what percent will result from combining two types of milk? July 16, 2014 9:49 AM   Subscribe

If I add 1/100 to 2/100, I get 3/100 or 3%. But if I add one hundred apples, one of which is red, to one hundred apples, two of which are red, I will have 3/200 or 1.5%. I'm very confused now as to what percent milk fat my milk will be if I combine the two types of milk (1% and 2%) together.
posted by windykites to Education (15 answers total)

Assuming the quantity of 1% milk is the same as the quantity of 2% milk, it would be 1.5%.
posted by barnoley at 9:51 AM on July 16, 2014 [2 favorites]

(Since you now have 3/200 which equals 1.5%.)
posted by barnoley at 9:51 AM on July 16, 2014

If you add 100 units of 1/100 stuff, and 100 units of 2/100 stuff, you end up with 3/200 -- not 3/100. Hence 1.5%
posted by wrok at 9:52 AM on July 16, 2014 [8 favorites]

But I don't understand. Adding fractions, you're not supposed to add the denominators together.
posted by windykites at 9:59 AM on July 16, 2014

If you add 100 units of 1/100 stuff, and 100 units of 2/100 stuff, you end up with 3/200 -- not 3/100. Hence 1.5%

Exactly. The milk is like the apple example.

Just saying 1% + 2% = 3% gives you the wrong answer, because it doesn't account for how much of each type of milk you are combining. The final mixture is half 1% and half 2%, so it has a fat content of 0.5*1% + 0.5*2% = 0.5% + 1% = 1.5%.

(This method also works for combining unequal volumes. 1 cup of 1% plus 3 cups of 2% is 1/4 1% and 3/4 2%, so it has fat concentration 0.25*1% + 0.75*2% = 1.75%)
posted by Pre-Taped Call In Show at 10:01 AM on July 16, 2014 [1 favorite]

2 totally different sums. If you add 1 cent + 2 cents you have 3 cents - 3% - of an imagined 100. If you add 100 cents to 100 cents you have 200 cents. The denominator is different.
posted by Hobo at 10:02 AM on July 16, 2014

Yes but in adding 1/100 and 2/100 you are adding the wrong fractions. You should be adding 1/200 to 2/200. The denominator must represent the total final quantity of milk.
posted by Pre-Taped Call In Show at 10:04 AM on July 16, 2014 [23 favorites]

You are adding 100 units of milk, of which 1 unit is fat, to another 100 units of milk, of which 2 units are fat. So now you have 200 units of milk, of which 3 units are fat. Your new fat content is 3/200, or 1.5%.
posted by Sternmeyer at 10:06 AM on July 16, 2014 [4 favorites]

(otherwise mixing together 50 cups of 2% milk would give you a big bucket of 100% fat!)
posted by Pre-Taped Call In Show at 10:07 AM on July 16, 2014 [11 favorites]

What you're missing is that your two examples (1/100 + 2/100 = 3/100, and 1% milk + 2% milk = 1.5% milk) are not actually the same situation. In the first example, 1/100 and 2/100 are taken to be complete quantities -- as in, 0.01 pounds + 0.02 pounds = 0.03 pounds. In the second example, 1% and 2% are in essence each a prefix (or ratio) that goes with some other quantity. 1% milk isn't telling you how much TOTAL fat is in the carton, it's telling you the ratio.

The solution with all the steps written out is instead done like Sternmeyer posted as I was writing this. This is actually the reason why we are so insistent on students writing out all their steps in math/science/engineering classes -- jumping to the end can take you past something important.

(Or to write out all the steps in full-bore pedantic mode: I have 100 ounces of 1% milk and 100 ounces of 2% milk. How much fat do I have? I have 3 ounces (1% of 100 + 2% of 100). If I put everything together I end up with 3 ounces of fat in 200 ounces of milk, which would be 1.5%.)
posted by range at 10:11 AM on July 16, 2014 [2 favorites]

Let's say this (grams = g):

1% milk = 1g of fat in 100ml of milk

2% milk = 2g of fat in 100ml of milk

Combine them, and you have 3g of fat in your total volume of 200ml of milk.

3g/200ml = 0.015, or 1.5%
posted by extramundane at 10:11 AM on July 16, 2014 [2 favorites]

The trick is to realize you shouldn't be adding fractions at all. You're not adding fractions you're adding a known quantity of milk and fat. If you have a gallon of milk at 1% that's 0.01 gallons of fat and 0.99 gallons of milk. You combine it with another gallon of milk at 2% that's 0.02 gallons of fat and 0.98 gallons of milk. Combine them both and you get 0.03 gallons of fat and 1.97 gallons of milk. That's 0.03 gallons of fat out of 2 gallons of liquid. That gives you 1.5% fat out of the total volume of liquid.
posted by Green With You at 10:15 AM on July 16, 2014 [3 favorites]

All the answers above are correct. Here's a "sanity check" you can apply to the problem: is it possible that you could mix two kinds of low-fat milk and end up with whole milk? No! Mixing is a kind of averaging process; the result can't have a higher concentration of fat than the highest-concentration input. So if you've added 1/100 to 2/100, you know you've done something wrong.

Pre-Taped Call In Show has the shortcut for averaging ratios above. It's handy to know, so I'll give one example of how I used it the other day (for real). I wanted to make ice cream from a recipe that calls for 1 cup of heavy cream (36% fat) and 3 cups half-and-half (12% fat; "half and half" is a misnomer, by the way). I wanted to make this from the cream and nonfat milk I already had on hand, so I worked out the fat content of the mixture specified by the recipe:

(1/4)(0.36) + (3/4)(0.12) = 0.18

and determined that 2 cups heavy cream and 2 cups nonfat milk would give the same result:

(1/2)(0.36) + (1/2)(0.00) = 0.18.

(No doubt someone with surpassing knowledge of dairy products will be along to tell me why this mixture isn't really the same thing as the cream + half-and-half. My ice cream was delicious, though.)
posted by aws17576 at 10:37 AM on July 16, 2014 [3 favorites]

Ohh, I get it now. OK, thanks metafilter!
posted by windykites at 10:48 AM on July 16, 2014

! Mixing is a kind of averaging process;

To extend this, suppose you average your 1% and 2% milk:

(1/100 + 2/100)/2 = (3/2)/100

Which is your 1.5 % again.
posted by leahwrenn at 5:23 PM on July 16, 2014

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