Training Fractions Comparing and Ordering Fractions
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Comparing and Ordering Fractions

18 min Fractions

Comparing and Ordering Fractions

Comparing and ordering fractions is essential whenever you need to decide which of two quantities is larger or arrange a set of values from least to greatest. Unlike whole numbers, fractions cannot be compared by simply looking at the digits — three-fourths is larger than five-eighths even though 5 is bigger than 3.

The most reliable method is to convert fractions to a common denominator and then compare the numerators. Alternatively, you can convert fractions to decimals or use cross-multiplication as a shortcut.

This lesson gives you multiple strategies for comparing fractions so you can choose the one that fits each situation best.

Method 1: Common Denominator

Rewrite with the same denominator, then compare numerators.

Example 1

Compare $\dfrac{3}{5}$ and $\dfrac{2}{3}$.

LCD = 15. $\;\dfrac{9}{15}$ vs $\dfrac{10}{15}$. Since $9 < 10$: $\dfrac{3}{5} < \dfrac{2}{3}$.

Method 2: Cross-Multiplication

Compare $\dfrac{a}{b}$ and $\dfrac{c}{d}$: if $a \times d < c \times b$ then $\dfrac{a}{b} < \dfrac{c}{d}$.

Example 2

Compare $\dfrac{5}{8}$ and $\dfrac{7}{12}$.

$5 \times 12 = 60$ vs $7 \times 8 = 56$. $60 > 56$, so $\dfrac{5}{8} > \dfrac{7}{12}$.

Method 3: Convert to Decimals

Example 3

Compare $\dfrac{4}{7}$ and $\dfrac{5}{9}$.

$\dfrac{4}{7} \approx 0.571$ and $\dfrac{5}{9} \approx 0.556$. So $\dfrac{4}{7} > \dfrac{5}{9}$.

Ordering Multiple Fractions

Example 4

Order least to greatest: $\dfrac{1}{2}$, $\dfrac{3}{8}$, $\dfrac{5}{12}$.

LCD = 24: $\dfrac{12}{24}$, $\dfrac{9}{24}$, $\dfrac{10}{24}$.

$$\frac{3}{8} < \frac{5}{12} < \frac{1}{2}$$

Practice Problems

1. Compare: $\dfrac{4}{5}$ ____ $\dfrac{7}{10}$
2. Compare: $\dfrac{5}{6}$ ____ $\dfrac{7}{8}$
3. Order least to greatest: $\dfrac{1}{3}$, $\dfrac{2}{5}$, $\dfrac{3}{10}$
4. Compare: $\dfrac{11}{15}$ ____ $\dfrac{3}{4}$
5. Order greatest to least: $\dfrac{5}{8}$, $\dfrac{2}{3}$, $\dfrac{7}{12}$
6. Is $\dfrac{9}{13}$ greater than $\dfrac{2}{3}$?
7. Compare: $2\dfrac{3}{4}$ ____ $2\dfrac{5}{7}$
8. Order: $\dfrac{3}{4}$, $\dfrac{4}{5}$, $\dfrac{5}{6}$, $\dfrac{6}{7}$
9. Which is closest to 1: $\dfrac{5}{6}$ or $\dfrac{7}{8}$?
10. Compare: $\dfrac{13}{20}$ ____ $\dfrac{7}{11}$
Show Answer Key

1. $>$ (since $\dfrac{8}{10} > \dfrac{7}{10}$)

2. $<$ (LCD 24: $\dfrac{20}{24} < \dfrac{21}{24}$)

3. $\dfrac{3}{10} < \dfrac{1}{3} < \dfrac{2}{5}$

4. $<$ (LCD 60: $\dfrac{44}{60} < \dfrac{45}{60}$)

5. $\dfrac{2}{3} > \dfrac{5}{8} > \dfrac{7}{12}$

6. Yes — cross: $27 > 26$

7. $>$ (LCD 28: $\dfrac{21}{28} > \dfrac{20}{28}$)

8. $\dfrac{3}{4} < \dfrac{4}{5} < \dfrac{5}{6} < \dfrac{6}{7}$

9. $\dfrac{7}{8}$ (distance $\dfrac{1}{8} < \dfrac{1}{6}$)

10. $>$ (cross: $143 > 140$)

Multiplying and Dividing Fractions Word Problems with Fractions