More or less than a half

In your book, Maths Mela (Chapter 2), we compare fractions with one half. A paratha cut into 8 equal pieces with 4 of them eaten is 4/8, which is exactly half. A paratha cut into 6 pieces with 3 eaten is 3/6, also exactly half. A fraction is exactly one half when its numerator is half of its denominator: 4 is half of 8, and 3 is half of 6.

A quick check: double the numerator. If you get the denominator, the fraction is exactly 1/2. For 7/14: 7 + 7 = 14, so 7/14 = 1/2. The same works for 10/20, 8/16, 5/10 and 6/12. If the double is LESS than the denominator, the fraction is less than half: for 3/9, 3 + 3 = 6, less than 9, so 3/9 is less than half. If the double is MORE, the fraction is more than half: for 5/8, 5 + 5 = 10, more than 8.

Now we can use 1/2 to compare two fractions. In your book, one child ate 5/8 paratha and another ate 3/6 paratha. Who ate more? 3/6 is exactly half. 5/8 is more than half, because half of 8 pieces is 4 and there are 5. So 5/8 > 3/6.

The same trick helps with fractions that look hard to compare. Take 2/9 and 4/7. Half of 9 is 4 and a half, and 2 is much less, so 2/9 is less than half. Half of 7 is 3 and a half, and 4 is more, so 4/7 is more than half. So 2/9 < 4/7. This works when one fraction is below 1/2 and the other is above it (or exactly 1/2).