Question

A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

• A. 44 km/hr, 8 km/hr
• B. 6 km/hr, 4 km/hr
• C. 10 km/hr, 2 km/hr
• D. 14 km/hr, 5 km/hr

Answer 1

The correct option is C 10 km/hr, 2 km/hr

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr. Then,
speed upstream = (x - y) km/hr
speed downstream = (x + y) km/hr

Now, time taken to cover 32 km upstream
$=\frac{32}{x-y}hr$

Time taken to cover 36 km downstream
$=\frac{36}{x+y}hrs$

$\therefore \frac{32}{x-y}+\frac{36}{x+y}=7...\left(1\right)$

Time taken to cover 40km upstream
$=\frac{40}{x-y}$

Time taken to cover 48 km downstream
$=\frac{48}{x+y}$

$\therefore \frac{40}{x-y}+\frac{48}{x+y}=9...\left(2\right)$

Putting $\frac{1}{x-y}=uand\frac{1}{x+y}=v$ in equations (1) and (2), we get

32u + 36v = 7 $\Rightarrow$ 32u + 36v - 7 = 0 ...(3)
40u + 48v = 9 $\Rightarrow$ 40u + 48v - 9 = 0 ...(4)
By cross - multiplication, we get

$\frac{u}{36\times (-9)-48\times (-7)}=\frac{-v}{32\times (-9)-40\times (-7)}=\frac{1}{32\times 48-40\times 36}\Rightarrow \frac{u}{-324+336}=\frac{-v}{-288+280}=\frac{1}{1536-1440}\Rightarrow \frac{u}{12}=\frac{v}{8}=\frac{1}{96}\Rightarrow u=\frac{12}{96}andv=\frac{8}{96}\Rightarrow u=\frac{1}{8}andv=\frac{1}{12}$

$Now,u=\frac{1}{8}\Rightarrow \frac{1}{x-y}=\frac{1}{8}\Rightarrow x-y=8$ ...(5)

and, $v=\frac{1}{12}\Rightarrow \frac{1}{x+y}=\frac{1}{12}\Rightarrow x+y=12$ ...(6)

On solving equations (5) and (6), we get
2x = 20
$\Rightarrow$ x = 10;
On substituting x = 10 in equation (5), we get
10 - y = 8
$\Rightarrow$ y = 10 - 8 = 2
Hence, speed of the boat in still water is 10km/hr and speed of the stream is 2 km/hr.