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Weighted Average - Birdseed

This topic has 4 expert replies and 1 member reply
melguy Master | Next Rank: 500 Posts
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Weighted Average - Birdseed

Post Tue Nov 15, 2016 4:59 am
A feed store sells two varieties of birdseed : Brand A, which is 40% millet and 60% sunflower and brand B which is 65% millet and 35% sunflower. If a customer purchases a mix of the two types of birdseed that is 50% millet, what percent of the mix is brand A?

(Source : Manhattan, Book#2 FDP :reproduced as is. No answer choices are given).

Please help me with this problem. Is it possible to solve it using alligation method? Or any other weighted average tricks anyone can suggest (i.e. weighted average will be more towards A or B)

Thanks!

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Post Sat Nov 26, 2016 5:53 am
melguy wrote:
A feed store sells two varieties of birdseed : Brand A, which is 40% millet and 60% sunflower and brand B which is 65% millet and 35% sunflower. If a customer purchases a mix of the two types of birdseed that is 50% millet, what percent of the mix is brand A?
It would be nice if this question came with some answer choices, because it's quite possible that we could just use a bit of number sense to answer the question in a few seconds.

Brand A is 40% millet
Brand B is 65% millet
We want a combined mixture with 50% millet.

If we were to combine EQUAL amounts of Brand A birdseed and Brand B birdseed, then the percentage of millet in the new mixture would be the AVERAGE of the two brands.
That is, the percentage of millet in the new mixture = (40% + 65%)/2 = (105%)/2 = 52.5%
In other words, if the new mixture were comprised of 50% Brand A and 50% Brand B, then that new mixture would 52.5% millet.
That's already pretty close to our desire percentage of 50%

To get that 52.5% millet DOWN to 50% millet, we a little need less of the 65% mixture (Brand B) and little more of the 40% mixture (Brand A).
So, the correct answer will have a little more than 50% of Brand A in the mixture. So, I'd look for an answer choice like 55% or 60%.

So, if the answer choices looked like this...
A) 25%
B) 45%
C) 50%
D) 60%
E) 75%
... I'd confidently choose D and move on.

However, if the answer choices looked like this...
A) 40%
B) 45%
C) 50%
D) 55%
E) 60%
... I'd probably have to go back and actually do the work (as the experts in this thread have already done)
Although I'd already know that the correct answer is either D or E, in which case I might just test those answer choices.

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- Weighted Averages: https://www.gmatprepnow.com/module/gmat-statistics/video/805

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Matt@VeritasPrep GMAT Instructor
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Post Fri Nov 25, 2016 3:53 pm
I usually find these easiest algebraically.

Say we've got a ounces of brand A and b ounces of brand B.

Taken on their own, the seeds have .4a and .65b ounces of millet, respectively.

Added together, the seed mix has .5*(a + b) ounces of millet.

Since these represent the same thing, we can say that

.4a + .65b = .5*(a + b)

and get a ratio of a to b, from which the answer is a cinch.

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Post Tue Nov 15, 2016 5:04 am
For the alligation approach, check here:
http://www.beatthegmat.com/dividing-1000-15-t287033.html

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Marty Murray Legendary Member
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Post Tue Nov 15, 2016 5:31 am
Brand A Percentages: Millet - 40 Sunflower - 60

Brand B Percentages: Millet - 65 Sunflower - 35

Desired Percentage: Millet - 50

Notice, you don't have to worry about the Sunflower percentage. You can base your answer entirely on Millet percentages.

The greater the percentage of Brand A in the final mix, the closer the Millet percentage in the final mix will be to 40.

The greater the percentage of Brand B in the final mix, the closer the Millet percentage in the final mix will be to 65.

Initial Logical Conclusion: Since 50 is closer to 40 - difference of 10 - than to 65 - difference of 15 - there will be more Brand A than Brand B in the final mix.

So the question is how much more.

Alligation Method:

Total Distance Between 40 and 65: 65 - 40 = 25

Distance Between 40 and 50: 50 - 40 = 10

Distance Between 65 and 50: 65 - 50 = 15

Fractions Of Total Distance Between Desired Percentage And Endpoint Percentages:

10/25 = 2/5 15/25 = 3/5

50 is 2/5 of the way from 40 to 65.

50 is 3/5 of the way from 65 to 40.

So the final mix will be 3/5 Brand A, to make the Millet percentage 3/5 of the way toward that of Brand A.

It will also be 2/5 Brand B, to make the Millet percentage 2/5 of the way toward that of Brand B.

Percent Brand A: 3/5 = 60%

Answer: 60%

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Post Tue Nov 15, 2016 10:07 am
Hi melguy,

This prompt is a Weighted Average question, so you can set up the Average Formula and work through it algebraically if you like.

We're told about 2 Brands of seed:
1) Brand A = 40% millet and 60% sunflower
2) Brand B = 65% millet and 35% sunflower

We're told that mixing a certain amount of each led to a 50% millet mixture. We're asked for the PERCENT of this final mix that is BRAND A.

To start, here's the formula:

A = number of pounds of A
B = number of pounds of B

[.4(A) + .65(B)] / (A+B) = .5

.4A + .65B = .5A + .5B
.15B = .1A
15B = 10A
3B = 2A

3/2 = A/B

This means that for every 3 pounds of A, there are 2 pounds of B. Out of every 5 total pounds, 3 of those pounds are A.

3/5 = 60%

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