To convert \( 6 \frac{7}{20} \) to a decimal using long division, we can first convert the mixed number into an improper fraction.
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Convert \( 6 \frac{7}{20} \) to an improper fraction: \[ 6 \frac{7}{20} = \frac{(6 \times 20) + 7}{20} = \frac{120 + 7}{20} = \frac{127}{20} \]
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Now, we will divide 127 by 20 using long division:
- 20 goes into 127 six times (since \( 20 \times 6 = 120 \)).
- Subtract 120 from 127 to find the remainder: \[ 127 - 120 = 7 \]
- Now, we have \( 6 \) as the whole number, and the remainder is \( 7 \). To continue with decimal places, we can add a decimal point and a zero (making it 70) to our dividend:
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Divide 70 by 20:
- 20 goes into 70 three times (since \( 20 \times 3 = 60 \)).
- Subtract 60 from 70: \[ 70 - 60 = 10 \]
- Add another zero to get 100.
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Divide 100 by 20:
- 20 goes into 100 five times (since \( 20 \times 5 = 100 \)).
- Subtract 100 from 100: \[ 100 - 100 = 0 \]
Putting this all together, we find: \[ 6 \frac{7}{20} = 6.35 \]
So, the final answer is 6.35.