Comparing Proper Fractions to Twelfths (A) - Free Printable

To compare fractions, we need to determine which one is larger, smaller, or if they are equal. Here is the step-by-step logic for solving these problems:

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

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 bigger (because the pieces are larger).
* *Example:* $1/2$ vs $1/8$. Cutting a pizza into 2 pieces gives bigger slices than cutting it into 8. So, $1/2 > 1/8$.

3. Different Numerators and Denominators: We can use "cross-multiplication" to check. Multiply the top of the first fraction by the bottom of the second, and vice versa. Compare those two results.
* *Example:* Compare $2/3$ and $3/5$.
* $2 \times 5 = 10$
* $3 \times 3 = 9$
* Since $10 > 9$, the first fraction ($2/3$) is larger. So, $2/3 > 3/5$.

4. Simplifying First: Sometimes, you can simplify a fraction to make it easier to compare.
* *Example:* $6/10$ simplifies to $3/5$. Then compare $3/5$ with the other fraction.

Here are the solutions for each problem on the sheet, verified using these methods:

Row 1:
1. $1/2$ vs $4/5$: Cross multiply ($1\times5=5$, $2\times4=8$). $5 < 8$, so <
2. $1/2$ vs $1/2$: They are identical. =
3. $1/4$ vs $7/8$: Convert $1/4$ to $2/8$. $2/8 < 7/8$. <
4. $1/10$ vs $1/8$: Same top, smaller bottom is bigger. $1/8$ is bigger. <
5. $4/9$ vs $1/6$: Cross multiply ($4\times6=24$, $9\times1=9$). $24 > 9$. >

Row 2:
6. $1/5$ vs $1/9$: Same top, smaller bottom is bigger. $1/5$ is bigger. >
7. $5/7$ vs $2/11$: Cross multiply ($5\times11=55$, $7\times2=14$). $55 > 14$. >
8. $1/4$ vs $2/3$: Cross multiply ($1\times3=3$, $4\times2=8$). $3 < 8$. <
9. $1/2$ vs $7/9$: Cross multiply ($1\times9=9$, $2\times7=14$). $9 < 14$. <
10. $10/12$ vs $1/9$: Simplify $10/12$ to $5/6$. Compare $5/6$ vs $1/9$. Cross multiply ($5\times9=45$, $6\times1=6$). $45 > 6$. >

Row 3:
11. $3/6$ vs $4/8$: Simplify both. $3/6 = 1/2$. $4/8 = 1/2$. They are equal. =
12. $6/10$ vs $2/5$: Simplify $6/10$ to $3/5$. Compare $3/5$ vs $2/5$. $3 > 2$. >
13. $4/5$ vs $5/8$: Cross multiply ($4\times8=32$, $5\times5=25$). $32 > 25$. >
14. $1/2$ vs $3/5$: Cross multiply ($1\times5=5$, $2\times3=6$). $5 < 6$. <
15. $1/9$ vs $3/4$: Cross multiply ($1\times4=4$, $9\times3=27$). $4 < 27$. <

Row 4:
16. $5/6$ vs $8/9$: Cross multiply ($5\times9=45$, $6\times8=48$). $45 < 48$. <
17. $1/2$ vs $8/9$: Cross multiply ($1\times9=9$, $2\times8=16$). $9 < 16$. <
18. $2/11$ vs $3/5$: Cross multiply ($2\times5=10$, $11\times3=33$). $10 < 33$. <
19. $4/10$ vs $4/5$: Same top, smaller bottom is bigger. $4/5$ is bigger. <
20. $3/10$ vs $2/3$: Cross multiply ($3\times3=9$, $10\times2=20$). $9 < 20$. <

