Compare fractions worksheet with cross-multiplication examples and exercises for students.

Problem Overview:

The task involves comparing fractions using the method of cross-multiplication. The goal is to determine whether one fraction is greater than, less than, or equal to another by multiplying the numerator of one fraction by the denominator of the other and comparing the results.

Steps to Solve:

1. Cross-Multiply: For two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \):
- Multiply the numerator of the first fraction by the denominator of the second: \( a \times d \).
- Multiply the numerator of the second fraction by the denominator of the first: \( c \times b \).
- Compare the two products:
- If \( a \times d > c \times b \), then \( \frac{a}{b} > \frac{c}{d} \).
- If \( a \times d < c \times b \), then \( \frac{a}{b} < \frac{c}{d} \).
- If \( a \times d = c \times b \), then \( \frac{a}{b} = \frac{c}{d} \).

2. Apply the Method: Use the cross-multiplication technique to compare each pair of fractions given in the worksheet.

---

Solution for Each Pair:

#### 1. \( \frac{3}{4} \) vs. \( \frac{5}{8} \)
- Cross-multiply:
- \( 3 \times 8 = 24 \)
- \( 5 \times 4 = 20 \)
- Compare: \( 24 > 20 \)
- Conclusion: \( \frac{3}{4} > \frac{5}{8} \)

#### 2. \( \frac{2}{9} \) vs. \( \frac{3}{6} \)
- Cross-multiply:
- \( 2 \times 6 = 12 \)
- \( 3 \times 9 = 27 \)
- Compare: \( 12 < 27 \)
- Conclusion: \( \frac{2}{9} < \frac{3}{6} \)

#### 3. \( \frac{3}{5} \) vs. \( \frac{3}{5} \)
- Cross-multiply:
- \( 3 \times 5 = 15 \)
- \( 3 \times 5 = 15 \)
- Compare: \( 15 = 15 \)
- Conclusion: \( \frac{3}{5} = \frac{3}{5} \)

#### 4. \( \frac{2}{6} \) vs. \( \frac{2}{4} \)
- Cross-multiply:
- \( 2 \times 4 = 8 \)
- \( 2 \times 6 = 12 \)
- Compare: \( 8 < 12 \)
- Conclusion: \( \frac{2}{6} < \frac{2}{4} \)

#### 5. \( \frac{3}{8} \) vs. \( \frac{4}{6} \)
- Cross-multiply:
- \( 3 \times 6 = 18 \)
- \( 4 \times 8 = 32 \)
- Compare: \( 18 < 32 \)
- Conclusion: \( \frac{3}{8} < \frac{4}{6} \)

#### 6. \( \frac{3}{8} \) vs. \( \frac{4}{12} \)
- Cross-multiply:
- \( 3 \times 12 = 36 \)
- \( 4 \times 8 = 32 \)
- Compare: \( 36 > 32 \)
- Conclusion: \( \frac{3}{8} > \frac{4}{12} \)

#### 7. \( \frac{6}{9} \) vs. \( \frac{2}{3} \)
- Cross-multiply:
- \( 6 \times 3 = 18 \)
- \( 2 \times 9 = 18 \)
- Compare: \( 18 = 18 \)
- Conclusion: \( \frac{6}{9} = \frac{2}{3} \)

#### 8. \( \frac{1}{5} \) vs. \( \frac{1}{10} \)
- Cross-multiply:
- \( 1 \times 10 = 10 \)
- \( 1 \times 5 = 5 \)
- Compare: \( 10 > 5 \)
- Conclusion: \( \frac{1}{5} > \frac{1}{10} \)

#### 9. \( \frac{5}{9} \) vs. \( \frac{3}{5} \)
- Cross-multiply:
- \( 5 \times 5 = 25 \)
- \( 3 \times 9 = 27 \)
- Compare: \( 25 < 27 \)
- Conclusion: \( \frac{5}{9} < \frac{3}{5} \)

#### 10. \( \frac{2}{6} \) vs. \( \frac{4}{12} \)
- Cross-multiply:
- \( 2 \times 12 = 24 \)
- \( 4 \times 6 = 24 \)
- Compare: \( 24 = 24 \)
- Conclusion: \( \frac{2}{6} = \frac{4}{12} \)

#### 11. \( \frac{3}{4} \) vs. \( \frac{3}{6} \)
- Cross-multiply:
- \( 3 \times 6 = 18 \)
- \( 3 \times 4 = 12 \)
- Compare: \( 18 > 12 \)
- Conclusion: \( \frac{3}{4} > \frac{3}{6} \)

#### 12. \( \frac{1}{4} \) vs. \( \frac{8}{12} \)
- Cross-multiply:
- \( 1 \times 12 = 12 \)
- \( 8 \times 4 = 32 \)
- Compare: \( 12 < 32 \)
- Conclusion: \( \frac{1}{4} < \frac{8}{12} \)

---

Final Answers:

1. \( \frac{3}{4} > \frac{5}{8} \)
2. \( \frac{2}{9} < \frac{3}{6} \)
3. \( \frac{3}{5} = \frac{3}{5} \)
4. \( \frac{2}{6} < \frac{2}{4} \)
5. \( \frac{3}{8} < \frac{4}{6} \)
6. \( \frac{3}{8} > \frac{4}{12} \)
7. \( \frac{6}{9} = \frac{2}{3} \)
8. \( \frac{1}{5} > \frac{1}{10} \)
9. \( \frac{5}{9} < \frac{3}{5} \)
10. \( \frac{2}{6} = \frac{4}{12} \)
11. \( \frac{3}{4} > \frac{3}{6} \)
12. \( \frac{1}{4} < \frac{8}{12} \)

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
1. > & 2. < & 3. = \\
4. < & 5. < & 6. > \\
7. = & 8. > & 9. < \\
10. = & 11. > & 12. <
\end{array}
}
\]