Comparing fractions

Mukta and Shabnam ask which group gets more chikki each: 1 chikki divided between 2 children, or 5 chikkis divided among 8 children? They must compare 1/2 and 5/8. Since 1/2 = 4/8, and 4/8 < 5/8, the second group gets more. When the denominators are the same, the fractional units are the same, so just compare the numerators: 1/5 < 2/5 and 3/7 < 4/7.

When the numerators are the same, the smaller denominator gives the bigger fraction: 3 slices of a pizza cut into 5 is more than 3 slices of one cut into 8, so 3/5 > 3/8. Next: 1 chikki between 2 children, or 4 chikkis among 7 children? Now they compare 1/2 and 4/7. Mukta writes 1/2 = 4/8. If the same 4 chikkis are shared by more children, each share is less, so 4/7 > 4/8. And 4/8 = 1/2, so 4/7 > 1/2: the group of 7 gets more.

When neither the numerators nor the denominators match, first make the fractional units the same. Shabnam and Mukta compare 3 glasses of sugarcane juice among 4 children with 7 glasses among 10. Using the product of the denominators, 4 × 10 = 40: 3/4 = 30/40 and 7/10 = 28/40. Since 30/40 > 28/40, 3/4 > 7/10. (15/20 and 14/20 would also work.)

Any common multiple of the denominators works. For 4/5 and 7/9, use 45: 4/5 = 36/45 and 7/9 = 35/45, so 4/5 > 7/9. So there are two steps:

• Change the fractions to equivalent fractions with the same denominator.
• Compare the numerators.

The same steps put a list of fractions in ascending order (smallest first) or descending order (biggest first).

A shortcut you may see is cross-multiplying: for 2/5 and 1/4, compare 2 × 4 = 8 with 1 × 5 = 5. It is the same two steps in disguise: 2/5 = 8/20 and 1/4 = 5/20, without writing the 20. So 2/5 > 1/4.