Pipes & Cistern - 01

Explanation:-

$$\eqalign{ & {\text{Net part filled in 1 hour}} \cr & {\text{ = }}\left( {\frac{1}{8} - \frac{1}{{12}}} \right) = \frac{1}{{24}} \cr & \therefore {\text{The tank will be filled in 24 hours}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{First pipe fill the tank in 1 hour}} \cr & {\text{ = }}\frac{1}{{12}}{\text{part of tank}} \cr & {\text{Second pipe fill the tank in 1 hour}} \cr & {\text{ = }}\frac{1}{{16}}{\text{part of tank}} \cr & {\text{Third pipe empty the tank in 1 hour}} \cr & {\text{ = }}\frac{1}{{30}}{\text{part of tank}} \cr & {\text{When all three pipes are opened simultaneously,}} \cr & {\text{part of the tank filled in 1 hour}} \cr & = \frac{1}{{12}} + \frac{1}{{16}} - \frac{1}{{30}} \cr & {\text{LCM of 12, 16, and 30 = 240}} \cr & {\text{ = }}\frac{{20 + 15 - 8}}{{240}} = \frac{{27}}{{240}} \cr & \therefore {\text{Required time taken by all the three pipes}} \cr & {\text{ = }}\frac{{240}}{{27}} = \frac{{80}}{9} = 8\frac{8}{9}\,{\text{Hours}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Rate of flow of water}} \cr & {\text{ = }}x{\text{ cm/minutes}} \cr & \therefore {\text{Valume of water that flowed in the in1 minutes}} \cr & {\text{ = 5}} \times {\text{4 }} \times x{\text{ = 20}}x\,\,{\text{cu}}{\text{.cm}} \cr & \therefore {\text{Valume of water that flowed in the in}} \cr & {\text{6 hours 18 minutes }} \cr & {\text{i}}{\text{.e}}{\text{. }}\left( {6 \times 60 + 18} \right) = 378\,\min \cr & = \,{\text{ 2}}x \times 378{\text{ cu}}{\text{.cm}} \cr & {\text{According to question, }} \cr & {\text{20}}x \times 378 = 700 \times 400 \times 450 \cr & \Rightarrow x = \left( {\frac{{700 \times 400 \times 450}}{{20 \times 378}}} \right){\text{cm /minutes}} \cr & \Rightarrow x = \left( {\frac{{700 \times 400 \times 450 \times 60}}{{100000 \times 20 \times 378}}} \right){\text{km/hours}} \cr & \Rightarrow x{\text{ = 10 km/hours}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Work done by the waste pipe in 1 minute }} \cr & {\text{ = }}\frac{1}{{15}} - \left( {\frac{1}{{20}} + \frac{1}{{24}}} \right) \cr & = \left( {\frac{1}{{15}} - \frac{{11}}{{120}}} \right) \cr & = - \frac{1}{{40}}\left[ { - \,{\text{Nagetive sign means emptying}}} \right] \cr & \therefore {\text{Volume of }}\frac{1}{{40}}{\text{ part = 3 galons}} \cr & {\text{Volume of whole tank}} \cr & {\text{ = }}\left( {3 \times 40} \right){\text{gallons}} \cr & {\text{ = 120 gallons}}{\text{.}} \cr}$$

Explanation:-

$$\eqalign{ & \left( {{\text{A + B}}} \right){\text{'s 1 hour work}} \cr & {\text{ = }}\left( {\frac{1}{{12}} + \frac{1}{{15}}} \right) = \frac{9}{{60}} = \frac{3}{{20}} \cr & \left( {{\text{A + C}}} \right){\text{'s 1 hour work}} \cr & {\text{ = }}\left( {\frac{1}{{12}} + \frac{1}{{20}}} \right) = \frac{8}{{60}} = \frac{2}{{15}} \cr & {\text{Part filled in 2 hrs}} \cr & {\text{ = }}\left( {\frac{3}{{20}} + \frac{2}{{15}}} \right) = \frac{{17}}{{60}}, \cr & {\text{Part filled in 6 hrs}} \cr & {\text{ = }}\left( {3 \times \frac{{17}}{{60}}} \right) = \frac{{17}}{{20}} \cr & {\text{Remaining part}} \cr & {\text{ = }}\left( {1 - \frac{{17}}{{20}}} \right) = \frac{3}{{20}} \cr & {\text{Now it is the turn of A and B and}} \cr & \frac{3}{{20}}{\text{ part is filled by A and B in 1 hour}}{\text{.}} \cr & \therefore {\text{Total time taken to fill tank}} \cr & {\text{ = }}\left( {6 + 1} \right){\text{hrs}} \cr & {\text{ = 7 hrs}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Part filled by (A + B) in 1 hour}} \cr & {\text{ = }}\left( {\frac{1}{{20}} + \frac{1}{{30}}} \right) = \frac{1}{{12}} \cr & {\text{So, A and B together can fill the}} \cr & {\text{tank in 12 hrs, }}\frac{1}{3}\,{\text{ part is filled by}} \cr & {\text{(A + B) in }}\left( {\frac{1}{3} \times 12} \right){\text{ = 4 hrs}} \cr & {\text{Since the leak empties one - third water,}} \cr & {\text{so time taken to fill the tank}} \cr & {\text{ = Time taken by (A + B) to fill the whole tank + }} \cr & {\text{ Time taken by (A + B) to fill one - third tank}} \cr & {\text{ = (12 + 4)}} \cr & {\text{ = 16 hrs}}{\text{.}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Capacity of the tank}} \cr & {\text{ = }}\left( {12 \times 13.5} \right){\text{litres = 162 litres}} \cr & {\text{Capacity of each bucket}} \cr & {\text{ = 9 litres}} \cr & {\text{Number of buckets needed}} \cr & {\text{ = }}\left( {\frac{{162}}{9}} \right) = 18 \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let the slower pipe alone }} \cr & {\text{fill the tank in }}x{\text{ minutes}} \cr & {\text{Then,}} \cr & {\text{Faster pipe alone will fill it}} \cr & {\text{ in }}\frac{x}{3}{\text{ minutes}} \cr & \therefore \frac{1}{x} + \frac{3}{x} = \frac{1}{{36}} \cr & \Rightarrow \frac{4}{x} = \frac{1}{{36}} \cr & \Rightarrow x = 144 \cr & {\text{So slower pipe alone will fill}} \cr & {\text{ the tank in 144 min}}{\text{.}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Work done by the leak in 1 hour}} \cr & {\text{ = }}\left( {\frac{1}{2} - \frac{3}{7}} \right) = \frac{1}{{14}} \cr & \therefore {\text{Leak will empty the tank in 14 hours }}{\text{.}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Net part filled in 1 hour}} \cr & = \frac{2}{3} - \left( {\frac{2}{9} + \frac{1}{3}} \right) \cr & = \left( {\frac{2}{3} - \frac{5}{9}} \right) \cr & = \frac{1}{9} \cr & \therefore \,\frac{1}{9}\,:\,\frac{1}{2}\,::\,1\,:\,x \cr & {\text{or}}\,\,\,x = \left( {\frac{1}{2} \times 9} \right) = 4\frac{1}{2}{\text{ hours}} \cr & {\text{So, the tank will be emptied in}} \cr & {\text{ = }}4\frac{1}{2}{\text{ hours}}. \cr}$$