Question

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Tommy rides in a motorboat against a river current for 25 km. Then he returns to his starting point by floating down river on a raft. Tommy travels 10 hours less on the motorboat than on the raft. Find the speed of the river current if the speed of the motorboat in still water is 12 km/h.

Answer 1

The speed of current is $\rm 2 \:km/h$.

What is the relation between distance,speed and time?

The relationship between distance, time and speed can be presented as:

$\begin{aligned} \rm Distance &= Speed \times Time\\\\Speed &= \dfrac{\rm Distance}{Time} \\\\\rm Time &= \dfrac{\rm Distance}{\rm Speed} \end$

Calculations:

Given:

$\begin{aligned} \rm Distance\: traveled\: on\: motorboat &= 25\:km\\\rm Speed\:of\:motorboat\:in\:still\:water &= 12\:km/h \end$  

 

Let the speed of the current be $\rm x$

 

The speed of the boat traveling against the current is ${(12 -\rm x) km/h}$

 

Time spent on motorboat is 10 hours less than time spent on raft.

Therefore:

$\rm \dfrac{25km}{x\:km/h} - \dfrac{25km}{(12 - x)km/h} = 10$

The equation obtained will be:

$\dfrac{25}{\rm x} - \dfrac{25}{(12-\rm x)} - = 10$

 

Multiplying both sides of the equation by $\rm x(12 - x )$

$\begin{aligned} \rm 25(12 - x) - 25x &= 10\times \rm x(12 - x)\\\\300 - 25x - 25x &= 120x - 10x^2\\\\300 - 50x &= 120x - 10x^2\\\\10x^2 - 120x - 50x + 300 &= 0\\\\10x^2 - 170x +300 &= 0\end$

Dividing the equation by 10

$\rm x^2 - 17x + 30 = 0$

On factoring the equation,

$\rm =x^2 - 2x -15x + 30 = 0\\\\=x(x - 2) -15(x - 2)\\\\=(x - 2)(x - 15)$

The values of x is 2 and 15.

Since speed of current i.e  $(12 -\rm x)km/h$ cannot be negative, $\rm x = 2$ is the only acceptable value.

 Therefore the speed of current is 2 km/h.

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