Converting Repeating Decimal to Fraction

To express the repeating decimal $5.97\bar3$ as a vulgar fraction, we follow these steps:

• Let the given number be $x$. So, $x = 5.97333...$
• Multiply $x$ by 100 to move the decimal point just before the repeating part starts:

  $100x = 597.333... \quad (1)$

• Multiply $x$ by 1000 to move the decimal point one place after the start of the repeating digit:

  $1000x = 5973.333... \quad (2)$

• Subtract equation (1) from equation (2) to eliminate the repeating decimal part:

  $1000x - 100x = 5973.333... - 597.333...$

  $900x = 5376$

• Solve for $x$:

  $x = \frac{5376}{900}$

• Simplify the fraction. Both numerator and denominator are divisible by 4:

  $x = \frac{5376 \div 4}{900 \div 4} = \frac{1344}{225}$

• The fraction $\frac{1344}{225}$ can be further simplified by dividing by 3 (since $1+3+4+4=12$ and $2+2+5=9$, both divisible by 3):

  $x = \frac{1344 \div 3}{225 \div 3} = \frac{448}{75}$

• Convert the improper fraction $\frac{448}{75}$ into a mixed number. Divide 448 by 75:

  $448 \div 75 = 5 \text{ with a remainder of } 448 - (5 \times 75) = 448 - 375 = 73$

  So, $x = 5\frac{73}{75}$.

Matching with Options

Now, let's check the given options:

• Option A: $5\frac{219}{225}$. To check if this equals $5\frac{73}{75}$, simplify the fractional part $\frac{219}{225}$. Both are divisible by 3:

  $\frac{219 \div 3}{225 \div 3} = \frac{73}{75}$

  Thus, $5\frac{219}{225} = 5\frac{73}{75}$.

Therefore, the correct representation of $5.97\bar3$ as a vulgar fraction is $5\frac{219}{225}$.