$\mathrm{}$\text{Express} $5.97\overline{3}$as a vulgar fraction.

The correct answer is:
$5\frac{219}{225}$

Step-by-Step Explanation:

To express the repeating decimal number $5.97\overline{3}$ as a vulgar fraction, we let:
$x=5.973333...$
Here, only the digit 3 repeats infinitely.

First, we multiply both sides of the equation by 100 to move the non-repeating decimal part (97) to the left of the decimal point:
$100x=597.3333...$ (Equation 1)

Next, we multiply the original equation by 1000 to place one of the repeating digits to the left of the decimal point:
$1000x=5973.3333...$ (Equation 2)

Now, subtract Equation 1 from Equation 2 to eliminate the repeating fractional part:
$1000x-100x=5973.3333...-597.3333...$

Simplifying both sides:
$900x=5376$

Solving for $x$:
$x=\frac{5376}{900}$

We convert the improper fraction into a mixed fraction by dividing 5376 by 900:
$5376=(5\times 900)+876$

This gives:
$x=5\frac{876}{900}$

Finally, we simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:
$876÷4=219$

$900÷4=225$

Therefore, expressing the repeating decimal as a vulgar mixed fraction results in:
$x=5\frac{219}{225}$