventually got Beersmith (mostly use it only for recipes). I like doing the math myself so that I can keep tweaking the formulae for better practical results, and also thereby have a much better understanding of what actually goes on during the brew. I do many other calculations in addition to what you have reported.
 
I'm saving this article for quick reference, thanks!
Correct me if I'm wrong...
1) FG=1.0147, ABV=5.67
2) 5.5 gal
3) Att.=91%
 
I read the title and thought, "beer math? Sounds like a way to take the fun out of brewing." But seriously. It's a good article. Thanks Brett.
 
I should be working, but I love math so I will bust out my slide ruler. Yup, I am ancient.
1. FG=1.017; ABV =5.3%
2. Final volume 4.76gal
3. Attenuation 91%
Would not want to do it by hand all the time, but thanks for the formulas behind the brewing software.
Is there a formula for converting final gravity refractometer readings to specific gravity that accounts for the alcohol that is present?
Thanks Brett... your article and the homework probably staved off my Alzheimerʻs a year or two.
 
Simple rule of thumb is the ABV = OG.
So, if your OG is 50 (1.050), your ABV will be 5.0%.
This rule assumes 76% attenuation, which is a good assumption. For you mathies out there, it's pretty easy to prove this, go for it.
 
Nice job class! @Bottoms_Up was the first to get all the questions right!
@Thetonymartin I am going to give you partial credit as I think you accidentally swapped the '7' and the '4' in the hundredths and thousandths column. Carrying that number forward would produce your stated ABV of 5.67%
Answers with work:
1) We have an OG 58, Attenuation 70%
FG = OG *(1-Attenuation)
FG = 58 * (1-.7)
FG = 58 * .3 = 17.4
FG = 1.0174
ABV = (OG - FG) * .131
ABV = (58 - 17.4)*.131
ABV = (40.6)*.131
ABV = 5.32
2) 7 gallons pre-boil, 70 Preboil Gravity, 89 OG
Preboil Gravity * gallons = OG x Gallons
7*70 = x * 89
490 = x * 89
490/89 = x
x = 5.51
3) Using the information from question 2 (sorry if that wasn't clear), 89 OG and 8 FG
Attenuation = (89-8)/89
81/89 = Attenuation
Attenuaiton = 91%
 
LavaSteam,
You asked: Is there a formula for converting final gravity refractometer readings to specific gravity that accounts for the alcohol that is present?

The answer is yes. In fact, there is more than one formula. I use the cubic approximation.
And here it is: FG = 1 - 0.0044993*(C15/C16) + 0.0117741*(D15/C16) + 0.000275806*(C15/C16)^2 - 0.00127169*(D15/C16)^2 - 0.00000727999*(C15/C16)^3 + 0.0000632929*(D15/C16)^3
C15 is OG in Brix
C16 is ~1.04
D15 is your refractometer reading in Brix
Sorry for the spreadsheet-type format. I don't know slide rule format.
 
Nice article. I've definitely spent a lot of time understanding calculations in hopes of making my brew days less stressful when I miss all my expectations.
 
I love these types of articles. Great job Brett. I love how we have apps and all to help us on brew day, but I like to be able to do the number crunching in my head too.
More of these please!
 
@Singletrack
thanks for the excel formula formula. Iʻll put the slide ruler away and plug it in to excel.
cheers
 
@passedpawn challenge accepted
FG = OG - ( OG * Atten.)
ABV = (OG - FG) * .131
Substitute FG:
ABV = (OG - (OG - (OG*Atten.))) * .131
ABV = (OG - OG + (OG*Atten.)) * .131
ABV = (OG * Atten.) * .131
Taking your assumption of 76% attenuation:
ABV = OG * 0.76 * 0.131
ABV = OG * 0.10
ABV = 0.1 * OG
If OG = 50, then ABV = 0.1 * 50 = 5 %
I checked my last 4 brews. One was close, one was fairly close, two were off by 1%. So it's a nice rule of thumb if you don't care if it's exactly right, or forget to measure. Estimated vs. actual will have a wider difference the further away actual attenuation is from 76%, and the larger the OG.
For example, here's my brews:
OG | FG | Atten. | Est. ABV | Measured ABV | Difference
69 | 08 | 88% | 6.9% | 8.0% | +1.1%
68 | 19 | 72% | 6.8% | 6.4% | -0.4%
74 | 19 | 74% | 7.4% | 7.2% | -0.2%
76 | 11 | 86% | 7.6% | 8.5% | +1.0%
 
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