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$Convert the following repeating decimal to a fraction in simplest form. .4 bar(1) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.4 \overline{1}$ $\newline$ Answer:

Full solution

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

\newline

.4 \overline{1}

\newline

Answer:

Write $x$ as repeating decimal: Let $x$ equal the repeating decimal $0.41$ with the $1$ repeating. We write this as: $\newline$ $x = 0.411111...$
Multiply $x$ by $10$ : To convert the repeating decimal to a fraction, we need to isolate the repeating part. To do this, we multiply $x$ by $10$ , since there is one digit before the repeating sequence: $\newline$ $10x = 4.11111\ldots$
Subtract $x$ from $10x$ : Now we subtract the original $x$ from $10x$ to get rid of the repeating part: $\newline$ $10x - x = 4.11111... - 0.41111...$ $\newline$ This simplifies to: $\newline$ $9x = 4 - 0.4$
Simplify the equation: Simplify the right side of the equation: $9x = 3.6$
Divide both sides by $9$ : Now, we divide both sides by $9$ to solve for $x$ : $x = \frac{3.6}{9}$
Simplify the fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $0.1$ for the numerator and $1$ for the denominator: $\newline$ $x = \frac{(3.6 / 0.1)}{(9 / 1)}$
Perform the division: Perform the division: $x = \frac{36}{9}$
Simplify the fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $9$ : $x = \frac{36}{9}$ $x = 4$