Question

A boatman rows his boat 35 km upstream and 55 km down stream in 12 hours. he can row 30 km. upstream and 44 km downstream in 10 hours. find the speed of he stream and that of the boat in still water. hence find the total time taken by the boat man to row 50 cm upstream and 77 km downstream

Answer 1

The answer is attached

Answer 2

Speed of the ‘boat’ in still water = 8 Km/hr & Speed of the ‘stream’ = 3 km/hr.  

Total time taken by boat man to row 77 km downstream and 50 km upstream = 17 hours  

Solution:

Given: A boatman rows his boat 55 km downstream and 35 km upstream in  12 hrs he can row 44 km downstream and 30 km upstream in 10 hrs.

Assume that the speed of boat in still water = x kmph and speed of current = y kmph

Hence, the upstream speed = Speed of boat in still water - speed of current

                   = x - y  

Downstream speed = Speed of boat in still water + speed of current

             = x + y  

We know that $\text { Time }=\frac{\text { Distance }}{\text { Speed }}$

A boat goes 35 km upstream and 55 km downstream in 12 hours.  

$12=\frac{35}{x-y}+\frac{55}{x+y} \ldots \ldots \ldots \ldots (1)$

It covers 30 km upstream and 44 km downstream in 10 hours.

$10=\frac{30}{x-y}+\frac{44}{x+y} \ldots \ldots \ldots \ldots(2)$

Let us take $\frac{1}{x-y}=a \text { and } \frac{1}{x+y}=b$, we get

12 = 35a + 55b  - [Substituting in equation (1)] …………(3)

10 = 30a + 44b   - [Substituting in equation (2)]

2(5) = 2(15a + 22b)

5 = 15a + 22b ……………..(4)

Subtracting (3) & (4) and multiplying equation (3) by 2 and equation (4) by 5,

70a + 110b = 24

75a + 110b = 25

        -5a    = -1

$a=\frac{1}{5} \text { i.e. } \frac{1}{x-y}=\frac{1}{5}$.

Hence, x - y = 5

Replacing $a=\frac{1}{5}\ in\ equation\ (4)$,

$5=15 a+22 b$

$5=15\left(\frac{1}{5}\right)+22 b$

$5=3+22 b$

$22b = 2$

$b=\frac{1}{11}$

Hence, $b=\frac{1}{x+y}=\frac{1}{11} \text { i.e. } \mathrm{x}+\mathrm{y}=11$

Now, x + y = 11 & x - y = 5, On adding we get,

$2x = 16$

$x=\frac{16}{2}=8$

& on subtracting x + y = 11 & x - y = 5, we get,

$2y = 6$

$y=\frac{6}{2}=3$

Hence, Speed of the boat in still water = 8 Km/hr & Speed of the stream = 3 km/hr.  

Total time taken by boat man to row 77 km downstream and 50 km upstream  $\frac{50}{5}+\frac{77}{11}=10+7=17\ hours$