To convert the fraction \( \frac{7}{13} \) into a decimal, we can perform the long division of 7 by 13.
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13 goes into 70 five times (since \( 13 \times 5 = 65 \)).
- Subtract \( 65 \) from \( 70 \): \[ 70 - 65 = 5 \]
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Bring down a 0, making it 50. 13 goes into 50 three times (since \( 13 \times 3 = 39 \)).
- Subtract \( 39 \) from \( 50 \): \[ 50 - 39 = 11 \]
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Bring down another 0, making it 110. 13 goes into 110 eight times (since \( 13 \times 8 = 104 \)).
- Subtract \( 104 \) from \( 110 \): \[ 110 - 104 = 6 \]
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Bring down another 0, making it 60. 13 goes into 60 four times (since \( 13 \times 4 = 52 \)).
- Subtract \( 52 \) from \( 60 \): \[ 60 - 52 = 8 \]
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Bring down another 0, making it 80. 13 goes into 80 six times (since \( 13 \times 6 = 78 \)).
- Subtract \( 78 \) from \( 80 \): \[ 80 - 78 = 2 \]
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Bring down another 0, making it 20. 13 goes into 20 one time (since \( 13 \times 1 = 13 \)).
- Subtract \( 13 \) from \( 20 \): \[ 20 - 13 = 7 \]
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Now we are back to the remainder of 7, which will start the cycle over again.
The decimal representation of \( \frac{7}{13} \) is therefore:
\[ 0.\overline{538461} \]
So, \( \frac{7}{13} = 0.\overline{538461} \).