Convert the repeating decimal 0.2(7) to a vulgar fraction Here 0.2(7) means 0.2777… with only the digit 7 recurring after the first decimal place. Express this recurring decimal as a single simplified fraction.

Introduction / Context:
This problem asks you to convert a recurring (repeating) decimal into a vulgar fraction (a simple fraction in lowest terms). The decimal is written as 0.2(7), which means 0.2777… with only the 7 repeating endlessly. Understanding how to separate the non-repeating and repeating parts is the key skill tested here.

Given Data / Assumptions:

• The decimal is 0.2(7) = 0.2777…
• Only the 7 repeats; the first digit 2 after the decimal is non-repeating.
• We must produce the fraction in simplest form.

Concept / Approach:
Let x denote the repeating decimal. Use the standard technique: isolate the repeating part by shifting decimals using powers of 10. Because the repeating block has length 1 (only one digit repeats), multiplying by 10 moves the decimal one place; the non-repeating prefix (2) must be handled carefully. Subtracting two equations eliminates the repeating tail.

Step-by-Step Solution:
Let x = 0.2777…Multiply by 10 to shift one decimal place: 10x = 2.7777…Subtract: (10x − x) = 2.7777… − 0.2777… = 2.5Therefore 9x = 2.5, so x = 2.5 / 9 = 25/90 = 5/18.

Verification / Alternative check:
Convert 5/18 back to a decimal: 5 ÷ 18 = 0.2777…, which matches 0.2(7). The non-repeating digit 2 appears once, and the digit 7 repeats indefinitely, confirming correctness.

Why Other Options Are Wrong:
5/19 ≈ 0.263… (too small); 7/19 ≈ 0.368… (too large); 3/17 ≈ 0.176… (far too small); 7/25 = 0.28 (terminating, not recurring 7’s). Only 5/18 equals 0.2777…

Common Pitfalls:
Readers often treat 0.2(7) as if both digits 2 and 7 are repeating (i.e., 0.(27)), which is a different number. Always verify which digits are in the repeating block.

Final Answer:
5/18