Math worksheet for comparing proper fractions using inequality symbols.

To compare two fractions, we need to figure out which one is bigger, smaller, or if they are equal. Here is the step-by-step logic used to solve these problems:

1. Same Denominator (Bottom Number): If the bottom numbers are the same, just look at the top numbers. The fraction with the bigger top number is larger.
* *Example:* $\frac{3}{8}$ vs $\frac{5}{8}$. Since $5 > 3$, then $\frac{5}{8} > \frac{3}{8}$.

2. Same Numerator (Top Number): If the top numbers are the same, look at the bottom numbers. The fraction with the *smaller* bottom number is actually larger (because the pieces are bigger).
* *Example:* $\frac{1}{2}$ vs $\frac{1}{8}$. Half of a pizza is much bigger than one-eighth of a pizza. So, $\frac{1}{2} > \frac{1}{8}$.

3. Different Numbers: If both numbers are different, we can use "cross-multiplication" or find a common denominator. A quick trick is cross-multiplying:
* Multiply the top of the first fraction by the bottom of the second.
* Multiply the top of the second fraction by the bottom of the first.
* Compare those two results.

Let's apply this to every problem on the sheet:

Row 1:
1. $\frac{1}{2}$ vs $\frac{4}{5}$: Cross multiply: $1\times5=5$ and $2\times4=8$. Since $5 < 8$, $\frac{1}{2} < \frac{4}{5}$.
2. $\frac{1}{2}$ vs $\frac{1}{2}$: They are identical. $\frac{1}{2} = \frac{1}{2}$.
3. $\frac{1}{4}$ vs $\frac{7}{8}$: Convert $\frac{1}{4}$ to eighths ($\frac{2}{8}$). $\frac{2}{8} < \frac{7}{8}$. $\frac{1}{4} < \frac{7}{8}$.
4. $\frac{1}{10}$ vs $\frac{1}{8}$: Same top number. Smaller bottom means bigger fraction. $\frac{1}{10} < \frac{1}{8}$.
5. $\frac{4}{9}$ vs $\frac{1}{6}$: Common denominator is 18. $\frac{8}{18}$ vs $\frac{3}{18}$. $\frac{4}{9} > \frac{1}{6}$.

Row 2:
6. $\frac{1}{5}$ vs $\frac{1}{9}$: Same top number. Smaller bottom is bigger. $\frac{1}{5} > \frac{1}{9}$.
7. $\frac{5}{7}$ vs $\frac{2}{11}$: Cross multiply: $55$ vs $14$. $\frac{5}{7} > \frac{2}{11}$.
8. $\frac{1}{4}$ vs $\frac{2}{3}$: Cross multiply: $3$ vs $8$. $\frac{1}{4} < \frac{2}{3}$.
9. $\frac{1}{2}$ vs $\frac{7}{9}$: Cross multiply: $9$ vs $14$. $\frac{1}{2} < \frac{7}{9}$.
10. $\frac{10}{12}$ vs $\frac{1}{9}$: $\frac{10}{12}$ is almost 1. $\frac{1}{9}$ is small. $\frac{10}{12} > \frac{1}{9}$.

Row 3:
11. $\frac{3}{6}$ vs $\frac{4}{8}$: Both simplify to $\frac{1}{2}$. $\frac{3}{6} = \frac{4}{8}$.
12. $\frac{6}{10}$ vs $\frac{2}{5}$: Simplify $\frac{6}{10}$ to $\frac{3}{5}$. $\frac{3}{5} > \frac{2}{5}$. $\frac{6}{10} > \frac{2}{5}$.
13. $\frac{4}{5}$ vs $\frac{5}{8}$: Cross multiply: $32$ vs $25$. $\frac{4}{5} > \frac{5}{8}$.
14. $\frac{1}{2}$ vs $\frac{3}{5}$: Cross multiply: $5$ vs $6$. $\frac{1}{2} < \frac{3}{5}$.
15. $\frac{1}{9}$ vs $\frac{3}{4}$: $\frac{1}{9}$ is tiny, $\frac{3}{4}$ is big. $\frac{1}{9} < \frac{3}{4}$.

Row 4:
16. $\frac{5}{6}$ vs $\frac{8}{9}$: Common denominator 18. $\frac{15}{18}$ vs $\frac{16}{18}$. $\frac{5}{6} < \frac{8}{9}$.
17. $\frac{1}{2}$ vs $\frac{8}{9}$: $\frac{1}{2}$ is half. $\frac{8}{9}$ is almost whole. $\frac{1}{2} < \frac{8}{9}$.
18. $\frac{2}{11}$ vs $\frac{3}{5}$: Cross multiply: $10$ vs $33$. $\frac{2}{11} < \frac{3}{5}$.
19. $\frac{4}{10}$ vs $\frac{4}{5}$: Same top number. Smaller bottom is bigger. $\frac{4}{10} < \frac{4}{5}$.
20. $\frac{3}{10}$ vs $\frac{2}{3}$: Cross multiply: $9$ vs $20$. $\frac{3}{10} < \frac{2}{3}$.

