Equivalent Fraction Patterns worksheet with answer key, designed for math practice.

Problem Overview:

The task involves filling in missing equivalent fractions in a series of fraction patterns. Equivalent fractions are fractions that represent the same value but are expressed with different numerators and denominators. The key to solving these problems is understanding how to scale both the numerator and the denominator by the same factor.

Solution Explanation:

To solve each problem, we need to identify the scaling factor applied to the original fraction and apply it consistently to find the missing values. Let’s go through the steps for a few examples:

#### General Approach:
1. Identify the Original Fraction: Start with the given fraction.
2. Determine the Scaling Factor: Look at how the numerator or denominator changes from one fraction to the next.
3. Apply the Scaling Factor: Use the identified factor to fill in the missing values.

#### Detailed Solutions for Selected Problems:

##### Problem 1:
Original fraction: \( \frac{8}{9} \)
Pattern: \( \frac{8}{9} = \frac{16}{18} = \frac{24}{27} = \frac{32}{36} = \frac{40}{45} = \frac{48}{54} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{24}{27} \)

##### Problem 2:
Original fraction: \( \frac{1}{6} \)
Pattern: \( \frac{1}{6} = \frac{2}{12} = \frac{3}{18} = \frac{4}{24} = \frac{5}{30} = \frac{6}{36} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{2}{12} \)

##### Problem 3:
Original fraction: \( \frac{2}{9} \)
Pattern: \( \frac{2}{9} = \frac{4}{18} = \frac{6}{27} = \frac{8}{36} = \frac{10}{45} = \frac{12}{54} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{10}{45} \)

##### Problem 4:
Original fraction: \( \frac{4}{6} \)
Pattern: \( \frac{4}{6} = \frac{8}{12} = \frac{12}{18} = \frac{16}{24} = \frac{20}{30} = \frac{24}{36} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{20}{30} \)

##### Problem 5:
Original fraction: \( \frac{1}{2} \)
Pattern: \( \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = \frac{6}{12} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{5}{10} \)

##### Problem 6:
Original fraction: \( \frac{3}{6} \)
Pattern: \( \frac{3}{6} = \frac{6}{12} = \frac{9}{18} = \frac{12}{24} = \frac{15}{30} = \frac{18}{36} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{9}{18} \)

##### Problem 7:
Original fraction: \( \frac{5}{7} \)
Pattern: \( \frac{5}{7} = \frac{10}{14} = \frac{15}{21} = \frac{20}{28} = \frac{25}{35} = \frac{30}{42} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{15}{21} \)

##### Problem 8:
Original fraction: \( \frac{3}{4} \)
Pattern: \( \frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16} = \frac{15}{20} = \frac{18}{24} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{15}{20} \)

##### Problem 9:
Original fraction: \( \frac{4}{5} \)
Pattern: \( \frac{4}{5} = \frac{8}{10} = \frac{12}{15} = \frac{16}{20} = \frac{20}{25} = \frac{24}{30} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{16}{20} \)

##### Problem 10:
Original fraction: \( \frac{2}{3} \)
Pattern: \( \frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{10}{15} = \frac{12}{18} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{8}{12} \)

##### Problem 11:
Original fraction: \( \frac{5}{6} \)
Pattern: \( \frac{5}{6} = \frac{10}{12} = \frac{15}{18} = \frac{20}{24} = \frac{25}{30} = \frac{30}{36} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{25}{30} \)

##### Problem 12:
Original fraction: \( \frac{8}{10} \)
Pattern: \( \frac{8}{10} = \frac{16}{20} = \frac{24}{30} = \frac{32}{40} = \frac{40}{50} = \frac{48}{60} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{40}{50} \)

##### Problem 13:
Original fraction: \( \frac{3}{8} \)
Pattern: \( \frac{3}{8} = \frac{6}{16} = \frac{9}{24} = \frac{12}{32} = \frac{15}{40} = \frac{18}{48} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{18}{48} \)

##### Problem 14:
Original fraction: \( \frac{3}{5} \)
Pattern: \( \frac{3}{5} = \frac{6}{10} = \frac{9}{15} = \frac{12}{20} = \frac{15}{25} = \frac{18}{30} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{15}{25} \)

##### Problem 15:
Original fraction: \( \frac{9}{10} \)
Pattern: \( \frac{9}{10} = \frac{18}{20} = \frac{27}{30} = \frac{36}{40} = \frac{45}{50} = \frac{54}{60} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{45}{50} \)

##### Problem 16:
Original fraction: \( \frac{2}{7} \)
Pattern: \( \frac{2}{7} = \frac{4}{14} = \frac{6}{21} = \frac{8}{28} = \frac{10}{35} = \frac{12}{42} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{4}{14} \)

##### Problem 17:
Original fraction: \( \frac{5}{8} \)
Pattern: \( \frac{5}{8} = \frac{10}{16} = \frac{15}{24} = \frac{20}{32} = \frac{25}{40} = \frac{30}{48} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{25}{40} \)

##### Problem 18:
Original fraction: \( \frac{6}{10} \)
Pattern: \( \frac{6}{10} = \frac{12}{20} = \frac{18}{30} = \frac{24}{40} = \frac{30}{50} = \frac{36}{60} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{30}{50} \)

##### Problem 19:
Original fraction: \( \frac{2}{4} \)
Pattern: \( \frac{2}{4} = \frac{4}{8} = \frac{6}{12} = \frac{8}{16} = \frac{10}{20} = \frac{12}{24} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{12}{24} \)

##### Problem 20:
Original fraction: \( \frac{2}{10} \)
Pattern: \( \frac{2}{10} = \frac{4}{20} = \frac{6}{30} = \frac{8}{40} = \frac{10}{50} = \frac{12}{60} \)

- The numerator and denominator are scaled by multiplying by 2, 3, 4, 5, and 6 respectively.
- Missing fraction: \( \frac{4}{20} \)

Final Answer:

The missing equivalent fractions are filled in as follows:

\[
\boxed{
\begin{array}{ll}
1. & \frac{24}{27} \\
2. & \frac{2}{12} \\
3. & \frac{10}{45} \\
4. & \frac{20}{30} \\
5. & \frac{5}{10} \\
6. & \frac{9}{18} \\
7. & \frac{15}{21} \\
8. & \frac{15}{20} \\
9. & \frac{16}{20} \\
10. & \frac{8}{12} \\
11. & \frac{25}{30} \\
12. & \frac{40}{50} \\
13. & \frac{18}{48} \\
14. & \frac{15}{25} \\
15. & \frac{45}{50} \\
16. & \frac{4}{14} \\
17. & \frac{25}{40} \\
18. & \frac{30}{50} \\
19. & \frac{12}{24} \\
20. & \frac{4}{20} \\
\end{array}
}
\]