3) \( \frac{7}{6} \div \frac{5}{6}-\left(\frac{1}{30}\right) \)
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The Deep Dive
To solve the expression \( \frac{7}{6} \div \frac{5}{6}-\left(\frac{1}{30}\right) \), we start by dividing the two fractions. Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{7}{6} \div \frac{5}{6} = \frac{7}{6} \times \frac{6}{5} = \frac{7 \times 6}{6 \times 5} = \frac{7}{5} \] Now we subtract \( \frac{1}{30} \) from \( \frac{7}{5} \). To do that, we need a common denominator. The least common multiple of 5 and 30 is 30. So we convert \( \frac{7}{5} \): \[ \frac{7}{5} = \frac{7 \times 6}{5 \times 6} = \frac{42}{30} \] Now we can subtract: \[ \frac{42}{30} - \frac{1}{30} = \frac{42 - 1}{30} = \frac{41}{30} \] Therefore, the final result is: \[ \frac{41}{30} \]