Difference between largest and smallest values

[mak_mak_mak]

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

[liferocks]

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

[asamaverick]

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.

[kstv]

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