To find the decimal expansion of \( \frac{5}{11} \) using long division, we divide 5 by 11.
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Set up the long division: 5 is less than 11, so we start with 0. We put a decimal point and add a zero to make it 50.
\[ 11 ) 5.00 \]
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Divide: 11 goes into 50 four times (since \( 11 \times 4 = 44 \)). Write 4 in the quotient.
\[ 11 ) 5.00 - 44 -------- 6 \]
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Bring down another 0: Now we have 60.
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Repeat the division: 11 goes into 60 five times (since \( 11 \times 5 = 55 \)). Write 5 in the quotient.
\[ 11 ) 5.00 - 44 -------- 60 - 55 -------- 5 \]
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Bring down another 0: Now we have 50 again.
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Repeat the process: 11 goes into 50 four times again. You will notice that this creates a repeating cycle.
\[ 11 ) 5.00 - 44 -------- 60 - 55 -------- 50 - 44 -------- 60 \]
Continuing this process will show that:
- The decimal representation of \( \frac{5}{11} \) is \( 0.454545\ldots \), where "45" is the repeating part.
Thus, the decimal expansion of \( \frac{5}{11} \) is:
\[ 0.\overline{45} \]