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The image asks which fractions are equivalent to the repeating decimal \( 0.17\overline{7} \). To check this, we need to convert the decimal into a fraction and then compare it with the options provided.

### Step 1: Converting \( 0.17\overline{7} \) into a fraction
Let \( x = 0.17\overline{7} \).

1. Multiply \( x \) by 10 to shift the decimal:
   \[
   10x = 1.7\overline{7}
   \]

2. Now, subtract \( x = 0.17\overline{7} \) from \( 10x = 1.7\overline{7} \):
   \[
   10x - x = (1.7\overline{7}) - (0.17\overline{7}) \Rightarrow 9x = 1.6
   \]

3. Solve for \( x \):
   \[
   x = \frac{1.6}{9} = \frac{16}{90}
   \]
   Simplifying \( \frac{16}{90} \) gives \( \frac{8}{45} \).

So, \( 0.17\overline{7} = \frac{8}{45} \).

### Step 2: Verifying the options
- **\( \frac{177}{1000} \):** This fraction is approximately \( 0.177 \), which is close to but not exactly \( 0.17\overline{7} \), so it **is not** correct.
- **\( \frac{17}{100} \):** This fraction is exactly \( 0.17 \), but it is a terminating decimal, not a repeating one, so it **is not** correct.
- **\( \frac{8}{45} \):** This matches the exact conversion we did above, so it **is** correct.
- **\( \frac{16}{90} \):** Simplifying this gives \( \frac{8}{45} \), which is equivalent, so it **is** correct.

### Correct answers:
- \( \frac{8}{45} \)
- \( \frac{16}{90} \)

Would you like more details or have any further questions?

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Here are 5 related questions:
1. How can you convert any repeating decimal to a fraction?
2. What is the process for simplifying fractions?
3. How do you check if two fractions are equivalent?
4. Can all repeating decimals be converted into fractions?
5. What are the differences between terminating and repeating decimals?

**Tip:** When converting repeating decimals, multiplying by powers of 10 helps eliminate the repeating part!

Which fractions are equivalent to 0.17̅? Select all that apply.

Learn how to determine which fractions are equivalent to the repeating decimal 0.17̅ by converting the decimal to a fraction and verifying equivalent forms. This problem is suitable for students in Grades 7-9 and covers key concepts like decimal-to-fraction conversion and fraction simplification.

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