IMO the easiest way to solve this is to just convert all of the numbers into decimals. I was already familiar with 2/3 (everyone should know this one!), 5/8 and 4/7. Took 10s to convert the other two. From there it was pretty easy to spot the greatest and smallest number. Then it's just a matter of subtracting the two fractions. All of this can be done within a minute.

If someone does know, could anybody please elaborate for those.

it is a shortcut

\(\frac{2}{3}\) = 0.66 = 66%.and so on...........

For instance, \(\frac{1}{4}\) we do know that is 0.25 or 25% (if the problem is talking about percentage).

Hope this helps

There are various ways to compare two values..
Here, the first step would be to find some similarity in numerators or denominators.

(I) get them into same numerator or same denominator
There are three numbers with multiple of 2, and 8 is the maximum, so get all in terms of 8..
\(\frac{2}{3}=\frac{2*4}{3*4}; \frac{8}{11}; \frac{4}{7}=\frac{4*2}{2*7}\)...
when numerators are same, bigger the denominator, smaller the number..
So, \(14>12>11..... \frac{8}{11}>\frac{2}{3}=\frac{2*4}{3*4}>\frac{4}{7}=\frac{4*2}{2*7}\)..

Let me check 5/8 with 4/7 and 8/11 by second way..
(II) Get the fractions in form of \(\frac{a+x}{b+x}\)
all three fractions have a difference of 3 in numerator and denominator that is d-n=3..
so let me write all of them in same terms.. \(\frac{4}{7};\frac{4+1}{7+1};\frac{4+4}{7+4}\)....
In these three if the gap is same bigger the number, bigger the fraction so \(\frac{4}{7}<\frac{4+1}{7+1}<\frac{4+4}{7+4}\)

Now you have largest as 8/11 and smallest as 4/7 and we have to compare 9/13...
when you compare 4/7, you can convert it into 4*2/7*2=8/14 ... now 8/14 has a bigger numerator and a smaller denominator than 9/13, so 8/14 is smaller..
(III) Cross-multiply - You can cross-multiply too to find the answer .. 8/11 vs 9/13 .. 8*13 vs 9*11 = 104 vs 99 .. since 104 is bigger, 8/11 is bigger as 104 represents the numerator 8.

so range = \(\frac{8}{11} - \frac{4}{7}=\frac{56-44}{77}=\frac{12}{77}\)

C

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