The denominator of a non-zero ratio is one less than twice the numerator. If the sum of the ratio and its reciprocal is $2 \frac{4}{15},$ find the ratio.

$A$ motor boat whose speed is $18\, km/h$ in still water takes $1\, hour$ more to go $24\, km$ upstream than to return downstream to the same spot. Find the speed of the stream. (in $ km/h$)

The sum of two numbers is $15$. If the sum of their reciprocals is $\frac{3}{10}$,then find the numbers.

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$AB = 2x + 3$,$BC = x + 2$,and $AC = 3x - 1$. Find the value of $x$.

For going to a city $B$ from city $A$,there is a route via city $C$ such that $AC \perp CB$,$AC = 2x \, km$ and $CB = 2(x+7) \, km$. It is proposed to construct a $26 \, km$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway (in $km$).

The speed of a motorboat in still water is $9\,km/h$. In a river,it covers $12\,km$ downstream and the same distance upstream in a total time of $3\,hours$. Find the speed of the current of the river. (The speed of the current of the river is less than the speed of the motorboat in still water.) (in $,km/h$)