To determine what fraction of the book James read each night, we start with the information given:
- James read a total of [tex]\(\frac{6}{7}\)[/tex] of the book over 8 nights.
We need to find the fraction of the book he read each night. To do this, we divide the total fraction of the book he read by the number of nights he spent reading:
[tex]\[
\text{Fraction read per night} = \frac{\frac{6}{7}}{8}
\][/tex]
Dividing a fraction by a whole number involves multiplying the fraction by the reciprocal of the whole number. So:
[tex]\[
\frac{\frac{6}{7}}{8} = \frac{6}{7} \times \frac{1}{8}
\][/tex]
Multiplying these fractions together:
[tex]\[
\frac{6 \times 1}{7 \times 8} = \frac{6}{56}
\][/tex]
Next, we simplify [tex]\(\frac{6}{56}\)[/tex] by finding the greatest common divisor (GCD) of 6 and 56. The GCD is 2, so we divide both the numerator and the denominator by 2:
[tex]\[
\frac{6 \div 2}{56 \div 2} = \frac{3}{28}
\][/tex]
Therefore, the fraction of the book that James read each night is:
[tex]\[
\frac{3}{28}
\][/tex]
From the given options, the correct answer is:
[tex]\[
\boxed{\frac{3}{28}}
\][/tex]
