Question

a boy covers half of his journey with a uniform speed of 'u' and other half with a uniform speed of 'v'.What is average speed for the whole journey?

Answer 1

$$\begin{gathered} \mathrm{Let} \\ \mathrm{Total}\mathrm{distance}\mathrm{of}\mathrm{journey}=2\mathrm{x} \\ \mathbf{First}\mathbf{}\mathbf{half}\mathbf{}\mathbf{of}\mathbf{}\mathbf{journey}\mathbf{:} \\ \mathrm{Distance}=\mathrm{x} \\ \mathrm{Time}\mathrm{taken},{\mathrm{t}}_1=\frac{\mathrm{x}}{\mathrm{u}} \\ \mathbf{Second}\mathbf{}\mathbf{half}\mathbf{}\mathbf{of}\mathbf{}\mathbf{journey}\mathbf{:} \\ \mathrm{Distance}=\mathrm{x} \\ \mathrm{Time}\mathrm{taken},{\mathrm{t}}_2=\frac{\mathrm{x}}{\mathrm{v}} \\ \mathbf{Average}\mathbf{}\mathbf{speed}\mathbf{:} \\ \mathrm{v}=\frac{\mathrm{Total}\mathrm{distance}}{\mathrm{total}\mathrm{time}} \\ =\frac{2\mathrm{x}}{\frac{\mathrm{x}}{\mathrm{u}}+\frac{\mathrm{x}}{\mathrm{v}}} \\ =\frac{2\mathrm{x}}{\mathrm{x}\left({\displaystyle \frac{\mathrm{v}+\mathrm{u}}{\mathrm{uv}}}\right)} \\ \mathbf{=}\frac{\mathbf{2}\mathbf{uv}}{\mathbf{u}\mathbf{+}\mathbf{v}} \end{gathered}$$