The question in the image asks how to express the repeating decimal $1.27\overline{7}$ (where the 7 repeats) as a fraction.

Steps to Convert $1.27\overline{7}$ to a Fraction:

1. Let $x = 1.27\overline{7}$, meaning $x = 1.277777\ldots$ (where the 7 repeats indefinitely).
2. Multiply both sides of the equation by 10 to shift the decimal point past the repeating part: $10x = 12.77\overline{7}$
3. Now subtract the original $x = 1.27\overline{7}$ from this new equation: $10x - x = 12.77\overline{7} - 1.27\overline{7}$ This simplifies to: $9x = 11.5$
4. Solve for $x$: $x = \frac{11.5}{9} = \frac{115}{90}$
5. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD = 5): $x = \frac{115 \div 5}{90 \div 5} = \frac{23}{18}$

Thus, $1.27\overline{7} = \frac{23}{18}$.

Comparing the options:

From the available options, none exactly match $\frac{23}{18}$, but the fraction $\frac{5}{18}$ is present, which could be a simpler version of the answer presented. The correct answer, however, is likely a typo or rounding mistake. You should verify this with the platform.

Would you like any further explanation or details on this? Here are some related questions:

1. How do you convert non-repeating decimals into fractions?
2. What is the general method to convert repeating decimals into fractions?
3. Why do we subtract equations when converting repeating decimals?
4. How can you simplify fractions after converting from decimals?
5. What are some tricks for recognizing repeating decimals from their fraction equivalents?

Tip: Always check if you can simplify the resulting fraction further after conversion!