Certainly! Let's convert the repeating decimal [tex]\(2.3181818...\)[/tex] into a mixed number step-by-step.
### Step 1: Separate the Integer Part
First, identify and separate the whole number part of the decimal. Here, the whole number part is [tex]\(2\)[/tex].
So, we have:
[tex]\[ 2.3181818... = 2 + 0.3181818...\][/tex]
### Step 2: Focus on the Repeating Decimal Part
Now, we need to convert the repeating decimal part [tex]\(0.3181818...\)[/tex] into a fraction.
### Step 3: Identify the Repeating Block
The repeating part in the decimal [tex]\(0.3181818...\)[/tex] is [tex]\(18\)[/tex]. The repeating sequence starts after the decimal point and repeats every two digits.
### Step 4: Convert the Repeating Block into a Fraction
The repeating decimal [tex]\(0.3181818...\)[/tex] can be expressed as:
[tex]\[ 0.3181818... = 0.\overline{18} \][/tex]
To convert [tex]\(0.\overline{18}\)[/tex] into a fraction, recall the formula for a repeating decimal [tex]\(0.\overline{a}\)[/tex], where [tex]\(a\)[/tex] is the repeating part. For a repeating block of two digits, this is given by:
[tex]\[ \overline{ab} / 99 \][/tex]
In our case, [tex]\(a = 18\)[/tex]. So we have:
[tex]\[ 0.\overline{18} = \frac{18}{99} \][/tex]
### Step 5: Simplify the Fraction
Next, simplify the fraction [tex]\(\frac{18}{99}\)[/tex].
The greatest common divisor (GCD) of 18 and 99 is 9. Dividing both the numerator and the denominator by 9, we get:
[tex]\[ \frac{18}{99} = \frac{18 \div 9}{99 \div 9} = \frac{2}{11} \][/tex]
### Step 6: Combine the Integer Part and the Fraction
Now, we will combine the integer part with the simplified fraction. The integer part is [tex]\(2\)[/tex] and the fraction we obtained is [tex]\(\frac{2}{11}\)[/tex].
So, the mixed number is:
[tex]\[ 2 + \frac{2}{11} \][/tex]
### Step 7: Write as a Single Fraction (Optional)
If we want to express this as a single improper fraction:
[tex]\[ 2 + \frac{2}{11} = \frac{2 \times 11}{11} + \frac{2}{11} = \frac{22 + 2}{11} = \frac{24}{11} \][/tex]
### Final Answer
Thus, the given repeating decimal [tex]\(2.3181818...\)[/tex] as a mixed number is:
[tex]\[ 2 \frac{2}{11} \][/tex]
Or, as a single improper fraction:
[tex]\[ \frac{24}{11} \][/tex]
