P can complete a work in 12 days working 8 hours a day. Q can complete the same work in 8 days working 10 hours a day. If both P and Q work together, working 8 hours a day, in how many days can they complete the work?

Explanation:-

$$\eqalign{ & {\text{P}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}} \cr & = \,\left( {12 \times 8} \right){\text{hrs}}{\text{.}} = 96\,{\text{hrs}}{\text{.}} \cr & {\text{Q}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}} \cr & = \left( {8 \times 10} \right){\text{hrs}}{\text{.}} = 80\,{\text{hrs}}{\text{.}} \cr & \therefore {\text{P's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{{96}}\,{\text{and}} \cr & \therefore {\text{Q's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{{80}} \cr & \left( {{\text{P + Q}}} \right){\text{'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = \left( {\frac{1}{{96}} + \frac{1}{{80}}} \right) = \frac{{11}}{{480}} \cr & {\text{So,}}\,{\text{both}}\,{\text{P}}\,{\text{and}}\,{\text{Q}}\,{\text{will}}\,{\text{finish}}\,{\text{the}}\,{\text{work}} \cr & = \left( {\frac{{480}}{{11}}} \right){\text{hrs}}{\text{.}} \cr & \therefore {\text{Number}}\,{\text{of}}\,{\text{days}}\,{\text{of}}\,{\text{8}}\,{\text{hours}}\,{\text{each}} \cr & \left( {\frac{{480}}{{11}} \times \frac{1}{8}} \right) = \frac{{60}}{{11}}{\text{days}} = 5\frac{5}{{11}}{\text{days}} \cr}$$