a) $A = \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}}$ $ = \frac{1}{{4 \times 5}} + \frac{1}{{5 \times 6}} + \frac{1}{{6 \times 7}} + \frac{1}{{7 \times 8}} + \frac{1}{{8 \times 9}} + \frac{1}{{9 \times 10}}$ $ = \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} + \frac{1}{6} - \frac{1}{7} + \frac{1}{7} - \frac{1}{8} + \frac{1}{8} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{10}}$ $ = \frac{1}{4} - \frac{1}{{10}} = \frac{3}{{20}}$   b) B = $\frac{3}{{1 \times 3}} + \frac{3}{{3 \times 5}} + \frac{3}{{5 \times 7}} + ..... + \frac{3}{{99 \times 101}}$ $ = \frac{3}{2} \times \left( {\frac{2}{{1 \times 3}} + \frac{2}{{3 \times 5}} + \frac{2}{{5 \times 7}} + .... + \frac{2}{{99 \times 101}}} \right)$ $ = \frac{3}{2} \times \left( {1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + .... + \frac{1}{{99}} - \frac{1}{{101}}} \right)$ $ = \frac{3}{2} \times \left( {1 - \frac{1}{{101}}} \right)$ $ = \frac{3}{2} \times \frac{{100}}{{101}} = \frac{{150}}{{101}}$