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What is the greatest number of identical bouquets that...

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What is the greatest number of identical bouquets that...

by MFaulkner » Thu Jun 03, 2010 10:10 am
What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal)

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Source: OG Quant Review 2nd Edition
Answer: E

Here is my question (Spoiler alert)
So, the answer is found by finding the greatest common factor of 21 and 91->
21=(7)(3) and 91=(7)(13)
GCF = 7

The explanation then says that 7 bouquets can be made, each with 3 white tulips and 13 red tulips.

However, I just cannot conceptualize why the GCF is the answer to this problem. I am still unclear with the explanation and would have never guessed to have taken the GCF to solve the problem. Can someone please shed some light for me?

Thanks,
Michael
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by Patrick_GMATFix » Thu Jun 03, 2010 11:27 am
MFaulkner wrote:I just cannot conceptualize why the GCF is the answer to this problem.
Hi Michael,

Any common factor would tell you one way in which the flowers can be broken down into identical bouquets. For simplicity's sake, suppose we had 30 whites and 42 reds. The common factors of 30 and 42 (other than 1) are 2, 3, and 6. This means that these flowers can be broken down into any of the following identical bouquets:
  • 2 bouquets of 15 whites and 21 reds
  • 3 bouquets of 10 whites and 14 reds
  • 6 bouquets of 5 whites and 7 reds
So if we're interested in how to group flowers into identical bouquets, we're really interested in finding a common factor to the number of whites and the number of reds. Notice from the example above that if we want a lot of bouquets, then we should use a greater common factor since the common factor dictates how many bouquets will be formed.

Thus to answer your question, which asks for the greatest number of identical bouquets possible, we need the greatest common factor of 21 and 91. This is why the answer is GCF(21,91) = 7.

Hope that makes sense,
-Patrick
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by MFaulkner » Thu Jun 03, 2010 11:32 am
^^Great explanation! Thank you very much.

Even though I am very comfortable with finding the GCF and LCM, I have never conceptualized as it is required for this problem.

Again, great explanation.
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by Patrick_GMATFix » Thu Jun 03, 2010 11:38 am
*tip of the hat
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by mcdesty » Tue Aug 14, 2012 4:25 pm
If you can't immediately conceptualize this then you can solve it like a word problem(Not as effective as the explanation above)

21/x=91/y (No flower can be left out)

Cross multiply: y/x = 91/21 = 13/3

For every 13 red we have 3 white...

Using the smaller number, we can see that we must have 7

GCF works best though...
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