Connect with experts and get insightful answers on IDNLearn.com. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
Sagot :
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]
### 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]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
