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Percents and Weighted Averages

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by GMATGuruNY » Sat Oct 02, 2010 2:17 am
ruchisharma wrote:Cereal A is 10% sugar by weight. Cereal B is 2% sugar by weight. To make a new mixture which is 4 % sugar, what should be the ratio of A and B, by weight?
The following method is called alligation. It's an easy way to handle weighted average problems.

To combine a 10% mixture with a 2% mixture to get a combined percentage of 4%:

The proportion needed of each starting percentage is the positive difference between the other 2 percentages.

Proportion needed of 10% mixture = 4-2 = 2.
Proportion needed of 2% mixture = 10-4 = 6.

A:B = 2:6 = 1:3

Another approach:

Since the GMAT is a multiple choice test, we also could plug in the answer choices, one of which would say that the ratio of A(10%) to B(2%) = 1:3.

Plug in A = 100.
Plug in B = 300.
10% of A = 10.
2% of B = 6.
Total sugar = 10+6 = 16.
A+B = 100+300 = 400.
16/400 = 8/200 = 4/100 = 4%. Success!
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by Arcane66 » Sat Oct 02, 2010 11:27 am
Wouldn't noticing that all the question is asking for is a basic linear equation make solving a lot easier? Your methods are amazing, Mitch, but I don't know how you know all these rules and formulas. hah. Anyway, I just did .1x+.02y=.04x+.04y. I did this because you don't know the mixtures of either of them but you don't have to. I used x for a and y for b. I did separate variables because the question doesn't say there's a relationship between the two and assuming there is one is going to lead you down the wrong path. I said .04x+.04y because the resulting mixture is going to be 4% of both (basically the total of .04(x+y)). Then you just solve for x/y in the equation and you get 1/3.
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by GMATGuruNY » Sat Oct 02, 2010 2:22 pm
Arcane66 wrote:Wouldn't noticing that all the question is asking for is a basic linear equation make solving a lot easier? Your methods are amazing, Mitch, but I don't know how you know all these rules and formulas. hah. Anyway, I just did .1x+.02y=.04x+.04y. I did this because you don't know the mixtures of either of them but you don't have to. I used x for a and y for b. I did separate variables because the question doesn't say there's a relationship between the two and assuming there is one is going to lead you down the wrong path. I said .04x+.04y because the resulting mixture is going to be 4% of both (basically the total of .04(x+y)). Then you just solve for x/y in the equation and you get 1/3.
The algebra that you used is a perfectly fine and efficient way to solve the problem. Nice work.

Whether the algebra is easier depends on the test-taker. The biggest danger with algebra -- and I love algebra -- is that errors can be made unwittingly.

For those prone to making errors, alligation and plugging in the answers might be safer methods. I personally find alligation to be the quickest way to solve.
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by Arcane66 » Sat Oct 02, 2010 8:41 pm
GMATGuruNY wrote:
Arcane66 wrote:Wouldn't noticing that all the question is asking for is a basic linear equation make solving a lot easier? Your methods are amazing, Mitch, but I don't know how you know all these rules and formulas. hah. Anyway, I just did .1x+.02y=.04x+.04y. I did this because you don't know the mixtures of either of them but you don't have to. I used x for a and y for b. I did separate variables because the question doesn't say there's a relationship between the two and assuming there is one is going to lead you down the wrong path. I said .04x+.04y because the resulting mixture is going to be 4% of both (basically the total of .04(x+y)). Then you just solve for x/y in the equation and you get 1/3.
The algebra that you used is a perfectly fine and efficient way to solve the problem. Nice work.

Whether the algebra is easier depends on the test-taker. The biggest danger with algebra -- and I love algebra -- is that errors can be made unwittingly.

For those prone to making errors, alligation and plugging in the answers might be safer methods. I personally find alligation to be the quickest way to solve.
Hey, thanks for the response. You truly are smart, man. I hadn't heard of some of your techniques until you bring them up.
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