how do solve mixture problems?

Question

I need to show this problem step by step.
At twilight Coffee shop, Manage Bella buys only half and half (12% fat) milk and skim (0% fat) milk to simplify inventory.
She is rushing out of town and needs to quickly leave instructions for her employees on how to obtain 2% fat milk for customers.
A) For one set of instruction, she would like to end with 1 gallon of 2% fat milk. How much half and half and skim milk must she mix together to obtain 1 gallon of 2% fat milk?

Answer 1

First define the variables.

s = the amount of skim milk

h = the amount of half and half



Second, rewrite the problem as an equation.

There are two things you know.

1. You want to add skim milk with a 0% milk fat and half and half with a 12% milk fat to get to 2% milk fat

s * 0% + h * 12% = 2%

2. You want to have 1 gallon (or 16 cups) in the end

s + h = 1 gallon = 16 cups



Let's solve the first problem first.

This is easy because since the skim milk has no milk fat that varriable falls away

s * 0% + h * 12% = 2%

h * 12% = 2%

h = 2% / 12%

h = 1/6

So half and half needs to be 1/6 of the mixture.



Now that we know that we can move to the second equation.

We want 16 cups total so 1/6 of 16 cups is 2 2/3 cups

To get the amount of skim milk s + 2 2/3 cups = 16 cups or s = 13 1/3 cups.



Tell your workers to mix 1 part half and half with 5 parts skim milk or 2 2/3 cups of half and half with 13 1/3 cups of skim milk to get 2% milk.

Answer 2

For simplicity call "half & half" milk H, skimmed milk S, and the 2% mixture milk M.



H @ 12%, S @ 0%, M @ 2%



Note that H + S = M, because the total amount of milk you've put in must come back out!



12%H + 0%S = 2%M to get the percentages right

12H + 0S = 2M simplified

You want one gallon of M, so let M = 1

12H + 0S = 2(1)

12H = 2

H = 2/12 = 1/6 = 0.1666666...



I said that H + S = M

(1/6) + S = 1

S = 1 - (1/6) = (5/6) = 0.833333...



She must mix 0.1666... gallons of half & half with 0.8333... gallons of skimmed milk to make 1 gallon of 2% special milk.



Same principle whenever you have a question about mixtures.

Answer 3

IDK if this is the official way to solve the problem, but the way I'd approach it is to think first how much fat am I going to need at the end? How much, in liquid measure, is there in a gallon of 2% fat milk? In other words, what is 2% of a gallon? Let's do it in ounces, b/c we don't know yet if it would even make a full cup. So step one would be how many ounces in a gallon.



1 gal. = 128 oz. (You get this either from looking up the conversion online, or by multiplying 32 oz. in a qt. times 4 qts. in a gallon.)



Then step two, to find out how many oz. of fat would make it 2%, you multiply the 128 oz. by .02 (2%) which gives you 2.56 oz.



Now, the skim milk doesn't provide any of the fat, so you need to get it all from the half and half. Step three is how much half and half gives 2.56 oz. of fat. You need to write a formula for that. (12% of how much [half & half] equals 2.56. oz.) Written mathematically that is .12x = 2.56



Step four is solve the equation: Dividing both sides by .12 (to get plain x on the left), which gives you x = 21 1/3 oz. (If your teacher usually likes the answer in decimals you can go two decimal places & call it 21.33 oz.) So you need that much half & half in the gallon to give you the 2.56 oz. of fat you need.



Now that you have all the fat you need in the gallon, step five is to find out how much skim milk you need to fill up the gallon. That is 128 oz. - 21 1/3 oz. = 106 2/3 oz. or if your teacher usually wants the answer in decimals, you could round it to two places and call it 106.67 oz. of skim milk.



You can't just copy and paste what I wrote b/c there is not supposed to be all those words I used to teach you how to do the problem. Just pick out the math part of it, and write each step in numbers.



ON EDIT: If you divide the answer I got in oz. by 8 to get the amount in cups, you will see that it matches what the other two people who solved the problem said. Theirs has more of the problem written in math, so I'd go w/ one of theirs.