Difference between largest and smallest values

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mak_mak_mak: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Wed Apr 07, 2010 11:06 am

by mak_mak_mak » Thu May 20, 2010 6:31 am

Hello ALL,
What is the positive difference between the largest and the smallest of the fractions below.

5/9 , 5/12 , 23/48, 11/24, 3/7 .

This question is taken from KAPLAN, but i am looking forward for the explanation and strategy.

Regards
MAK

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Source: Beat The GMAT — Problem Solving | Thu May 20, 2010 6:31 am

liferocks: Legendary Member; Posts: 576; Joined: Sat Mar 13, 2010 8:31 pmFollowed by: 1 member; Thanked: 97 times

by liferocks » Thu May 20, 2010 7:00 am

we have to make the denominator same to compare the values
here denominator will be D=48*3*7..or in other words LCM of 9,12,48,24,7

so the fractions are in the order in the question
560/D,420/D, 503/D,482/D,432/D

so positive difference between the largest and the smallest of the fractions is 560-420/D=140/48*3*7=5/36

"If you don't know where you are going, any road will get you there."
Lewis Carroll

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asamaverick: Master | Next Rank: 500 Posts; Posts: 106; Joined: Mon Sep 28, 2009 9:29 amFollowed by: 2 members; Location: Boston, MA; Thanked: 26 times; GMAT Score:700

by asamaverick » Thu May 20, 2010 7:23 am

If we do not want to do the long multiplications, there is another approach.

Here is how I look at the nos:
5/9 - Slightly greater than 0.5 (in fact the correct value is 0.5555, not required to calculate this though)
5/12 - Slightly less than 0.5
23/48 - Slightly less than 0.5
11/24 - Slightly less than 0.5
3/7 - Slightly less than 0.5

It is obvious that 5/9 is the largest of the given set.

To identify the smallest, convert the remaining 4 to a form so that the denominators are same (or as close as possible).
5/12 = 20/48 (Multiplying by 4)
23/48 (No need to convert this)
11/24 = 22/48
3/7 = 21/49 (Multiplying by 7)

So the smallest is either 20/48 or 21/49

As a general rule for every x, y (both x & y positive integers & x < y), it follows that x/y < (x+1)/(y+1). Using this, we can deduce that 20/48 is the smaller of the above two and hence the smallest of the lot.

So the positive difference (between largest and smallest) = 5/9 - 5/12 = 20/36 - 15/36 = 5/36.

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kstv: Legendary Member; Posts: 610; Joined: Fri Jan 15, 2010 12:33 amFollowed by: 2 members; Thanked: 47 times

by kstv » Thu May 20, 2010 8:49 am

5/9 , 5/12 , 23/48, 11/24, 3/7
compare 5/12, 23/48 & 11/24
or 20/48, 23/48 & 22/48 making the denominator 48
5/9 & 5/12 , numerator is the same
5/12 < 5/9, so the larger denominator results in a smaller no
5/12 & 3/7 cross multiply 5*7 < 3*12
5/12 is the smallest
5/9 is the only no > 1/2 in all others the numerator is < 1/2 of the denominator

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