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Mixture Problem............OG Problem

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Mixture Problem............OG Problem

by Mo2men » Fri Aug 04, 2017 1:54 pm

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A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%

OA: A
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by DavidG@VeritasPrep » Fri Aug 04, 2017 2:39 pm
Mo2men wrote:A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%

OA: A
We've got a weighted average here. The amount of 20% coffee is four times the amount of the 60% coffee. (400 pounds to 100 pounds.) We know that if we plot 20%, the overall percent, and 60% on the number line, the overall percent will be closer to 20% than 60%, and we can designate the distance from each group to the overall percent as x and 4x.

20-------Total Average--------------------60
Gap: x------------------------------4x--------

The distance from 20 to 60 is x + 4x = 5x. So 5x = 60 -20 = 40. If 5x = 40, x = 8. Because 20 is 'x' away from the overall average, we know that the overall average is 20 + 8 = 28. The answer is A
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by DavidG@VeritasPrep » Fri Aug 04, 2017 2:42 pm
Mo2men wrote:A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%

OA: A
Or just do the arithmetic. Total coffee = 400 + 100 = 500

decaf coffee = 400*(.20) = 80 from what's initially in stock
100* (.6) = 60 from the new purchase.
Total decaf = 80 + 60 = 140

140/500 = 280/1000 = 28%. the answer is A
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by Mo2men » Fri Aug 04, 2017 2:48 pm
DavidG@VeritasPrep wrote:
Mo2men wrote:A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%

OA: A
Or just do the arithmetic. Total coffee = 400 + 100 = 500

decaf coffee = 400*(.20) = 80 from what's initially in stock
100* (.6) = 60 from the new purchase.
Total decaf = 80 + 60 = 140

140/500 = 280/1000 = 28%. the answer is A
Thanks David.

My aim was the alligation method as presented. I felt I did something wrong while solving the question
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by GMATGuruNY » Sat Aug 05, 2017 3:02 am
Mo2men wrote:A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%
Alternate approach:

(20% decaf)/(60% decaf) = (400 pounds)/(100 pounds) = 4/1.
The mixture contains 4 pounds 20% decaf for every 1 pound 60% decaf.
Thus, the average decaf percentage for every 5 pounds = (4*20 + 1*60)/5 = 140/5 = 28.

The correct answer is A.
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by Matt@VeritasPrep » Sun Aug 06, 2017 10:53 pm
One approach:

We want ((20% of 400) + (60% of 100)) / 500, or (.2 * 400 + .6 * 100) / 500. Multiplying by 10/10 gives us (2 * 400 + 6 * 100) / 5000, or 1400/5000, or 14/50, or 28/100, or 28%.
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by Matt@VeritasPrep » Sun Aug 06, 2017 10:55 pm
Another approach:

We know 80% of the coffee is 20% decaf, and that 20% of the coffee is 60% decaf. Since we have four times as much of the 20% as we do of the 60%, the average must be (4/5) of the way from 60 to 20. 60 - (4/5) * (60 - 20) = 28, so that's our answer: 28%.
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by Matt@VeritasPrep » Sun Aug 06, 2017 10:56 pm
A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

One last approach!

If we want to find 20% of 400, we can write this as (20 / 100) * 400, or 20 * (400 / 100), or 80. Similarly, 60% of 100 is just 60.

That leaves us with 140 pounds of decaf. We know we've got 500 total, so 140/500, or 14/50, or 28/100, or 28%.
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by ceilidh.erickson » Mon Aug 07, 2017 9:17 am
With weighted average / mixture problems, the first question I always ask is "what do I have MORE of?" The final weighted average will need to be closer to that.

Here, if we have much more of the 20% mixture than the 60% mixture, the weighted average will be much closer to 20 than 60.

The halfway point between 20 and 60 would be 40: that's what we'd get if we had equal amounts of each mixture. Since we have more of the 20% stuff, our answer will have to be quite a bit less than 40. Eliminate E and probably D.

Ok... that didn't get us all the way to the answer on this question! The other answer choices are pretty close together, so I wouldn't feel comfortable just estimating. From there, we'd have to follow one of the strategies outlined by other posters. But on other questions, this thought process might get us all the way to a right answer, or at least help us to formulate a good guess.

For examples, see:
https://www.beatthegmat.com/seed-mixture-t21603.html
https://www.beatthegmat.com/mixture-prob ... 90121.html

For more on weighted averages, see:
https://www.beatthegmat.com/mgmat-5th-ed ... tml#789804
https://www.manhattanprep.com/gmat/blog ... -problems/
https://www.manhattanprep.com/gmat/blog ... roblems-2/
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by Scott@TargetTestPrep » Mon Aug 14, 2017 6:37 am
Mo2men wrote:A grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated. If the grocer buys another 100 pounds of coffee of which 60 percent is decaffeinated, what percent, by weight, of the grocer's stock of coffee is decaffeinated?

A. 28%
B. 30%
C. 32%
D. 34%
E. 40%

OA: A
Since the grocer has 400 pounds of coffee in stock, 20 percent of which is decaffeinated, 0.2 x 400 = 80 pounds are decaffeinated.

Since the grocer buys another 100 pounds of coffee, 60 percent of which is decaffeinated, the grocer has 60 more pounds of decaffeinated coffee.

Thus, the percentage of decaffeinated coffee is (80 + 60)/(400 + 100) = 140/500 = 28/100 = 28%.

Answer: A

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