Equivalent Fraction Patterns worksheet with answer key for math practice.

It looks like you've uploaded an image of a completed worksheet titled "Equivalent Fraction Patterns", which asks students to fill in missing equivalent fractions in sequences. The worksheet includes 20 problems, and the answer key is already filled in on the right-hand side.

Since this is a completed answer key, I’ll walk through how to solve one or two of the problems as examples to explain the pattern and method used. Then, I'll summarize the general strategy for solving such fraction equivalence problems.

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🔍 Example Problem: #1

> Fill in the missing equivalent fraction:
>
> $$
> \frac{8}{9} = \frac{16}{18} = \frac{24}{?} = \frac{32}{36} = \frac{40}{45} = \frac{48}{54}
> $$

#### Step-by-step solution:

We are given:
- $ \frac{8}{9} $
- $ \frac{16}{18} $ → This is $ \frac{8 \times 2}{9 \times 2} $, so it's equivalent.
- Next: $ \frac{24}{?} $

We see that numerator increases by 8 each time:
- 8 → 16 → 24 → 32 → 40 → 48

So the pattern is multiplying the original fraction $ \frac{8}{9} $ by increasing whole numbers:
- $ \frac{8}{9} \times 1 = \frac{8}{9} $
- $ \frac{8}{9} \times 2 = \frac{16}{18} $
- $ \frac{8}{9} \times 3 = \frac{24}{27} $
- $ \frac{8}{9} \times 4 = \frac{32}{36} $
- etc.

So the missing denominator when numerator is 24 should be:
$$
\frac{24}{x} = \frac{8}{9} \Rightarrow x = \frac{24 \times 9}{8} = 27
$$

✔ So, $ \frac{24}{27} $ is correct.

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🔍 Example Problem: #3

> $$
> \frac{2}{9} = \frac{4}{18} = \frac{6}{27} = \frac{8}{36} = \frac{10}{?} = \frac{12}{54}
> $$

Let’s check the pattern:
- Numerators: 2, 4, 6, 8, ?, 12 → increasing by 2
- Denominators: 9, 18, 27, 36, ?, 54 → increasing by 9

So:
- $ \frac{2}{9} \times 1 = \frac{2}{9} $
- $ \times 2 = \frac{4}{18} $
- $ \times 3 = \frac{6}{27} $
- $ \times 4 = \frac{8}{36} $
- $ \times 5 = \frac{10}{45} $
- $ \times 6 = \frac{12}{54} $

So the missing term is $ \frac{10}{45} $ → ✔ Matches answer key.

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🧠 General Strategy for Solving These Problems:

1. Identify the base fraction: Usually the first one (e.g., $ \frac{8}{9} $).
2. Check the pattern: Are numerators and denominators increasing linearly?
3. Look at multiplication factor: Each new fraction is the base fraction multiplied by an integer (1, 2, 3...).
- For example: $ \frac{8}{9} \times 3 = \frac{24}{27} $
4. Use cross-multiplication or scaling:
- If numerator is known, use: $ \frac{a}{b} = \frac{c}{d} \Rightarrow d = \frac{b \cdot c}{a} $
5. Ensure all fractions are equivalent — same value.

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✔ Summary of Answer Key (for verification):

| Problem | Missing Fraction |
|--------|------------------|
| 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} $ |

All answers follow the pattern of multiplying the base fraction by successive integers.

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💡 Tip:

To check if two fractions are equivalent:
- Cross-multiply: $ \frac{a}{b} = \frac{c}{d} $ iff $ a \cdot d = b \cdot c $

For example: Is $ \frac{24}{27} = \frac{8}{9} $?
→ $ 24 \cdot 9 = 216 $, $ 27 \cdot 8 = 216 $ → Yes!

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✔ Final Note: The worksheet is correctly solved. All answers in the key are accurate and follow the pattern of equivalent fractions generated by scaling up the original fraction.

Let me know if you'd like a printable version or a step-by-step walkthrough for any specific problem!