Thank you very much

Notice all have numerators of 5.

Change 1/2 to 5/10 first.


The denominator represents dividing by a certain number of parts.
The numerator represents what is being divided.

The closer the number of parts we divide 5 by, the closer the size of each part will be.


So from the options , we know that 5/9 and 5/11 can be shortlisted since 9 and 11 are closest to 10 (only 1 away)

Next, how to know if 5/9 or 5/11 is closer to 5/10?

Compare 5/9, 5/10 and 5/11 , 5/10 separately.

Subtract the smaller fraction from the bigger one.


① 5/9 - 5/10

= 50/90 - 45/90
= 5/90
= 1/18

5/10 - 5/11

= 55/110 - 50/110
= 5/110
= 1/22

The difference of 1/22 is smaller than 1/18. So 5/11 is closer to 5/10 than 5/9 is to 5/10.

Another thing to realise :


Since the more parts we divide something into, the smaller the size of each part becomes,

the difference between the part size and the subsequent part size (where the total is divided into 1 more part than before) is getting smaller and smaller.


Eg.


1/2 - 1/3 = 3/6 - 2/6 = 1/6
1/3 - 1/4 = 4/12 - 3/12 = 1/12
1/4 - 1/5 = 5/20 - 4/20 = 1/20
1/5 - 1/6 = 6/30 - 5/30 = 1/30

We can see the difference decreasing.

This can be summarised for the general case :

m/n - m/(n+1)
= m(n+1) / (n)(n+1) - mn / (n)(n+1)
= (mn + m - mn) / (n)(n+1)
= m/n(n+1)

Where m is the numerator and n is the number of parts we choose to divide by for the bigger fraction.

As n increases, the difference decreases since the denominator is now bigger .


Once this is understood, we will automatically pick 5/11 as the answer when choosing between 5/9 and 5/11 without even having to calculate.