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Q.
A motor boat covers the distance between two spots on the river in $t_{1}=8\, hr$ and $t _{2}=12\, hr$ downstream and upstream respectively. The time required for the boat to cover this distance in still water will be-
Solution:
Let $s$ be the distance between that two spots.
Also assume that the velocity of the motor
boat in still water is $v$ and the velocity of flow of water is $u$.
Then, for downward journey,
$s / t_{1}=v+u\,\,\,...(A)$
For upward journey,
$s / t_{2}=v-u\,\,\,...(B)$
Adding eq. (A) to (B),
$s / t_{1}+s / t_{2}=2 v$
or $t=\frac{s}{v}=\frac{2 t_{1} t_{2}}{\left(t_{1}+t_{2}\right)}=\frac{2 \times 8 \times 12}{(8+12)} $
$=9.6 \,hr$