Comparing fractions

Two fractions sit in front of you: $\frac{3}{5}$ and $\frac{4}{7}$. Which is bigger? You cannot just glance , bigger numbers don't always mean a bigger fraction. There are three reliable techniques.

Concept

Method 1: Same denominator

If two fractions have the same denominator, the one with the bigger numerator is bigger.

• $\frac{3}{8}$ vs $\frac{5}{8}$: same eighths, so $\frac{5}{8} > \frac{3}{8}$ (five pieces beats three pieces of equal size).

Method 2: Same numerator

If two fractions have the same numerator but different denominators, the one with the smaller denominator is bigger. Why? With a smaller denominator each piece is larger, so the same number of bigger pieces is more.

• $\frac{3}{5}$ vs $\frac{3}{8}$: same number of pieces but fifths are bigger than eighths. So $\frac{3}{5} > \frac{3}{8}$.

Method 3: Different numerators and denominators , find a common denominator

This is the universal method. To compare $\frac{a}{b}$ and $\frac{c}{d}$:

1. Find a common multiple of $b$ and $d$ (the simplest choice is $b \times d$; better is the LCM of $b$ and $d$).
2. Convert both fractions to that common denominator.
3. Compare the new numerators.

Example. Compare $\frac{3}{5}$ and $\frac{4}{7}$.

• Common denominator $= 35$.
• $\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$.
• $\frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35}$.
• $21 > 20$, so $\frac{3}{5} > \frac{4}{7}$.

Shortcut: cross-multiplication

To compare $\frac{a}{b}$ and $\frac{c}{d}$, compute $a \times d$ vs $b \times c$. The bigger product indicates the bigger fraction.

For $\frac{3}{5}$ and $\frac{4}{7}$:

• $3 \times 7 = 21$ and $5 \times 4 = 20$. $21 > 20$. So $\frac{3}{5} > \frac{4}{7}$.

This is just method $3$ written with less ink , but it's quick.

Be careful with mixed comparisons

$\frac{7}{6}$ is greater than $1$ (it has more parts than the denominator). $\frac{5}{6}$ is less than $1$. So $\frac{7}{6} > \frac{5}{6}$ obviously , but always sanity-check whether a fraction is greater or less than $1$.

Worked examples

Example 1. Compare $\frac{2}{3}$ and $\frac{3}{4}$.

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

Example 2. Compare $\frac{5}{6}$ and $\frac{7}{9}$.

• Cross-multiply: $5 \times 9 = 45$ and $6 \times 7 = 42$. $45 > 42$.
• So $\frac{5}{6} > \frac{7}{9}$.

Example 3. Arrange in increasing order: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{4}$.

• Common denominator $= 12$.
• $\frac{1}{2} = \frac{6}{12}$, $\frac{2}{3} = \frac{8}{12}$, $\frac{3}{4} = \frac{9}{12}$, $\frac{1}{4} = \frac{3}{12}$.
• Order: $\frac{3}{12} < \frac{6}{12} < \frac{8}{12} < \frac{9}{12}$.
• So $\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}$.

Example 4. Compare $\frac{5}{12}$ and $\frac{1}{2}$.

• $\frac{1}{2} = \frac{6}{12}$.
• $6 > 5$, so $\frac{1}{2} > \frac{5}{12}$.

Example 5. Without computing, decide: which is larger, $\frac{99}{100}$ or $\frac{100}{101}$?

• Both are very close to $1$, but each is less than $1$ by $\frac{1}{100}$ and $\frac{1}{101}$ respectively.
• $\frac{1}{100} > \frac{1}{101}$, so $\frac{99}{100}$ is farther from $1$. Hence $\frac{99}{100} < \frac{100}{101}$.

Try it yourself

1. Compare $\frac{3}{5}$ and $\frac{4}{9}$.
2. Compare $\frac{7}{10}$ and $\frac{2}{3}$.
3. Arrange in increasing order: $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}$.
4. Arrange in decreasing order: $\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{8}$.
5. Compare $\frac{11}{12}$ and $\frac{12}{13}$.
6. Which is larger: $\frac{7}{8}$ or $\frac{15}{16}$?
7. Without calculating, which is greater: $\frac{1}{1000}$ or $\frac{2}{2000}$?
8. Investigate: between $\frac{1}{2}$ and $\frac{1}{3}$ is there any fraction? Find one.

Activity

Race the fractions. With a friend, write any $5$ fractions on small cards (e.g., $\frac{3}{5}$, $\frac{4}{7}$, $\frac{2}{3}$, $\frac{5}{8}$, $\frac{7}{10}$). Shuffle them and try to arrange them in increasing order without doing any pencil work , just by mental cross-multiplying or by eye. Then verify by converting to a common denominator. How accurate was your eye?