To solve the given expression quickly, we need to simplify the sum of fractions:

\[ A = \frac{1}{2} + \frac{5}{6} + \frac{11}{12} + \frac{19}{20} + \frac{29}{30} + \frac{41}{42} + \frac{55}{56} + \frac{71}{72} + \frac{89}{90} + \frac{109}{110} \]

Notice that each fraction is of the form \(\frac{n(n+1)}{n(n+2)}\), which simplifies to:

\[ \frac{n(n+1)}{n(n+2)} = \frac{n+1}{n+2} \]

So, the expression becomes:

\[ A = \frac{1+1}{2+2} + \frac{5+1}{6+2} + \frac{11+1}{12+2} + \frac{19+1}{20+2} + \frac{29+1}{30+2} + \frac{41+1}{42+2} + \frac{55+1}{56+2} + \frac{71+1}{72+2} + \frac{89+1}{90+2} + \frac{109+1}{110+2} \]

Simplifying each term:

\[ A = \frac{2}{4} + \frac{6}{8} + \frac{12}{14} + \frac{20}{22} + \frac{30}{32} + \frac{42}{44} + \frac{56}{58} + \frac{72}{74} + \frac{90}{92} + \frac{110}{112} \]

Further simplification:

\[ A = \frac{1}{2} + \frac{3}{4} + \frac{6}{7} + \frac{10}{11} + \frac{15}{16} + \frac{21}{22} + \frac{28}{29} + \frac{36}{37} + \frac{45}{46} + \frac{55}{56} \]

Now, summing these fractions directly might be complex, but we can approximate or use a calculator for the exact sum.

\[ A \approx 0.5 + 0.75 + 0.857 + 0.909 + 0.9375 + 0.9545 + 0.9655 + 0.973 + 0.978 + 0.982 \]

Adding these approximate values:

\[ A \approx 0.5 + 0.75 + 0.857 + 0.909 + 0.9375 + 0.9545 + 0.9655 + 0.973 + 0.978 + 0.982 \approx 8.8065 \]

Thus, the approximate value of \( A \) is \( 8.8065 \). For an exact value, you would need to sum the fractions precisely.