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weighted averages

by meh » Fri Dec 13, 2013 11:19 am
A mixture of 20 grams of lean ground beef is mixed with 50 grams of fatty ground beef to create a ground beef mix with 8% fat. How many grams of fat does the lean ground beef have?

My method:

20 grams lean : 50 grams fatty : 70 grams total
x% fat : y% fat : 10% fat
(2 x (-diff) ) x (5 x (+diff) ) = 0
(10-5) = 5% (10+2)= 12%

20x.05 = 1 gram of fat.


Is there an easier way??
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by Stuart@KaplanGMAT » Fri Dec 13, 2013 12:03 pm
meh wrote:A mixture of 20 grams of lean ground beef is mixed with 50 grams of fatty ground beef to create a ground beef mix with 8% fat. How many grams of fat does the lean ground beef have?

My method:

20 grams lean : 50 grams fatty : 70 grams total
x% fat : y% fat : 10% fat
(2 x (-diff) ) x (5 x (+diff) ) = 0
(10-5) = 5% (10+2)= 12%

20x.05 = 1 gram of fat.


Is there an easier way??
Hi!

Did you omit something from the Q? Unless we know the % fat in the "fatty" ground beef, we can't answer the Q.
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by meh » Fri Dec 13, 2013 12:11 pm
No, but I did alter the question.

The original doesn't mention the amount of grams. It looks something like this...

A mixture of lean ground beef with 2% fat is mixed with fatty ground beef with 10% fat to create a ground beef mix with 8% fat. What is the ratio of lean ground to to fatty ground beef in the new mixture?

Method:

+2 differential (fatty) + -6 differential (lean) = 0

3=fatty ; lean = 1

Answer: 1:3

If we knew the total grams in the new mixture, we'd be able to calculate the amount of fatty ground beef in grams.
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by meh » Fri Dec 13, 2013 12:12 pm
No, but I did alter the question.

The original doesn't mention the amount of grams. It looks something like this...

A mixture of lean ground beef with 2% fat is mixed with fatty ground beef with 10% fat to create a ground beef mix with 8% fat. What is the ratio of lean ground to to fatty ground beef in the new mixture?

Method:

+2 differential (fatty) + -6 differential (lean) = 0

3=fatty ; lean = 1

Answer: 1:3

If we knew the total grams in the new mixture, we'd be able to calculate the amount of fatty ground beef in grams.
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by meh » Fri Dec 13, 2013 12:12 pm
Could there be another answer to the previous question I posted?
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by GMATGuruNY » Fri Dec 13, 2013 1:07 pm
A mixture of lean ground beef with 2% fat is mixed with fatty ground beef with 10% fat to create a ground beef mix with 8% fat. What is the ratio of lean ground to to fatty ground beef in the new mixture?
Fat percentage in the lean beef: 2%.
Fat percentage in the fatty beef: 10%.
Fat percentage in the mixture: 8%.

Let L = the lean beef and F = the fatty beef.
The following approach is called ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.

Step 1: Plot the 3 percentages on a number line, with the percentages for L and F on the ends and the percentage for the mixture in the middle.
L 2%-------------8%-----------10% F

Step 2: Calculate the distances between the percentages.
L 2%------6-----8%-----2------10% F

Step 3: Determine the ratio in the mixture.
The required ratio of L to F is the RECIPROCAL of the distances in red.
L:F = 2:6 = 1:3.
Additional problem:
How many pounds of fatty beef are needed to form 24 pounds of mixture?
Since L:F = 1:3, of every 4 pounds of mixture, 1 pound is L and 3 pounds are F.
Thus, in 24 pounds of mixture, F = (3/4) * 24 = 18 pounds.

For two similar problems, check here:

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by Stuart@KaplanGMAT » Fri Dec 13, 2013 11:04 pm
meh wrote:Could there be another answer to the previous question I posted?
Sure - based on the original question, both the lean and the fatty could be 8% (or many many other combinations) and generate a different # of grams of fat. You have 2 unknowns but only 1 equation, so there's an infinite number of solutions.

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by meh » Mon Dec 16, 2013 5:54 am
Sure - based on the original question, both the lean and the fatty could be 8%
That would be true, but one is noted as "lean ground beef" and the other "fatty ground beef" so we assume that fatty ground beef has a higher fat content. If this is the case, what other combination could there be?
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by Stuart@KaplanGMAT » Mon Dec 16, 2013 11:34 am
meh wrote:
Sure - based on the original question, both the lean and the fatty could be 8%
That would be true, but one is noted as "lean ground beef" and the other "fatty ground beef" so we assume that fatty ground beef has a higher fat content. If this is the case, what other combination could there be?
A mixture of 20 grams of lean ground beef is mixed with 50 grams of fatty ground beef to create a ground beef mix with 8% fat. How many grams of fat does the lean ground beef have?

The ratio of fat content for L:F is 2:5. So, as GGNY (does he have a real name - does anyone know it?) aptly noted, all we can conclude is that the fatty beef has 2.5 times as much weight in the mix as does the lean.

In line form:

Lean % ----5x---- Mixture 8% --2x-- Fatty %

And "x" could have any value at all (well, any value that keeps the Lean % greater than 0).

So, for exampl:
If x=1, lean would be 3% fat and fatty would be 10% fat.
If x=.5, lean would be 5.5% fat and fatty would be 9% fat.
If x=.1, lean would be 7.5% fat and fatty would be 8.2% fat.

And of course, each of those fat contents would give you a different answer to your question.
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