To express the repeating decimal [tex]\( 2.3\overline{14} \)[/tex] as a simplified fraction, follow these steps:
1. Initial Representation:
Let [tex]\( x = 2.31414141414\ldots \)[/tex]
2. Multiply to Align Decimals:
Multiply both sides by 100 (because two digits repeat) to shift the repeating part to the left:
[tex]\[ 100x = 231.41414141414\ldots \][/tex]
3. Subtract to Eliminate Repeats:
Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 100x - x = 231.41414141414\ldots - 2.31414141414\ldots \][/tex]
[tex]\[ 99x = 229.1 \][/tex]
4. Solve for x:
[tex]\[ x = \frac{229.1}{99} \][/tex]
5. Remove the Decimal Point:
Multiply numerator and denominator by 10 to remove the decimal point:
[tex]\[ x = \frac{2291}{990} \][/tex]
6. Simplify the Fraction:
Find the greatest common divisor (GCD) of the numerator and the denominator to simplify the fraction. The GCD of 2291 and 990 is 1.
7. Final Simplified Form:
Since the GCD is 1, the fraction is already in its simplest form:
[tex]\[ x = \frac{2291}{990} \][/tex]
Therefore, the simplified fraction for [tex]\( 2.3\overline{14} \)[/tex] is:
[tex]\[ \boxed{\frac{2291}{990}} \][/tex]
