To convert the repeating decimal [tex]\( 2.3\overline{14} \)[/tex] into a simplified fraction, follow these steps:
1. Identify the repeating part: The decimal [tex]\( 2.3\overline{14} \)[/tex] has a repeating part "14".
2. Express the decimal as a sum: We can break down the decimal into two parts:
- The non-repeating part: [tex]\( 2.3 \)[/tex]
- The repeating part: [tex]\( 0.\overline{14} \)[/tex]
3. Convert the non-repeating part to a fraction:
- [tex]\( 2.3 \)[/tex] can be written as [tex]\( 2 + 0.3 \)[/tex].
- [tex]\( 0.3 = \frac{3}{10} \)[/tex].
- So, [tex]\( 2.3 = 2 + \frac{3}{10} \)[/tex].
4. Convert the repeating part to a fraction:
- Let [tex]\( x = 0.\overline{14} \)[/tex].
- To eliminate the repeating part, multiply [tex]\( x \)[/tex] by [tex]\( 100 \)[/tex] (since "14" has two digits):
[tex]\( 100x = 14.\overline{14} \)[/tex].
- Subtract the original [tex]\( x \)[/tex] from this equation: [tex]\( 100x - x = 14.\overline{14} - 0.\overline{14} \)[/tex].
- This simplifies to: [tex]\( 99x = 14 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x = \frac{14}{99} \)[/tex].
5. Sum the fractions:
- [tex]\( 2.3\overline{14} = 2 + \frac{3}{10} + \frac{14}{99} \)[/tex].
6. Combine the fractions into a single fraction:
- First, find a common denominator for all parts. The common denominator of 1, 10, and 99 is 990.
- Convert 2 to a fraction with the denominator 990: [tex]\( 2 = \frac{2 \times 990}{990} = \frac{1980}{990} \)[/tex].
- Convert [tex]\( \frac{3}{10} \)[/tex] to a fraction with the denominator 990: [tex]\( \frac{3}{10} = \frac{3 \times 99}{10 \times 99} = \frac{297}{990} \)[/tex].
- Convert [tex]\( \frac{14}{99} \)[/tex] to a fraction with the denominator 990: [tex]\( \frac{14}{99} = \frac{14 \times 10}{99 \times 10} = \frac{140}{990} \)[/tex].
7. Add the fractions:
- [tex]\( \frac{1980}{990} + \frac{297}{990} + \frac{140}{990} = \frac{1980 + 297 + 140}{990} = \frac{2417}{990} \)[/tex].
Therefore, the simplified fraction representation of [tex]\( 2.3\overline{14} \)[/tex] is [tex]\( \boxed{\frac{2417}{990}} \)[/tex].
