A dairy needs 296 gallons of milk containing 6% butterfat. How many gallons each of milk containing 7% butterfat and milk containing 3% butterfat must be used to obtain the desired 296 gallons?

Answer 1

Let x = number of gallons of 3% butterfat milk

Then 296 - x = number of gallons of 7% butterfat milk

0.03x + 0.07(296 - x) = 0.06(296)

Solve for x.

Answer 2

(Gallons of milk. ) (%of buttermilk) = gallons of buttermilk

7% buttermilk. x .07 .07x

3% buttermilk. 296 - x .03 .03(296-x)

6% buttermilk. 296 .06 .06(296)

.07x + .03(296 - x) = .06(296)

.07x + 8.88 - .03x = 17.76

.04x = 8.88

x = 222 gallons of 7% buttermilk

296 - 222 = 74 gallons of 3 % buttermilk

Answer 3

Hi,

So I solved this with logic, although there may be a mathematical way to solve it. What you need is to figure out the ratio of 7% to 3% milk. We want 6% milk right?

So I gave it some thought and if you just did equal parts 7% and 3% it would not work. You would average those and get 5% milk. That would not help us.

So if you think about it, you need more 7% milk than 3% milk. Think, how far away is 7 from 6? It is one away. How far a way is 3 from 6? It is 3 away. This is where there may be a more elegant mathematical way to do it, but here is how I did it. Since 3 is 3 times as far away from 6 as 7 from 6, then we need 3 times as many 7% gallons as 3% gallons.

Now before we solve it, let's take an example to make sure we have the right ratio. Say you use this ratio to make 6% milk. You will need 3 gallons of 7% and 1 gallon of 3% (that way we are using the ratio of 3 times as many gallons of 7% milk as 3% milk).

Multiply 3 times 7 and get 21.

Multiply 1 times 3 and get 3.

Add 21 and 3 and get 24.

Now divide 24 by 4 because we are taking an average and get 6.

6% milk!

Now we have 4 gallons of 6% milk made from 3 gallons of 7% and 1 gallon of 3%.

This is all an example to show that we have the right ratio to make 6% milk. This really is the key to the problem! The ratio is 3 parts 7% milk to one part 3% milk.

Now lets apply this to the original problem. Let's call 3% milk x and then 7% milk 3x.

x + 3x = 296

4x = 296

x=74

That is the number of gallons of 3% milk. 74 gallons.

Now multiply that by 3 you get 222. That is the number of 7% gallons. 222 gallons.

Therefore, you will need 222 gallons of 7% milk and 74 gallons of 3% milk. Good luck and I am looking forward to seeing if any other experts have a different way of explaining it.