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$Convert the following repeating decimal to a fraction in simplest form. .5 bar(2) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.5 \overline{2}$ $\newline$ Answer:

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Partial sums of geometric series

Full solution

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

\newline

.5 \overline{2}

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Answer:

Representing Decimal as $x$ : Let $x$ represent the repeating decimal $0.5(2)$ . We write it as: $\newline$ $x = 0.52222...$
Shifting Decimal Point: To convert the repeating decimal to a fraction, we need to isolate the repeating part. To do this, we can multiply $x$ by $10$ to shift the decimal point to the right: $10x = 5.2222\ldots$
Subtracting Original from Shifted: Now we subtract the original $x$ from $10x$ to get rid of the repeating part: $\newline$ $10x - x = 5.2222... - 0.52222...$ $\newline$ This gives us: $\newline$ $9x = 4.7$
Solving for x: Now we solve for x by dividing both sides of the equation by $9$ : $x = \frac{4.7}{9}$
Expressing as Fraction: To express $4.7$ as a fraction, we write it as $\frac{47}{10}$ since $4.7$ is the same as $47$ tenths. Now we have: $\newline$ $x = \left(\frac{47}{10}\right) / 9$
Simplifying the Fraction: To simplify the fraction, we multiply the denominator by $10$ to combine the two fractions: $\newline$ $x = \frac{47}{(10 \times 9)}$ $\newline$ $x = \frac{47}{90}$
Checking for Further Simplification: We check if the fraction $\frac{47}{90}$ can be simplified further. Since $47$ is a prime number and does not share any common factors with $90$ other than $1$ , the fraction is already in its simplest form.

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