November 5th, 2013, 10:02 PM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  999 fractions
Question: Find 999 fractions between 1/3 and 1/2 such that the difference between pairs of numbers next to each other is the same. (Hint: Find convenient equivalent fractions for 1/3 and 1/2) My attempt: The difference 1/2  1/3 = 1/6. This difference has to be divided into 1000 parts to get 999 fractions which lie between 1/3 and 1/2. so each "step" between required fractions = (1/6)/1000 = 1/6000 Therefore required list is 2001/6000, 2002/6000,..., 2998/6000, 2999/6000 Am I correct? Is there a another/simpler way to solve this problem? Did I use the hint?? I am very confused... 
November 5th, 2013, 10:41 PM  #2 
Senior Member Joined: Oct 2013 From: Sydney Australia Posts: 126 Thanks: 0  Re: 999 fractions
It all looks good to me. That is how I would have done it. 
November 7th, 2013, 01:18 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,262 Thanks: 1958 
There are other correct answers as well. Can you find one?

November 17th, 2013, 08:56 PM  #4 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  Re: 999 fractions
To skipjack I tried the method of finding equivalent fractions (1/3)*(2/2) and (1/2)*(3/3) gives 2/6...3/6 Using factor 2/2 for the above we get 4/12 and 6/12. ONE fraction in between 5/12 Using factor 3/3 for the above we get 6/18 and 9/18. TWO fractions in between 7/18 and 8/18 So using 1000/1000, we get 2000/6000 to 3000/6000 with 999 fractions in between. This is the same answer I got using the other method as above Reducing the "step" size increases the number of fractions and increasing the "step" size decreases the number of fractions If there is another way to find the answer, I can't see it. Thanks 
November 17th, 2013, 11:32 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,262 Thanks: 1958  Re: 999 fractions
You correctly found what may well have been the expected answer. However, other answers are possible by giving a slightly different meaning to "numbers next to each other". The concept of "next" implies the numbers have been ordered, but it doesn't imply they have been put in ascending order. Even without the above change of interpretation, other answers are possible. The 999 fractions could, for example, be 5/12, 10/24, 15/36, ... , 4995/11988. As the question doesn't require the fractions to have different values, each pair could differ by zero. (The question doesn't explicitly state that the numbers 1/3 and 1/2 are included when determining which pairs are considered.) 

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