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Express 0.99999 in the form of
(p)/(q) where
p and
q are integers and
q!=0.

Express $0$ . $99999$ in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$ .

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Transformations of linear functions

Full solution

Q. Express

0

.

99999

in the form of

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

.

Assign Variable $x$ : Let $x$ equal the repeating decimal we want to express as a fraction. $\newline$ Set $x = 0.99999$ .
Multiply by $10^5$ : Multiply both sides of the equation by $10^5$ ( $100000$ ) to shift the decimal point five places to the right, since there are five $9$ 's after the decimal. $\newline$ $100000x = 99999$ .
Subtract Original Equation: Subtract the original equation $x = 0.99999$ from the new equation $100000x = 99999$ to get rid of the repeating decimal. $\newline$ $100000x - x = 99999 - 0.99999$ .
Perform Subtraction: Perform the subtraction on both sides of the equation. $\newline$ $99999x = 99999 - 0.99999$ . $\newline$ $99999x = 99998.00001$ .
Solve for x: Solve for x by dividing both sides of the equation by $99999$ . $\newline$ x = rac{99998.00001}{99999}.
Simplify Numerator: Recognize that the subtraction in the numerator $99998.00001$ is very close to $99998$ , and the small difference is due to the repeating decimal. Since we are looking for a fraction with integers, we can simplify the numerator to $99998$ . $x = \frac{99998}{99999}$ .
Check Simplest Form: Check that the fraction is in simplest form. Since $99998$ and $99999$ have no common factors other than $1$ , the fraction is already in simplest form.

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