Faster way (can be done mentally)

1/2 = 3/6

The numerators are all the same now so compare the denominators.

The closer to the denominator is to 6, the closer the fraction will be to 3/6 since each part will be closer in size.

For example, 1/5 is closer to 1/4 than to 1/2 as a fifth is closer in size to a fourth/quarter than to a half.

So we can already eliminate 3/4 and 3/8 from the start.

Now compare 3/5 and 3/7. Which is closer to 3/6?


The answer is 3/7. But why, you may ask?


Logic :

First consider dividing 1 into 2 parts. Then into 3 parts. Then 4. And so on.

1 ÷ 2 = ½ or 0.5
1 ÷ 3 = ⅓ or 0.333...
1 ÷ 4 = ¼ or 0.25
1 ÷ 5 = 1/5 or 0.2
1 ÷ 6 = 1/6 or 0.1666...
1 ÷ 7 = 0.142857142857...


And we can see, the more parts we divide into, the smaller each part becomes.

But also realise that the difference between each part gets smaller and smaller.

For example,

½ - ⅓ = 3/6 - 2/6 = 1/6
Or
0.5 - 0.333... = 0.1666...

1/3 - 1/4 = 4/12 - 3/12 = 1/12
Or
0.333... - 0.25 = 0.08333...

1/4 - 1/5 = 5/20 - 4/20 = 1/20
Or
0.25 - 0.2 = 0.05



Something like 1/999 is much closer to 1/1000 than 1/6 to 1/7.

Likewise, 1/6 and 1/7 will be closer than 1/5 and 1/6.

And this also means 3/6 and 3/7 are closer than 3/5 and 3/6.


Once you remember this logic, then when a similar question appears where the numerators can be made the same,

just choose the option with the closest denominator, and the denominator is bigger than that of the reference fraction.