Solution:
Ratio of times taken by Kamal and Bimal
$$\eqalign{ = 150:100 \cr = 3:2 \cr} $$
Suppose Bimal takes x days to do the work
$$\eqalign{ \Rightarrow 3:2::15:x \cr \Rightarrow x = \left( {\frac{{2 \times 15}}{3}} \right) \cr \Rightarrow x = 10{\text{ days}}{\text{.}} \cr} $$

Solution:
According to the question,
Let the remaining work is complete in D days
$${\text{4}}{{\text{5}}_{{\text{men}}}} \times {\text{1}}{{\text{6}}_{{\text{days}}}}$$    = $$\left( {{\text{4}}{{\text{5}}_{{\text{men}}}} \times {{\text{4}}_{{\text{days}}}}} \right)$$    + $${\left( {{\text{45}} + {\text{36}}} \right)_{{\text{men}}}}$$   $$ \times {\text{D}}$$
$$\eqalign{ \Rightarrow \frac{{540}}{{81}} = {\text{D}} \cr \Rightarrow {\text{D}} = 6\frac{2}{3}{\text{ days}} \cr} $$

Solution:
Ratio of time, taken by A and B = 160 : 100 = 8 : 5
If A takes 8 days, B takes 5 days
If A takes 12 days,
B takes $${\text{ = }}\frac{5}{8} \times 12 = \frac{{15}}{2} = 7\frac{1}{2}$$     days

Solution:
$$\eqalign{ \left( {3m + 8w} \right) \times \frac{{75}}{8} = \left( {9m + 12w} \right) \times \frac{{25}}{7} \cr 63m + 168w = 72m + 96w \cr 72w = 9m \cr \frac{m}{w} = \frac{8}{1} \cr {\text{Total work}} = \left( {3m + 8w} \right) \times \frac{{75}}{8} \cr = \left( {3 \times 8 + 8 \times 1} \right) \times \frac{{75}}{8} \cr = 4 \times 75 \cr {\text{Time taken by }}15w = \frac{{4 \times 75}}{{15 \times 1}} = 20{\text{ day}} \cr} $$

Solution:
$$\eqalign{ {\text{1 men's 1 day's work}} \cr = \frac{1}{{21 \times 3}} \cr = \frac{1}{{63}} \cr {\text{1 boy's 1 day's work}} \cr = \frac{1}{{21 \times 9}} \cr = \frac{1}{{189}} \cr \left( {{\text{5 men}} + {\text{6 boy}}} \right){\text{'s 1 day's work}} \cr = \frac{5}{{63}} + \frac{6}{{189}} \cr = \frac{5}{{63}} + \frac{2}{{63}} \cr = \frac{7}{{63}} \cr = \frac{1}{9} \cr} $$
Hence, 5 men's and 6 boy's together can do the work in 9 days.

Solution:
Ratio of times taken by A and B
$$\eqalign{ = 100:80 \cr = 5:4 \cr} $$
Suppose B takes x hours to do the work
$$\eqalign{ \Rightarrow 5:4::\frac{{15}}{2}:x \cr \Rightarrow x = \left( {\frac{{4 \times 15}}{{2 \times 5}}} \right) \cr \Rightarrow x = 6{\text{ hours}}{\text{}} \cr} $$

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