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Let's consider two divisions:
We use three dots at the end of a number to indicate that a pattern repeats itself endlessly.
$\text{4}\xf7\text{3}=\text{1}\text{.}\text{333}\dots $
Another way, aside from using three dots, of denoting an endlessly repeating pattern is to write a bar ( ¯ ) above the repeating sequence of digits.
$\text{4}\xf7\text{3}=\text{1}\text{.}\overline{3}$
The bar indicates the repeated pattern of 3.
Repeating patterns in a division can be discovered in two ways:
Carry out each division until the repeating pattern can be determined.
$\text{100}\xf7\text{27}$
$\begin{array}{c}\hfill 3.70370\\ \hfill 27\overline{)100.00000}\\ \hfill \underline{81}\\ \hfill 190\\ \hfill \underline{189}\\ \hfill 100\\ \hfill \underline{81}\\ \hfill 190\\ \hfill 189\end{array}$
When the remainder is identical to the dividend, the division is nonterminating. This implies that the pattern in the quotient repeats.
$\text{100}\xf7\text{27}=3\text{.}\text{70370370}\dots $ The repeating block is 703.
$\text{100}\xf7\text{27}=3\text{.}\overline{\text{703}}$
$\text{1}\xf7\text{9}$
We see that this “product, difference”pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats.
$1\xf79=0\text{.}\text{111}\dots $ The repeating block is 1.
$1\xf79=0\text{.}\overline{1}$
Divide 2 by 11 and round to 3 decimal places.
Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places.
$\begin{array}{c}\hfill .1818\\ \hfill 11\overline{)2.0000}\\ \hfill \underline{11}\\ \hfill 90\\ \hfill \underline{88}\\ \hfill 20\\ \hfill \underline{11}\\ \hfill 90\end{array}$
The number .1818 rounded to three decimal places is .182. Thus, correct to three decimal places,
$\text{2}\xf7\text{11}=\text{0}\text{.}\text{182}$
Divide 1 by 6.
We see that this “product, difference” pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats at the 6.
$1\xf76=0\text{.}1\overline{6}$
Carry out the following divisions until the repeating pattern can be determined.
For the following 20 problems, carry out each division until the repeating pattern is determined. If a repeating pattern is not apparent, round the quotient to three decimal places.
$8\xf7\text{11}$
$3\xf71\text{.}1$
$\text{10}\xf72\text{.}7$
$1\text{.}\text{11}\xf79\text{.}9$
$0\text{.}1\overline{\text{12}}$
$8\text{.}\text{08}\xf73\text{.}1$
$\text{51}\xf78\text{.}2$
$6\text{.}\overline{\text{21951}}$
$0\text{.}\text{213}\xf70\text{.}\text{31}$
$0\text{.}\text{009}\xf71\text{.}1$
$0\text{.}\text{00}\overline{\text{81}}$
$6\text{.}\text{03}\xf71\text{.}9$
$1\text{.}\text{55}\xf70\text{.}\text{27}$
$0\text{.}\text{333}\xf70\text{.}\text{999}$
$0\text{.}\overline{3}$
$0\text{.}\text{444}\xf70\text{.}\text{999}$
$0\text{.}\text{555}\xf70\text{.}\text{27}$
$2\text{.}0\overline{5}$
$3\text{.}8\xf70\text{.}\text{99}$
$8\xf7\text{11}$
$1\xf7\text{44}$
$0\text{.}7\xf70\text{.}9$ (Compare this with [link] .)
$\text{80}\xf7\text{110}$ (Compare this with [link] .)
$0\text{.}\overline{\text{72}}$
$0\text{.}\text{0707}\xf70\text{.}\text{7070}$
$1\xf70\text{.}\text{9999999}$
( [link] ) In the number 411,105, how many ten thousands are there?
1
( [link] ) Find the quotient, if it exists. $\text{17}\xf70$ .
( [link] ) Find the least common multiple of 45, 63, and 98.
4410
( [link] ) Subtract 8.01629 from 9.00187 and round the result to three decimal places.
( [link] ) Find the quotient. $\text{104}\text{.}\text{06}\xf7\text{12}\text{.}1$ .
8.6
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