Row 5:
21. $2/3$ vs $1/8$: Cross multiply ($2\times8=16$, $3\times1=3$). $16 > 3$. >
22. $2/3$ vs $7/10$: Cross multiply ($2\times10=20$, $3\times7=21$). $20 < 21$. <
23. $6/12$ vs $5/6$: Simplify $6/12$ to $1/2$. Compare $1/2$ vs $5/6$. Convert $1/2$ to $3/6$. $3/6 < 5/6$. <
24. $2/6$ vs $3/5$: Simplify $2/6$ to $1/3$. Cross multiply ($1\times5=5$, $3\times3=9$). $5 < 9$. <
25. $1/2$ vs $2/7$: Cross multiply ($1\times7=7$, $2\times2=4$). $7 > 4$. >

Row 6:
26. $2/5$ vs $4/8$: Simplify $4/8$ to $1/2$. Compare $2/5$ vs $1/2$. Cross multiply ($2\times2=4$, $5\times1=5$). $4 < 5$. <
27. $5/6$ vs $9/11$: Cross multiply ($5\times11=55$, $6\times9=54$). $55 > 54$. >
28. $2/8$ vs $5/10$: Simplify $2/8$ to $1/4$ and $5/10$ to $1/2$. $1/4 < 1/2$. <
29. $2/3$ vs $3/5$: Cross multiply ($2\times5=10$, $3\times3=9$). $10 > 9$. >
30. $4/8$ vs $1/5$: Simplify $4/8$ to $1/2$. Compare $1/2$ vs $1/5$. Same top, smaller bottom is bigger. >

Row 7:
31. $1/6$ vs $7/10$: Cross multiply ($1\times10=10$, $6\times7=42$). $10 < 42$. <
32. $3/4$ vs $1/8$: Convert $3/4$ to $6/8$. $6/8 > 1/8$. >
33. $8/9$ vs $3/4$: Cross multiply ($8\times4=32$, $9\times3=27$). $32 > 27$. >
34. $2/4$ vs $1/3$: Simplify $2/4$ to $1/2$. Compare $1/2$ vs $1/3$. Same top, smaller bottom is bigger. >
35. $3/7$ vs $2/3$: Cross multiply ($3\times3=9$, $7\times2=14$). $9 < 14$. <

Row 8:
36. $1/8$ vs $1/6$: Same top, smaller bottom is bigger. $1/6$ is bigger. <
37. $4/8$ vs $6/7$: Simplify $4/8$ to $1/2$. Compare $1/2$ vs $6/7$. Cross multiply ($1\times7=7$, $2\times6=12$). $7 < 12$. <
38. $4/11$ vs $5/11$: Same bottom, larger top is bigger. $5 > 4$. <
39. $1/9$ vs $1/6$: Same top, smaller bottom is bigger. $1/6$ is bigger. <
40. $1/6$ vs $2/4$: Simplify $2/4$ to $1/2$. Compare $1/6$ vs $1/2$. Same top, smaller bottom is bigger. <

Row 9:
41. $7/12$ vs $7/9$: Same top, smaller bottom is bigger. $7/9$ is bigger. <
42. $1/2$ vs $3/7$: Cross multiply ($1\times7=7$, $2\times3=6$). $7 > 6$. >
43. $3/10$ vs $1/2$: Convert $1/2$ to $5/10$. $3/10 < 5/10$. <
44. $1/5$ vs $1/3$: Same top, smaller bottom is bigger. $1/3$ is bigger. <
45. $7/11$ vs $3/6$: Simplify $3/6$ to $1/2$. Compare $7/11$ vs $1/2$. Half of 11 is 5.5. 7 is more than half. So $7/11 > 1/2$. >

Row 10:
46. $2/10$ vs $2/8$: Same top, smaller bottom is bigger. $2/8$ is bigger. <
47. $7/9$ vs $4/6$: Simplify $4/6$ to $2/3$. Convert $2/3$ to ninths ($6/9$). Compare $7/9$ vs $6/9$. $7 > 6$. >
48. $5/9$ vs $1/5$: Cross multiply ($5\times5=25$, $9\times1=9$). $25 > 9$. >
49. $2/9$ vs $1/6$: Find common denominator (18). $2/9 = 4/18$. $1/6 = 3/18$. $4 > 3$. >
50. $8/9$ vs $5/11$: Cross multiply ($8\times11=88$, $9\times5=45$). $88 > 45$. >

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