Row 5:
21. $\frac{2}{3}$ vs $\frac{1}{8}$: $\frac{2}{3}$ is big, $\frac{1}{8}$ is small. $\frac{2}{3} > \frac{1}{8}$.
22. $\frac{2}{3}$ vs $\frac{7}{10}$: Cross multiply: $20$ vs $21$. $\frac{2}{3} < \frac{7}{10}$.
23. $\frac{6}{12}$ vs $\frac{5}{6}$: $\frac{6}{12}$ is $\frac{1}{2}$. $\frac{5}{6}$ is bigger than half. $\frac{6}{12} < \frac{5}{6}$.
24. $\frac{2}{6}$ vs $\frac{3}{5}$: Simplify $\frac{2}{6}$ to $\frac{1}{3}$. Cross multiply $\frac{1}{3}$ vs $\frac{3}{5}$ ($5$ vs $9$). $\frac{2}{6} < \frac{3}{5}$.
25. $\frac{1}{2}$ vs $\frac{2}{7}$: Cross multiply: $7$ vs $4$. $\frac{1}{2} > \frac{2}{7}$.

Row 6:
26. $\frac{2}{5}$ vs $\frac{4}{8}$: $\frac{4}{8}$ is $\frac{1}{2}$. $\frac{2}{5}$ is less than half. $\frac{2}{5} < \frac{4}{8}$.
27. $\frac{5}{6}$ vs $\frac{9}{11}$: Cross multiply: $55$ vs $54$. $\frac{5}{6} > \frac{9}{11}$.
28. $\frac{2}{8}$ vs $\frac{5}{10}$: $\frac{2}{8}=\frac{1}{4}$, $\frac{5}{10}=\frac{1}{2}$. $\frac{2}{8} < \frac{5}{10}$.
29. $\frac{2}{3}$ vs $\frac{3}{5}$: Cross multiply: $10$ vs $9$. $\frac{2}{3} > \frac{3}{5}$.
30. $\frac{4}{8}$ vs $\frac{1}{5}$: $\frac{4}{8}$ is $\frac{1}{2}$. $\frac{4}{8} > \frac{1}{5}$.

Row 7:
31. $\frac{1}{6}$ vs $\frac{7}{10}$: $\frac{1}{6}$ is small. $\frac{1}{6} < \frac{7}{10}$.
32. $\frac{3}{4}$ vs $\frac{1}{8}$: $\frac{3}{4}$ is $\frac{6}{8}$. $\frac{3}{4} > \frac{1}{8}$.
33. $\frac{8}{9}$ vs $\frac{3}{4}$: Cross multiply: $32$ vs $27$. $\frac{8}{9} > \frac{3}{4}$.
34. $\frac{2}{4}$ vs $\frac{1}{3}$: $\frac{2}{4}$ is $\frac{1}{2}$. $\frac{2}{4} > \frac{1}{3}$.
35. $\frac{3}{7}$ vs $\frac{2}{3}$: Cross multiply: $9$ vs $14$. $\frac{3}{7} < \frac{2}{3}$.

Row 8:
36. $\frac{1}{8}$ vs $\frac{1}{6}$: Same top. Smaller bottom is bigger. $\frac{1}{8} < \frac{1}{6}$.
37. $\frac{4}{8}$ vs $\frac{6}{7}$: $\frac{4}{8}$ is half. $\frac{6}{7}$ is almost whole. $\frac{4}{8} < \frac{6}{7}$.
38. $\frac{4}{11}$ vs $\frac{5}{11}$: Same bottom. Bigger top is bigger. $\frac{4}{11} < \frac{5}{11}$.
39. $\frac{1}{9}$ vs $\frac{1}{6}$: Same top. Smaller bottom is bigger. $\frac{1}{9} < \frac{1}{6}$.
40. $\frac{1}{6}$ vs $\frac{2}{4}$: $\frac{2}{4}$ is $\frac{1}{2}$. $\frac{1}{6} < \frac{2}{4}$.

Row 9:
41. $\frac{7}{12}$ vs $\frac{7}{9}$: Same top. Smaller bottom is bigger. $\frac{7}{12} < \frac{7}{9}$.
42. $\frac{1}{2}$ vs $\frac{3}{7}$: Cross multiply: $7$ vs $6$. $\frac{1}{2} > \frac{3}{7}$.
43. $\frac{3}{10}$ vs $\frac{1}{2}$: $\frac{1}{2}$ is $\frac{5}{10}$. $\frac{3}{10} < \frac{1}{2}$.
44. $\frac{1}{5}$ vs $\frac{1}{3}$: Same top. Smaller bottom is bigger. $\frac{1}{5} < \frac{1}{3}$.
45. $\frac{7}{11}$ vs $\frac{3}{6}$: $\frac{3}{6}$ is $\frac{1}{2}$. $\frac{7}{11}$ is more than half ($\frac{5.5}{11}$). $\frac{7}{11} > \frac{3}{6}$.

Row 10:
46. $\frac{2}{10}$ vs $\frac{2}{8}$: Same top. Smaller bottom is bigger. $\frac{2}{10} < \frac{2}{8}$.
47. $\frac{7}{9}$ vs $\frac{4}{6}$: $\frac{4}{6}$ is $\frac{2}{3}$ which is $\frac{6}{9}$. $\frac{7}{9} > \frac{6}{9}$. $\frac{7}{9} > \frac{4}{6}$.
48. $\frac{5}{9}$ vs $\frac{1}{5}$: Cross multiply: $25$ vs $9$. $\frac{5}{9} > \frac{1}{5}$.
49. $\frac{2}{9}$ vs $\frac{1}{6}$: Common denominator 18. $\frac{4}{18}$ vs $\frac{3}{18}$. $\frac{2}{9} > \frac{1}{6}$.
50. $\frac{8}{9}$ vs $\frac{5}{11}$: $\frac{8}{9}$ is almost 1. $\frac{5}{11}$ is less than half. $\frac{8}{9} > \frac{5}{11}$.

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