This is another system of equations question, so let's call some variables.
You really only need to call one variable, since they give you the amount of regular hamburger meat required. I'm going to say that:
L = amount of lean meat required
Since we're dealing with mixing high fat/ low fat products, I'm going to write an equation to balance the total amount of fat content on both sides. You know three things:
1. You have L amount of 12% fat.
2. You have 339 pounds of 27% fat.
3. You want "a batch" of 20% fat.
It doesn't explicitly define how much is in a batch, but that's not a problem. The total amount of burger meat after all the mixing is done will be:
L + 339
Since that's how much you added to begin with.
So, let's state our equation.
L*0.12 + 339*0.27 = (L+339)*0.20
Since L is 12% fat, it will contribute L*0.12 pounds of fat to the mixture.
Since the 339 pounds of regular burger meat is 27% fat, it will contribute 339*0.27 pounds of fat to the mixture.
You want the total amount to be 20% fat, so you'll have (L+339)*0.20 pounds of fat in your mixture.
Now let's just solve for L. Simplify and distribute.
0.12L + 91.53 = 0.20L + 67.8
23.73 = 0.08L
L = 296.625
You'll need 296.625 pounds of extra lean 12% meat to bring your batch to 20%. You'll have a grand total of 635.625 pounds of 20% burger meat.
Hope my explanations made sense.