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$Convert the following repeating decimal to a fraction in simplest form. .1 bar(5) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.1 \overline{5}$ $\newline$ Answer:

Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.

\newline

.1 \overline{5}

\newline

Answer:

Define $x$ as repeating decimal: Let $x$ be the repeating decimal $0.1$ with a repeating $5$ , so we have: $\newline$ $x = 0.155555...$
Multiply by power of $10$ : To convert this repeating decimal to a fraction, we can use the following technique: Multiply $x$ by a power of $10$ that will move the decimal point to the right so that the repeating part lines up with the original decimal. Since we have one digit repeating, we multiply by $10$ : $10x = 1.55555...$
Subtract original equation: Now, subtract the original equation $x = 0.155555...$ from the new equation $10x = 1.55555...$ to get rid of the repeating part: $\newline$ $10x - x = 1.55555... - 0.155555...$ $\newline$ $9x = 1.4$
Solve for x: Now, solve for x by dividing both sides of the equation by $9$ : $x = \frac{1.4}{9}$
Simplify the fraction: To simplify the fraction, we can write $1.4$ as $\frac{14}{10}$ and then divide both numerator and denominator by the greatest common divisor (GCD) of $14$ and $9$ , which is $1$ : $x = \left(\frac{14}{10}\right) / 9$ $x = \frac{14}{10 \times 9}$ $x = \frac{14}{90}$
Final simplified fraction: Now, simplify the fraction by dividing both the numerator and the denominator by their GCD, which is $2$ : $x = \left(\frac{14}{2}\right) / \left(\frac{90}{2}\right)$ $x = \frac{7}{45}$