Let’s work through the problem step-by-step to convert the repeating decimal [tex]\(1.888\ldots\)[/tex] to a mixed number.
1. Assign the repeating decimal to a variable:
Let [tex]\(x = 1.888\ldots\)[/tex]
2. Multiply by 10 to shift the decimal point:
[tex]\(10x = 18.888\ldots\)[/tex]
3. Subtract the original [tex]\(x\)[/tex] from this new equation to eliminate the repeating part:
[tex]\(10x - x = 18.888\ldots - 1.888\ldots\)[/tex]
4. Simplify the subtraction:
[tex]\(10x - x = 16.991999999999997\)[/tex]
5. This gives us:
[tex]\(9x = 16.991999999999997\)[/tex]
6. Solve for [tex]\(x\)[/tex] by dividing both sides by 9:
[tex]\(x = \frac{16.991999999999997}{9}\)[/tex]
Thus, the value of [tex]\(x\)[/tex] is approximately:
[tex]\[ x = 1.8879999999999997 \][/tex]
However, to convert to the exact fraction form, we should recognize that [tex]\(x = 1.888\ldots\)[/tex] was the original repeating decimal we started with. To express this as a mixed number:
[tex]\[ 1.888\ldots = 1 + \frac{8}{9} \][/tex]
So, [tex]\(1.888\ldots\)[/tex] as a mixed number is [tex]\(1 \frac{8}{9}\)[/tex].
