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Express the following rational numbers in the form $\frac{p}{q}$ , $p$ , $q$ are integers, $q\neq 0$ . $\newline$ (i) $6.\overline{46}$ $\newline$ (ii) $0.1\overline{36}$ $\newline$ (iii) $3.\overline{146}$ $\newline$ (iv) $-5.\overline{12}$

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Full solution

Q. Express the following rational numbers in the form

\frac{p}{q}

,

p

,

q

are integers,

q\neq 0

.

\newline

(i)

6.\overline{46}

\newline

(ii)

0.1\overline{36}

\newline

(iii)

3.\overline{146}

\newline

(iv)

-5.\overline{12}

Convert to Fraction: Given the number $6.\overline{46}$ , let's denote it as $x$ .
$x = 6.464646...$
To convert this into a fraction, we will use the following method:
Let's multiply $x$ by $100$ to shift the decimal two places to the right.
$100x = 646.464646...$
Now, subtract the original $x$ from this new equation to get rid of the repeating decimals.
$100x - x = 646.464646... - 6.464646...$
$99x = 640$
Now, we can solve for $x$ by dividing both sides by $x$ $0$ .
$x$ $1$
Check for Simplification: Now let's check for any simplification. $640$ and $99$ have no common factors other than $1$ , so the fraction is already in its simplest form.
Convert to Fraction: For the number $0.1$ bar( $36$ ), let's denote it as $y$ . $y = 0.1363636...$ To convert this into a fraction, we will use the following method:First, multiply $y$ by $10$ to shift the non-repeating decimal to the left of the decimal point. $10y = 1.363636...$ Now, multiply $y$ by $1000$ to shift the repeating decimals three places to the right. $1000y = 136.363636...$ Subtract $36$ $0$ from $36$ $1$ to get rid of the repeating decimals. $36$ $2$ $36$ $3$ Now, we can solve for $y$ by dividing both sides by $36$ $5$ . $36$ $6$
Simplify Fraction: Let's simplify the fraction $\frac{135}{990}$ . Both the numerator and the denominator are divisible by $135$ . $\frac{135}{135} = 1$ $\frac{990}{135} = 7.4$ However, the denominator should be an integer, so we made a mistake in our simplification. Let's correct it. Both the numerator and the denominator are divisible by $45$ . $\frac{135}{45} = 3$ $\frac{990}{45} = 22$ So, the simplified fraction is $\frac{3}{22}$ .

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