Solution:
Number of pages typed by Gorge in 1 hour
$$\eqalign{ = \frac{{50}}{8} \cr = \frac{{25}}{4} \cr} $$
Number of pages typed by Sonia in 1 hour
$$\eqalign{ = \frac{{50}}{6} \cr = \frac{{25}}{3} \cr} $$
Number of pages typed by Gorge and Sonia together in 1 hour
$$\eqalign{ = \left( {\frac{{25}}{4} + \frac{{25}}{3}} \right) \cr = \left( {\frac{{75 + 100}}{{12}}} \right) \cr = \frac{{175}}{{12}} \cr \therefore {\text{Required time}} \cr = \left( {100 \div \frac{{175}}{{12}}} \right){\text{hours}} \cr = \left( {\frac{{100 \times 12}}{{175}}} \right){\text{ hours}} \cr = \frac{{48}}{7}{\text{ hours}} \cr = 6\frac{6}{7}{\text{ hours}} \cr} $$

Solution:
$$\eqalign{ 1\,woman's\,1\,day's\,work = \frac{1}{{70}} \cr {\text{1}}\,{\text{child's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{140}} \cr \left( {{\text{5}}\,{\text{women + 10}}\,{\text{children}}} \right){\text{'s}}\,{\text{day's}}\,{\text{work}} \cr = {\frac{5}{{70}} + \frac{{10}}{{140}}} = {\frac{1}{{14}} + \frac{1}{{14}}} = \frac{1}{7} \cr \therefore {\text{5}}\,{\text{women}}\,{\text{and}}\,{\text{10}}\,{\text{chidren}}\,{\text{will}}\,{\text{complete}} \cr {\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{7}}\,{\text{days}} \cr} $$

Solution:
$$\eqalign{ \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr = {\frac{1}{{15}} + \frac{1}{{10}}} = \frac{1}{6} \cr {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{in}}\,{\text{2}}\,{\text{days}} \cr = {\frac{1}{6} \times 2} = \frac{1}{3} \cr {\text{Remaining}}\,{\text{work}} \cr = {1 - \frac{1}{3}} = \frac{2}{3} \cr {\text{Now}},\,\frac{1}{{15}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr \therefore \frac{2}{3}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{a}}\,{\text{in}} \cr {15 \times \frac{2}{3}} = 10\,{\text{days}} \cr {\text{Hence,}}\,{\text{the}}\,{\text{total}}\,{\text{time}}\,{\text{taken}} \cr = {10 + 2} = 12\,{\text{days}} \cr} $$

Solution:
$$\eqalign{ {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{Ravi}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr = \frac{{32}}{6} = \frac{{16}}{3} \cr {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{Kumar}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr = \frac{{40}}{5} = 8 \cr {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{both}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr = {\frac{{16}}{3} + 8} = \frac{{40}}{3} \cr \therefore {\text{Time}}\,{\text{taken}}\,{\text{by}}\,{\text{both}}\,{\text{to}}\,{\text{type}}\,{\text{110}}\,{\text{pages}} \cr = {110 \times \frac{3}{{40}}} {\text{hours}} \cr = 8\frac{1}{4}\,{\text{hours}}\,{\text{(or)}}\,{\text{8}}\,{\text{hours}}\,{\text{15}}\,{\text{minutes}} \cr} $$

Solution:

7 days work of (A and B) = 7 × 10 = 70
Remaining work = 105 - 70 = 35
60% of remaining work done by A = $$\frac{{35 \times \frac{3}{5}}}{3}$$  = 7 days

Solution:
Let the time taken by the three friends together to do the work be x hours
Then, time taken by Anne alone = (x + 6) hours
Time taken by Bob alone = (x + 1) hours
Time taken by Bob alone = 2x hours
$$\therefore \frac{1}{{x + 6}} + \frac{1}{{x + 1}} + \frac{1}{{2x}} = \frac{1}{x}$$
$$ \Rightarrow $$ $$\frac{{2x\left( {x + 1} \right) + 2x\left( {x + 6} \right) + \left( {x + 1} \right)\left( {x + 6} \right)}}{{2x\left( {x + 6} \right)\left( {x + 1} \right)}}$$         $$ = $$ $$\frac{1}{x}$$
$$\eqalign{ \Rightarrow 5{x^2} + 21x + 6 = 2\left( {{x^2} + 7x + 6} \right) \cr \Rightarrow 3{x^2} + 7x - 6 = 0 \cr \Rightarrow \left( {x + 3} \right)\left( {3x - 2} \right) = 0 \cr \Rightarrow x = \frac{2}{3}{\text{ }}\left[ {\because x \ne - 3} \right] \cr \therefore {\text{Required times}} \cr = \frac{2}{3}{\text{ hours}} \cr = \left( {\frac{2}{3} \times 60} \right){\text{hours}} \cr = 40{\text{ minutes}} \cr} $$

Solution:
1^st method:
A and B complete a work in = 15 days
One day's work of (A + B) = $$\frac{1}{{15}}$$
B complete the work in = 20 days;
One day's work of B = $$\frac{1}{{20}}$$
Then, A's one day's work
$$\eqalign{ = \frac{1}{{15}} - \frac{1}{{20}} \cr = \frac{{4 - 3}}{6} \cr = \frac{1}{{60}} \cr} $$
Thus, A can complete the work in = 60 days.

2^nd method:
(A + B)'s one day's % work = $$\frac{{100}}{{15}}$$ = 6.66%
B's one day's % work = $$\frac{{100}}{{20}}$$ = 5%
A's one day's % work = 6.66 - 5 = 1.66%
Thus, A need = $$\frac{{100}}{{1.66}}$$ = 60 days to complete the work.