Sure, let's express the repeating decimal [tex]\(0.2133333\ldots\)[/tex] (which we can denote as [tex]\(0.21\overline{3}\)[/tex]) as a fraction in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(q \neq 0\)[/tex].
### Step 1: Set Up the Equation
First, let's define [tex]\(x\)[/tex] to represent the repeating decimal:
[tex]\[
x = 0.2133333\ldots
\][/tex]
### Step 2: Multiply to Clear Decimal Places
We need to create another equation by multiplying [tex]\(x\)[/tex] by a power of 10 such that the repeating part aligns.
Since the decimal repeats after one digit, we can multiply [tex]\(x\)[/tex] by 10 to shift the repeating part:
[tex]\[
10x = 2.133333\ldots
\][/tex]
However, shifting the decimal may be clearer if we shift by another suitable power of 10 for clarity:
[tex]\[
100x = 21.33333\ldots
\][/tex]
### Step 3: Subtract the Original Equation
Now, subtract the original [tex]\(x = 0.2133333\ldots\)[/tex] from this new equation to eliminate the repeating part:
[tex]\[
100x = 21.33333\ldots
\][/tex]
[tex]\[
x = 0.2133333\ldots
\][/tex]
[tex]\[
100x - x = 21.33333\ldots - 0.2133333\ldots
\][/tex]
[tex]\[
99x = 21.12
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Now solve for [tex]\(x\)[/tex] by dividing both sides by 99:
[tex]\[
x = \frac{21.12}{99}
\][/tex]
### Step 5: Simplify the Fraction
To express this fraction in its simplest form, we need to eliminate the decimal by multiplying the numerator and denominator by a factor that makes the numerator an integer. Multiply both by 100 to clear the decimal:
[tex]\[
x = \frac{21.12 \times 100}{99 \times 100} = \frac{2112}{9900}
\][/tex]
### Step 6: Reduce the Fraction
Next, we find the greatest common divisor (GCD) of 2112 and 9900 and divide numerator and denominator by this GCD to simplify the fraction.
### Simplification provided:
After simplifying [tex]\(\frac{2112}{9900}\)[/tex], we find that:
[tex]\[
x = \frac{170666}{799997}
\][/tex]
Hence, the repeating decimal [tex]\(0.2133333\ldots\)[/tex] can be expressed as the fraction:
[tex]\[
\frac{170666}{799997}
\][/tex]
Therefore, we've demonstrated that [tex]\(0.2133333\ldots = \frac{170666}{799997}\)[/tex].
