To simplify the expression \( \frac{6}{7} + \frac{3}{4} \times \frac{7}{30} \), we first handle the multiplication:
\[ \frac{3}{4} \times \frac{7}{30} = \frac{3 \times 7}{4 \times 30} = \frac{21}{120} \]
Next, we can simplify \( \frac{21}{120} \) by finding the greatest common divisor (GCD) of 21 and 120, which is 3:
\[ \frac{21 \div 3}{120 \div 3} = \frac{7}{40} \]
Now we go back to the original expression and substitute the simplified multiplication:
\[ \frac{6}{7} + \frac{7}{40} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 40 is 280.
Now we convert each fraction:
\[ \frac{6}{7} = \frac{6 \times 40}{7 \times 40} = \frac{240}{280} \]
\[ \frac{7}{40} = \frac{7 \times 7}{40 \times 7} = \frac{49}{280} \]
Now we can add these fractions:
\[ \frac{240}{280} + \frac{49}{280} = \frac{240 + 49}{280} = \frac{289}{280} \]
The fraction \( \frac{289}{280} \) cannot be simplified further since 289 is not divisible by 280.
Thus, the final simplified result of the expression \( \frac{6}{7} + \frac{3}{4} \times \frac{7}{30} \) is:
\[ \frac{289}{280} \]
or as a mixed number:
\[ 1 \frac{9}{280} \]
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