The recurring decimal representation 1.272727... is equivalent to

The correct option is $\frac{14}{11}$.

To convert a recurring decimal such as 1.272727... into a fraction, we can use algebra step-by-step.

Step 1: Define a variable for the given recurring decimal.

Let $x=\mathrm{1.272727...}$

Step 2: Multiply by a power of 10.

Since two digits ("27") are repeating continuously, we multiply both sides of the equation by 10² (which is 100):

$100x=\mathrm{127.272727...}$

Step 3: Subtract the original equation from the multiplied equation.

Subtracting $x=\mathrm{1.272727...}$ from $100x=\mathrm{127.272727...}$ cancels out the infinite repeating decimal part after the decimal point:

$100x-x=\mathrm{127.272727...}-\mathrm{1.272727...}$

$99x=126$

Step 4: Solve for $x$ and simplify the fraction.

$x=\frac{126}{99}$

Now, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 9:

$x=\frac{126÷9}{99÷9}=\frac{14}{11}$

Therefore, the recurring decimal 1.272727... is equal to $\frac{14}{11}$.