Sure! Let's solve the problem step-by-step, beginning by converting the repeating decimal [tex]\(1.888\ldots\)[/tex] into a fraction or mixed number.
Step 1: Let [tex]\(x = 1.888\ldots\)[/tex]
Step 2: Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[10x = 18.888\ldots\][/tex]
Step 3: Subtract the original equation ([tex]\(x\)[/tex]) from this new equation:
[tex]\[10x - x = 18.888\ldots - 1.888\ldots\][/tex]
Step 4: Simplify the subtraction:
[tex]\[9x = 17\][/tex]
Step 5: Solve for [tex]\(x\)[/tex] by dividing both sides of the equation by 9:
[tex]\[x = \frac{17}{9}\][/tex]
Step 6: Convert the fraction [tex]\(\frac{17}{9}\)[/tex] to a mixed number. To do this, divide the numerator (17) by the denominator (9):
[tex]\[17 \div 9 = 1 \text{ with a remainder of } 8\][/tex]
So, [tex]\(\frac{17}{9} = 1 \frac{8}{9}\)[/tex].
Therefore, the repeating decimal [tex]\(1.888\ldots\)[/tex] can be written as the mixed number [tex]\(1 \frac{8}{9}\)[/tex].
In summary:
1. Let [tex]\(x = 1.888\ldots\)[/tex]
2. [tex]\(10x = 18.888\ldots\)[/tex]
3. [tex]\(10x - x = 18.888\ldots - 1.888\ldots\)[/tex]
4. [tex]\(9x = 17\)[/tex]
5. [tex]\(x = \frac{17}{9}\)[/tex]
6. [tex]\(\frac{17}{9} = 1 \frac{8}{9}\)[/tex]
Thus, [tex]\(1.888\ldots = 1 \frac{8}{9}\)[/tex].
