Let’s solve the expressions step by step.

For the first expression: (9 7/9 - 1 5/18) : 3.06

Convert mixed numbers to improper fractions:

For (9 \frac{7}{9}):
[
9 \frac{7}{9} = \frac{9 \times 9 + 7}{9} = \frac{81 + 7}{9} = \frac{88}{9}
]For (1 \frac{5}{18}):
[
1 \frac{5}{18} = \frac{1 \times 18 + 5}{18} = \frac{18 + 5}{18} = \frac{23}{18}
]

Subtract the two fractions:

Find a common denominator. The least common multiple of 9 and 18 is 18.
[
\frac{88}{9} = \frac{88 \times 2}{9 \times 2} = \frac{176}{18}
]
Now we can subtract:
[
\frac{176}{18} - \frac{23}{18} = \frac{176 - 23}{18} = \frac{153}{18}
]

Now divide by 3.06: [
\text{Divide } \frac{153}{18} \text{ by } 3.06 \text{ (which is the same as multiplying by }\frac{1}{3.06})
]

To proceed, convert 3.06 to a fraction.
(3.06 = \frac{306}{100} = \frac{153}{50}) (dividing by 3)

Now:
[
\frac{153}{18} \div 3.06 = \frac{153}{18} \times \frac{50}{153} = \frac{50}{18}
]

Simplify ( \frac{50}{18} ): [
\frac{50}{18} = \frac{25}{9}
]

For the second expression: (1 \frac{9}{14} + \frac{10}{21} \cdot 12.3)

Convert mixed numbers to improper fractions:

For (1 \frac{9}{14}):
[
1 \frac{9}{14} = \frac{1 \times 14 + 9}{14} = \frac{14 + 9}{14} = \frac{23}{14}
]

Calculate (\frac{10}{21} \cdot 12.3):

Convert (12.3) into a fraction: (12.3 = \frac{123}{10})Now multiply:
[
\frac{10}{21} \cdot \frac{123}{10} = \frac{123}{21}
]

Add (\frac{23}{14}) and (\frac{123}{21}):

Find a common denominator for 14 and 21. The least common multiple is 42.Convert both fractions:
[
\frac{23}{14} = \frac{23 \times 3}{14 \times 3} = \frac{69}{42}
]
[
\frac{123}{21} = \frac{123 \times 2}{21 \times 2} = \frac{246}{42}
]Now add the two fractions:
[
\frac{69}{42} + \frac{246}{42} = \frac{69 + 246}{42} = \frac{315}{42}
]

Simplify the sum ( \frac{315}{42} ):

Divide both numerator and denominator by 21:
[
\frac{315 \div 21}{42 \div 21} = \frac{15}{2}
]Final Results:For the first expression, the result is ( \frac{25}{9} ).For the second expression, the result is ( \frac{15}{2} ) or ( 7 \frac{1}{2} ).