a man walks from A to B at the rate of 3km/hr., rests for half an hour and then returns to A at the rate of 5km/hr. if the whole journey takes 2hrs. and 20 mins., find the distance between A and B ?

Dear Student,

Please find below the solution to the asked query:

Given : a man walks from A to B at the rate of 3 km/hr., rests for half an hour and then returns to A at the rate of 5km/hr. if the whole journey takes 2hrs. and 20 mins., So 

Time taken in given journey =  2 hr + 20 minutes = 2 hr + $\frac{20}{60}$ hr = $\left(2 + \frac{1}{3}\right)\mathrm{hr} = \left(\frac{6 + 1}{3}\right)\mathrm{hr} = \frac{\mathbf{7}}{\mathbf{3}}\mathbf{ }\mathbf{hr}$( We know 1 hr  = 60 minutes )

Let distance between A and B = x 

And we know Time = $\frac{\mathrm{Distance}}{\mathrm{Speed}}$ , So from given condition we get :

$$\begin{gathered} \frac{x}{3} + \frac{1}{2} + \frac{x}{5} = \frac{7}{3} \\ \Rightarrow \frac{{\displaystyle x}}{{\displaystyle 3}} + \frac{{\displaystyle x}}{{\displaystyle 5}} = \frac{{\displaystyle 7}}{{\displaystyle 3}}- \frac{{\displaystyle 1}}{{\displaystyle 2}} \\ \Rightarrow \frac{{\displaystyle 5 x +\hspace{0.17em}3 x}}{{\displaystyle 15}} = \frac{{\displaystyle 14 - 3}}{{\displaystyle 6}} \\ \Rightarrow \frac{{\displaystyle 8 x}}{{\displaystyle 15}} = \frac{{\displaystyle 11}}{{\displaystyle 6}} \\ \Rightarrow x = \frac{{\displaystyle 11}}{{\displaystyle 6}} \times \frac{15}{8} \\ \Rightarrow x = \frac{{\displaystyle 165}}{{\displaystyle 48}} \\ \Rightarrow x =3.4375 = 3 \mathrm{km} + 0.4375 \mathrm{km} = 3 \mathrm{km} +\left( 0.4375 \times 1000\right)\mathrm{m} =\mathbf{ }\mathbf{3}\mathbf{ }\mathbf{km}\mathbf{ }\mathbf{+}\mathbf{437}\mathbf{.}\mathbf{5}\mathbf{m}\mathbf{ } \end{gathered}$$

Therefore,

Distance between A and B  =  3 km 437.5 m                                  ( Ans )


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