(3 7/30- 1 5/12): 18 1/6

To solve the expression \((3 \frac{7}{30} - 1 \frac{5}{12}) : 18 \frac{1}{6}\), we first need to perform the subtraction and then the division.

Step 1: Convert mixed numbers to improper fractions

1. For \(3 \frac{7}{30}\): \[ 3 \times 30 + 7 = 90 + 7 = 97 \Rightarrow \frac{97}{30} \]

2. For \(1 \frac{5}{12}\): \[ 1 \times 12 + 5 = 12 + 5 = 17 \Rightarrow \frac{17}{12} \]

3. For \(18 \frac{1}{6}\): \[ 18 \times 6 + 1 = 108 + 1 = 109 \Rightarrow \frac{109}{6} \]

Step 2: Perform the subtraction

Now subtract \(\frac{17}{12}\) from \(\frac{97}{30}\).

To do this, we need a common denominator. The least common multiple of \(30\) and \(12\) is \(60\).

• Convert \(\frac{97}{30}\) to a denominator of \(60\): \[ \frac{97}{30} = \frac{97 \times 2}{30 \times 2} = \frac{194}{60} \]

• Convert \(\frac{17}{12}\) to a denominator of \(60\): \[ \frac{17}{12} = \frac{17 \times 5}{12 \times 5} = \frac{85}{60} \]

Now perform the subtraction: \[ \frac{194}{60} - \frac{85}{60} = \frac{194 - 85}{60} = \frac{109}{60} \]

Step 3: Perform the division

Next, we need to divide \(\frac{109}{60}\) by \(\frac{109}{6}\).

Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{109}{60} \div \frac{109}{6} = \frac{109}{60} \times \frac{6}{109} \]

The \(109\) cancels: \[ \frac{6}{60} \]

Simplifying \(\frac{6}{60}\): \[ \frac{6 \div 6}{60 \div 6} = \frac{1}{10} \]

Conclusion

Thus, the final answer is: \[ \frac{1}{10} \]