Sure, let's convert the recurring decimal [tex]\( 0.126262626\ldots \)[/tex] to a fraction step-by-step.
### Step 1: Define the repeating decimal as a variable
Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[ x = 0.126262626\ldots \][/tex]
### Step 2: Eliminate the repeating part by shifting the decimal
To deal with the repeating portion, we need to shift the decimal point to the right. Since the repeating part is "262", we multiply [tex]\( x \)[/tex] by 1000 (three decimal places):
[tex]\[ 1000x = 126.262626\ldots \][/tex]
### Step 3: Subtract to remove the repeating decimal
Now, subtract the original [tex]\( x \)[/tex] from [tex]\( 1000x \)[/tex]:
[tex]\[ 1000x = 126.262626\ldots \][/tex]
[tex]\[ - x = 0.126262626\ldots \][/tex]
[tex]\[ 999x = 126.136 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Now, solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 999:
[tex]\[ x = \frac{126.136}{999} \][/tex]
### Step 5: Simplify the fraction
First, note that [tex]\( 126.136 \)[/tex] can be written as a fraction itself. This requires breaking it up into the whole number and the decimal part. Thus,
[tex]\[ 126.136 = 126 + 0.136 \][/tex]
Expressing [tex]\( 0.136 \)[/tex] as a fraction:
[tex]\[ 0.136 = \frac{136}{1000} = \frac{34}{250} = \frac{17}{125} \][/tex]
So, we get:
[tex]\[ 126.136 = 126 + \frac{17}{125} \][/tex]
Now, combine these into a single fraction. To do this, we need a common denominator:
[tex]\[ 126.136 = \frac{126 \times 125 + 17}{125} = \frac{15750 + 17}{125} = \frac{15767}{125} \][/tex]
Therefore,
[tex]\[ \frac{126.136}{999} = \frac{\frac{15767}{125}}{999} = \frac{15767}{125 \times 999} = \frac{15767}{124875} \][/tex]
### Step 6: Simplify the resulting fraction
We can attempt to simplify [tex]\(\frac{15767}{124875}\)[/tex] by finding the greatest common divisor (GCD). Using the Euclidean algorithm:
- The GCD of 15767 and 124875 is 1 (since 15767 is a prime factor).
Hence, the fraction is already in its simplest form:
[tex]\[ \boxed{\frac{15767}{124875}} \][/tex]
So, the recurring decimal [tex]\( 0.126262626\ldots \)[/tex] can be written exactly as the fraction [tex]\( \frac{15767}{124875} \)[/tex].
