Answer: Option D. -> $$1:29 \times {10^6}$$
5 cm on the map represents 1450 km
∴ 1 cm on the map represents $$\left( {\frac{{1450}}{5}} \right)$$  km = 290 km
Hence, the scale is 1 cm : 290 km i.e., $$1:29 \times {10^6}{\text{ }}$$
$$\left[ {\because 290{\text{km}} = \left( {29 \times {{10}^6}} \right){\text{cm}}} \right]$$

Answer: Option B. -> 12.5 m
Let the height of the building be x metres
Lees lengthy shadow, Less is the height (Direct proportion)
$$\eqalign{
& \therefore 40.25:28.75::17.5:x \cr
& \Leftrightarrow 40.25 \times x = 28.75 \times 17.5 \cr
& \Leftrightarrow x = \frac{{\left( {28.75 \times 17.5} \right)}}{{40.25}} \cr
& \Leftrightarrow x = 12.5 \cr} $$

Answer: Option D. -> 16
Let the required number of mats be x
More weavers , More mates ( Direct proportion )
More days, More mats ( Direct proportion )
\[\left. \begin{gathered}
{\text{Weavers }}4:8 \hfill \\
\,\,\,\,\,\,\,\,{\text{Days }}4:8 \hfill \\
\end{gathered} \right\}::4:x\]
$$\eqalign{
& \therefore 4 \times 4 \times x = 8 \times 8 \times 4 \cr
& \Leftrightarrow x = \frac{{\left( {8 \times 8 \times 4} \right)}}{{\left( {4 \times 4} \right)}} \cr
& \Leftrightarrow x = 16 \cr} $$

Answer: Option D. -> $${\text{7 hours}}$$
Since each maid would work with one mop,
So,we shall consider 1 maid and 1 mop as 1 unit
Let the required time be x hours
Less maids and mops, More time (Indirect proportion)
Less floor, Less time (Direct proportion)
\[\left. \begin{gathered}
{\text{Maids & Mops 3}}:7 \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Floors 7}}:3 \hfill \\
\end{gathered} \right\}::7:x\]
$$\eqalign{
& \therefore {\text{ }}3 \times 7 \times x = 7 \times 3 \times 7 \cr
& \Leftrightarrow x = \frac{{\left( {7 \times 3 \times 7} \right)}}{{\left( {3 \times 7} \right)}} \cr
& \Leftrightarrow x = 7 \cr} $$

Answer: Option A. -> 4 hours
Since each gardener would work with one grass mower,
So, we shall consider 1 gardener and 1 grass mower as one unit.
Let the required time be x hours
More gardeners and grass mowers, Less time (Indirect proportion)
More area, More time (Direct proportion)
\[\left. \begin{gathered}
{\text{Gardeners & grass mowers 8}}:4 \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Area 400}}:800 \hfill \\
\end{gathered} \right\}::4:x\]
$$\eqalign{
& \therefore {\text{ }}8 \times 400 \times x = 4 \times 800 \times 4 \cr
& \Leftrightarrow x = \frac{{\left( {4 \times 800 \times 4} \right)}}{{\left( {8 \times 400} \right)}} \cr
& \Leftrightarrow x = 4 \cr} $$

Answer: Option A. -> 225
Let the required number of bottles be x
More machines, More bottles produced (Direct proportion)
Less time, Less bottles produce (Direct proportion)
\[\left. \begin{gathered}
{\text{Machines 6}}:15 \hfill \\
\,\,\,\,\,\,{\text{Times 60}}:30 \hfill \\
\end{gathered} \right\}::180:x\]
$$\eqalign{
& \therefore {\text{ }}6 \times 60 \times x = 15 \times 30 \times 180 \cr
& \Leftrightarrow x = \frac{{\left( {15 \times 30 \times 180} \right)}}{{\left( {6 \times 60} \right)}} \cr
& \Leftrightarrow x = 225 \cr} $$

Answer: Option B. -> Rs. 9450
Let the weekly earning be Rs. xMore persons , More earning (Direct proportion)
Less working hours, Less earning (Direct proportion)
\[\left. \begin{gathered}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons 6}}:9 \hfill \\
{\text{Working hours 8}}:6 \hfill \\
\end{gathered} \right\}::8400:x\]
$$\eqalign{
& \therefore {\text{ }}6 \times 8 \times x = 9 \times 6 \times 8400 \cr
& \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 8400} \right)}}{{\left( {6 \times 8} \right)}} \cr
& \Leftrightarrow x = 9450 \cr} $$

Answer: Option D. -> 160 kg
Let the required quantity be x kg.
More workers, More quantity (Direct proportion)
More days, More quantity (Direct proportion)
\[\left. \begin{gathered}
{\text{Workers 5}}:8 \hfill \\
\,\,\,\,\,\,\,\,{\text{Days 3}}:5 \hfill \\
\end{gathered} \right\}::60:x\]
$$\eqalign{
& \therefore {\text{ }}5 \times 3 \times x = 8 \times 5 \times 60 \cr
& \Leftrightarrow x = \frac{{\left( {8 \times 5 \times 60} \right)}}{{\left( {5 \times 3} \right)}} \cr
& \Leftrightarrow x = 160 \cr} $$

Answer: Option E. -> None of these
Let the required number of days be x
Less people, More days (Indirect proportion)
Less quantity, Less days ( Direct proportion)
\[\left. \begin{gathered}
\,\,\,\,\,\,\,{\text{People 35}}:50 \hfill \\
{\text{Quantity 350}}:50 \hfill \\
\end{gathered} \right\}::30:x\]
$$\eqalign{
& \therefore {\text{ }}35 \times 350 \times x = 50 \times 50 \times 30 \cr
& \Leftrightarrow x = \frac{{\left( {50 \times 50 \times 30} \right)}}{{\left( {35 \times 350} \right)}} \cr
& \Leftrightarrow x = \frac{{300}}{{49}} \cr
& \Leftrightarrow x = 6\frac{6}{{49}} \cr} $$

Answer: Option B. -> 8 hours
Let the number of working hours per day be x
More persons, Less working hours ( Indirect proportion)
Less days, More working hours (Indirect proportion)
\[\left. \begin{gathered}
{\text{Persons 7}}:4 \hfill \\
{\text{Quantity 4}}:8 \hfill \\
\end{gathered} \right\}::7:x\]
$$\eqalign{
& \therefore {\text{ }}7 \times 4 \times x = 4 \times 8 \times 7 \cr
& \Leftrightarrow x = \frac{{\left( {4 \times 8 \times 7} \right)}}{{\left( {7 \times 4} \right)}} \cr
& \Leftrightarrow x = 8 \cr} $$