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Cashews cost 4.75 per pound and hazelnuts cost 4.50 per pound. What is larger, the number of pounds of cashews in a mixture of cashews and hazelnuts that costs $5.50 per pound, or 1.25? Alternatively, are they equal, or is it impossible to calculate?

My answer: I believe that 1.25 is larger. I conclude this because even if a 1.25lb mix were entirely composed of cashews, it would be more costly than $5.50/lb. Therefore, the mixture must have fewer than 1.25lbs of cashews, and 1.25 is greater.

The supposed answer: "There is no way to calculate the number of pounds of either nut in the mixture. We can calculate the ratio of the number of pounds of cashews to the number of pounds of hazelnuts required of the mixture to cost $5.50 per pound, but without knowing how many total pounds the mixture is, we cannot calculate the number of pounds of either component."

Please let me know if I'm wrong, but if I'm not, I think I need to email them.

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Is the 1.25 in the question 1.25lbs? And the prices in the question are presumably all in dollars? And how can any mixture cost $5.50 per pound? – SiliconCelery Jul 29 '12 at 20:05
@mixedmath Kaplan's second practice test (second set, set to more difficult questions based on my first set). – Matrym Jul 29 '12 at 20:16
@SiliconCelery You understood correctly and Gigili's edits make that clearer. – Matrym Jul 29 '12 at 20:17
Thanks for that - I used to consult for the correctness of a certain (different) publisher's GRE prep, and in particular did all their ratio work (as well as some others). I mention this only because it was absolutely stunning how awful the drafts given to me were... even on ratios. I can only imagine it's roughly the same there. – mixedmath Jul 29 '12 at 20:22
@mixedmath so you're in agreement that this question is flawed? :P – Matrym Jul 29 '12 at 20:31
up vote 3 down vote accepted

First of all, there is no way any mixture of them can cost \$5.50 per pound: if we have a mixture of $A$% cashews and $(100-A)$% hazelnuts, then $x$ pounds of the mixture costs $$x\text{ pounds}\cdot\left(\frac{A}{100}\cdot\frac{4.75\text{ dollars}}{\text{pound}}+\frac{100-A}{100}\cdot\frac{4.50\text{ dollars}}{\text{pound}}\right)$$ $$=4.75x\left(\frac{A}{100}\right)+4.5x\left(\frac{100-A}{100}\right)\text{ dollars}$$ $$\leq 4.75x\left(\frac{A}{100}\right)+4.75x\left(\frac{100-A}{100}\right)\text{ dollars}=4.75x\text{ dollars}$$ so the cost of the mixture is never more than \$4.75 per pound of mixture. Perhaps you, or the test writers, have made a typo in this respect.

But more importantly, there is no such thing as the

number of pounds of cashews in a mixture of cashews and hazelnuts

Suppose I tell you "spice" is a mixture consisting of 50% garlic and 50% salt (by weight). How many pounds of garlic are there in spice? The question doesn't make any sense - you can only talk about how many pounds of garlic there are in a specific pile of spice. The substance spice does not have any pounds of anything; it has ratios of its constituent ingredients.

Any information of the form "a mixture of cashews and hazelnuts costs $y$ dollars per pound" can only ever specify a ratio of ingredients. There is no number of pounds of cashews in a mixture; there is a number of pounds of cashews in a specified amount of a mixture.

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First, thanks for answering. But to address your point, let's say that spice and garlic each weigh the same. If you have two pounds of a mixture of 50% each, then you have one pound of each. I related the question accurately, although the question as written does seem to have flaws. – Matrym Jul 29 '12 at 20:19
@Matrym: The question, as you wrote it, refers to " the number of pounds of cashews in a mixture of cashews and hazelnuts that costs $5.50 per pound". You never said how many pounds of the mixture you have. – Zev Chonoles Jul 29 '12 at 20:23
Yes, but you must agree that even if the entire mixture were cashews, the price could not be $5.50 (or lower) AND be 1.25lbs (of cashews) or more. This price would be lower than the price of cashews. – Matrym Jul 29 '12 at 20:30
@Matrym: I do not understand what you're saying. Any mixture of the nuts cannot have a cost of more than 4.75 dollars per pound of mixture. But besides the issue of the cost, the question does not make sense because it does not even specify two different quantities of pounds of cashews to compare. – Zev Chonoles Jul 29 '12 at 20:37
@Matrym: Glad I could help! I'd still complain to them, because even claiming "impossible to calculate" is the right answer would, in my opinion, require that the question makes sense, but only that there is not enough information to answer it; I consider that to be very different from the question being nonsense. – Zev Chonoles Jul 29 '12 at 20:48

Let's suppose the question were written only slightly differently:

Cashews cost 4.75 per pound and hazelnuts cost 4.50 per pound. What is larger, the number of pounds of cashews in a mixture of cashews and hazelnuts that costs $4.60 per pound, or 1.25 pounds? Alternatively, are they equal, or is it impossible to calculate?

I changed only the dollar amount per pound and added the word "pounds" after $1.25$. I like this more because it's suggestive: 4.60 per pound is between 4.50 and 4.75, so it's possible, and it gives the illusion that there may be a way to proceed. Yet your analysis still wouldn't lead to the correct answer. So let's use this version.

If you knew that there were 2 pounds total of the mixture, then you could even calculate that there must be 1.2 pounds of hazelnuts and 0.8 pounds of cashews. This is the sort of analysis that you seem to have done.

The problem is that we don't know how many pounds there are. Perhaps there are 20 pounds of the mixture, in which case we'd have 12 pounds of hazelnut and 8 pounds of cashews.

So I have no idea whether or not there are more or less than 1.25 pounds of anything. Of course, since in the original statement of the problem you have an impossible price per pound, it has even less meaning. Perhaps the ratio would be 44.9%-55%-.1% on hazelnuts-cashews-diamonds. At least the price would be possible then.

To answer your question: both your analysis and the question itself are flawed. You should email them, but you should also learn that the question is very nearly reasonable.

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I realize now that the whole point of question was to give me seemingly useful information that was in fact insufficient to make a comparative assessment. Thanks! – Matrym Jul 29 '12 at 20:45

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