To solve the given expression, first perform the division of the fractions, and then add the result to the other fraction. Simplify the final result to its simplest form.
To solve the expression \(\frac{7}{8} + \frac{1}{4} \div \frac{6}{7}\), we first perform the division of the fractions \(\frac{1}{4}\) and \(\frac{6}{7}\). This is done by multiplying \(\frac{1}{4}\) by the reciprocal of \(\frac{6}{7}\):
\[
\frac{1}{4} \div \frac{6}{7} = \frac{1}{4} \times \frac{7}{6} = \frac{7}{24}
\]
Next, we add the result of the division \(\frac{7}{24}\) to the fraction \(\frac{7}{8}\):
\[
\frac{7}{8} + \frac{7}{24}
\]
To add these fractions, we need a common denominator. The least common multiple of 8 and 24 is 24. Convert \(\frac{7}{8}\) to have a denominator of 24:
\[
\frac{7}{8} = \frac{21}{24}
\]
Now, add the fractions:
\[
\frac{21}{24} + \frac{7}{24} = \frac{28}{24}
\]
Simplify \(\frac{28}{24}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
\[
\frac{28}{24} = \frac{7}{6}
\]
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