Ratio GRE question

Question

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.

Answer 1

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.

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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.

Answer 2

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.