Sure! Let's convert the repeating decimal [tex]\( 2.3\overline{14} \)[/tex] into a fraction.
First, we denote the repeating decimal by [tex]\( x \)[/tex]. So:
[tex]\[ x = 2.3\overline{14} \][/tex]
We can break this down into the whole part and the repeating decimal part:
[tex]\[ x = 2 + 0.3\overline{14} \][/tex]
### Step 1: Convert the repeating decimal part
Let's consider [tex]\( y = 0.3\overline{14} \)[/tex].
### Step 2: Express [tex]\( y \)[/tex] (the repeating part) as a fraction
Let [tex]\( y = 0.3\overline{14} \)[/tex].
Move the decimal point of [tex]\( y \)[/tex] such that it aligns with the repeating part. Multiply [tex]\( y \)[/tex] by 100 (since the repeating part is two digits):
[tex]\[ 100y = 31.4\overline{14} \][/tex]
It is difficult to solve directly due to the non-repeating leading digits. So rearranging it:
[tex]\[ y = 0.3 + z \][/tex]
where [tex]\( z = 0.\overline{14} \)[/tex].
### Step 3: Convert [tex]\( z \)[/tex] into a fraction
Let's denote [tex]\( z = 0.\overline{14} \)[/tex].
We multiply [tex]\( z \)[/tex] by 100 (since the repeating part is two digits):
[tex]\[ 100z = 14.\overline{14} \][/tex]
Now subtract [tex]\( z \)[/tex] from [tex]\( 100z \)[/tex]:
[tex]\[ 100z - z = 14.\overline{14} - 0.\overline{14} \][/tex]
[tex]\[ 99z = 14 \][/tex]
Solving for [tex]\( z \)[/tex]:
[tex]\[ z = \frac{14}{99} \][/tex]
### Step 4: Combine the parts
Substitute back [tex]\( z \)[/tex] into our expression for [tex]\( y \)[/tex]:
[tex]\[ y = 0.3 + \frac{14}{99} \][/tex]
Express 0.3 as a fraction:
[tex]\[ 0.3 = \frac{3}{10} \][/tex]
So we have:
[tex]\[ y = \frac{3}{10} + \frac{14}{99} \][/tex]
Find a common denominator and combine:
[tex]\[ y = \frac{3 \times 99 + 14 \times 10}{10 \times 99} \][/tex]
[tex]\[ y = \frac{297 + 140}{990} \][/tex]
[tex]\[ y = \frac{437}{990} \][/tex]
### Step 5: Simplify the fraction
We simplify [tex]\( \frac{437}{990} \)[/tex] by finding the greatest common divisor (gcd):
The gcd of 437 and 990 is 1, hence the fraction is already in its simplest form.
[tex]\[ y = \frac{437}{990} \][/tex]
### Step 6: Combine [tex]\( x = 2 + y \)[/tex]
Finally, consider the whole number part:
[tex]\[ x = 2 + y = 2 + \frac{437}{990} \][/tex]
Convert 2 to a fraction and combine:
[tex]\[ x = \frac{2 \times 990 + 437}{990} \][/tex]
[tex]\[ x = \frac{1980 + 437}{990} \][/tex]
[tex]\[ x = \frac{2417}{990} \][/tex]
Thus, the simplified fraction form of [tex]\( 2.3\overline{14} \)[/tex] is:
[tex]\[ \boxed{\frac{2417}{990}} \][/tex]
