Comparing fractions

Imagine two friends sharing pizzas. Aman eats $\frac{2}{3}$ of his and Bina eats $\frac{3}{4}$ of hers. Who ate more? To answer, we compare the fractions.

What is this?

When two fractions have the same denominator, comparing is easy. The one with the bigger numerator is bigger.

For example, $\frac{3}{5}$ and $\frac{2}{5}$. Both are out of 5 equal parts. 3 parts is more than 2 parts, so $\frac{3}{5} > \frac{2}{5}$.

When two fractions have the same numerator but different denominators, the trick flips. If you cut a roti into 2 pieces, each piece is bigger than if you cut it into 4 pieces. So $\frac{1}{2} > \frac{1}{4}$ , the smaller the denominator, the bigger the piece.

When both numerator and denominator are different, we change them to equivalent fractions with the same denominator and then compare. The common denominator can be any number both denominators divide into. The easiest one is their product, or the LCM.

For $\frac{2}{3}$ and $\frac{3}{4}$:

• $\frac{2}{3} = \frac{8}{12}$
• $\frac{3}{4} = \frac{9}{12}$
• 9 > 8, so $\frac{3}{4} > \frac{2}{3}$.

So Bina ate more.

Worked examples

Example 1. Which is bigger: $\frac{4}{7}$ or $\frac{5}{7}$?

• Same denominator → compare numerators.
• 5 > 4, so $\frac{5}{7} > \frac{4}{7}$.

Example 2. Which is bigger: $\frac{1}{3}$ or $\frac{1}{5}$?

• Same numerator → smaller denominator wins.
• 3 < 5, so $\frac{1}{3} > \frac{1}{5}$.

Example 3. Which is bigger: $\frac{3}{4}$ or $\frac{5}{8}$?

• Common denominator = 8.
• $\frac{3}{4} = \frac{6}{8}$.
• 6 > 5, so $\frac{3}{4} > \frac{5}{8}$.

Try it

1. Compare: $\frac{2}{9}$ and $\frac{5}{9}$.
2. Which is bigger: $\frac{1}{2}$ or $\frac{1}{6}$?
3. Compare: $\frac{2}{3}$ and $\frac{4}{6}$.
4. Arrange in increasing order: $\frac{1}{2}, \frac{1}{4}, \frac{1}{3}$.
5. Rina ate $\frac{3}{8}$ of a cake and Sumi ate $\frac{1}{4}$. Who ate more?

Hands-on

Two-bar test. On a sheet of paper, draw two rectangles of the same length. Divide the first into 3 parts, the second into 4 parts. Shade $\frac{2}{3}$ on one and $\frac{2}{4}$ on the other. Which shaded area is bigger? Now you know how to compare with your eyes.

Sweet share. Take 12 small sweets. Give $\frac{1}{3}$ to a friend, $\frac{1}{4}$ to a sibling. Count how many each got. Use the counts to confirm which fraction is bigger.