To convert the mixed number \( 6 \frac{7}{15} \) into a decimal using long division, follow these steps:
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Convert the mixed number to an improper fraction. \[ 6 \frac{7}{15} = \frac{6 \times 15 + 7}{15} = \frac{90 + 7}{15} = \frac{97}{15} \]
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Perform long division of 97 by 15.
- 15 goes into 97 a total of 6 times (since \( 15 \times 6 = 90 \)).
- Subtract \( 90 \) from \( 97 \) which leaves a remainder of \( 7 \).
\[ 97 - 90 = 7 \]
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Continue the division:
- Now, we add a decimal point and a zero to the remainder, making it \( 70 \).
- 15 goes into 70 a total of 4 times (since \( 15 \times 4 = 60 \)).
- Subtract \( 60 \) from \( 70 \) which leaves a remainder of \( 10 \).
\[ 70 - 60 = 10 \]
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Continue again:
- Add another zero, making it \( 100 \).
- 15 goes into 100 a total of 6 times (since \( 15 \times 6 = 90 \)).
- Subtract \( 90 \) from \( 100 \) which leaves a remainder of \( 10 \).
\[ 100 - 90 = 10 \]
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This remainder of \( 10 \) will continue to produce results of \( 6 \) when we bring down another zero (it will result in \( 100 \) again). Therefore, the cycle starts repeating.
Thus, the decimal representation of \( 6 \frac{7}{15} \): \[ 6.46666...\text{ (repeats)} \] is approximately \( 6.47 \) when rounded to two decimal places.
The correct response from the options given is: \[ \approx 6.47 \]
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