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For any integer k greater than 1, the symbol k* denotes the

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For any integer k greater than 1, the symbol k* denotes the

by II » Wed Jul 02, 2008 2:45 pm
For any integer k greater than 1, the symbol k* denotes the product of all the fractions of the form 1/t, where t is an integer between 1 and k, inclusive. What is the value of 5*/4* ?

A) 5 B) 5/4 C) 4/5 D) 1/4 E) 1/5

I am very interested getting views and feedback on the following:
- alternative approaches to solving this. Please illustrate your logic.
- guessing strategies you would use on this type of question, if you were running out of time, or if you simply had to guess on this question
- the traps and tricks which are built into this question (if you can see any)

Thanks in advance.
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by Ian Stewart » Wed Jul 02, 2008 3:15 pm
I don't know that there are any 'alternative approaches' here. This is one of the GMAT 'function' questions. We've never seen this * function before, and we can't really have any idea how to simplify an expression that includes it. We don't have much choice; we need to:

a) figure out what the * function does;
b) if asked to evaluate some expression involving the *, we need to get rid of the *, using the definition we worked out in step a).

5* = (1/1)*(1/2)*(1/3)*(1/4)*(1/5) = 1/5!
4* = 1/4!

Now we know 5*/4* = (1/5!)/(1/4!) = 4!/5! = 1/5

_________

Some may be tempted to choose 5/4 here, either because it looks similar to the question, or because they will incorrectly divide their fractions near the end of the calculation. I imagine others may find the definition of the * function somewhat abstract. Still others might make the incorrect 'deduction' that 5*/4* = (5/4)*, which would lead to confusion. Those are the 'traps' I can see in the question; perhaps there are others.

If forced to guess an answer within three seconds of reading the question, I'd notice that the question involves reciprocals ("fractions of the form 1/t"), so I'd choose something with 5 in the denominator- C or E. I'd rule out B because it looks too similar to the expression in the question.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by II » Wed Jul 02, 2008 3:23 pm
Ian Stewart wrote: Some may be tempted to choose 5/4 here, either because it looks similar to the question, or because they will incorrectly divide their fractions near the end of the calculation. I imagine others may find the definition of the * function somewhat abstract. Still others might make the incorrect 'deduction' that 5*/4* = (5/4)*, which would lead to confusion. Those are the 'traps' I can see in the question; perhaps there are others.
question.
Thanks Ian.
How would you avoid the traps and spot them in future similar questions questions ?
And what are the key-takeaways you would take forward from this type of question ?
I think one key take-away is to understand the procedure denoted by the strange symbol in these strange symbol formula type question. What is important here is that you follow each step in the procedure which the symbol signals.

Thanks again.
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by II » Wed Jul 02, 2008 4:02 pm
Ian Stewart wrote: Now we know 5*/4* = (1/5!)/(1/4!) = 4!/5! = 1/5
Also can you quickly tell us how you got from : (1/5!)/(1/4!) to 4!/5!

I simply did the following calculation:

5* / 4* = [(1/1)(1/2)(1/3)(1/4)(1/5)] / [(1/1)(1/2)(1/3)(1/4)]

The "(1/1)(1/2)(1/3)(1/4)" on the numerator and denominator cancel out, leaving 1/5.

Thanks.
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by beeparoo » Wed Jul 02, 2008 5:00 pm
II wrote:Also can you quickly tell us how you got from : (1/5!)/(1/4!) to 4!/5!

I simply did the following calculation:

5* / 4* = [(1/1)(1/2)(1/3)(1/4)(1/5)] / [(1/1)(1/2)(1/3)(1/4)]

You did the same as what Ian just demonstrated. The only difference is the notation.
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