A motor boat can travel at 15km/hr in still water. It traveled 63 km downstream in a river and then returned taking altogether 10 hr. What is the rate of flow of the river?

The boat travels 63 km downstream and then 63 km upstream. The total journey time is 10 hours.

Let the river flow at $x$ km/hr.

Downstream speed $= 15 + x$ km/hr

Upstream speed $= 15 - x$ km/hr

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Time taken downstream $= \dfrac{63}{15 + x}$ hours

Time taken upstream $= \dfrac{63}{15 - x}$ hours

Total time $= 10$ hours

$\dfrac{63}{15 + x} + \dfrac{63}{15 - x} = 10$

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$63\left[\dfrac{1}{15 + x} + \dfrac{1}{15 - x}\right] = 10$

$63\left[\dfrac{15 - x + 15 + x}{(15 + x)(15 - x)}\right] = 10$

$63\left[\dfrac{30}{225 - x^2}\right] = 10$

$\dfrac{1890}{225 - x^2} = 10$

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$1890 = 10(225 - x^2)$

$1890 = 2250 - 10x^2$

$10x^2 = 2250 - 1890$

$10x^2 = 360$

$x^2 = 36$

$x = 6$

Therefore, the rate of flow of the river is 6 km/hr.