A boat takes $$3 \frac{2}{3}$$ hours in going 40 km upstream and 30 km downstream and the speed of the stream is $$7 \frac{1}{2}$$​ km/h. How much time (in hours) will the boat take to travel 90 km in still water?

Correct option is C

Given:

Total time for the combined upstream and downstream journey =  $$3\frac{2}{3}$$​ hours = $$\frac{11}{3}$$​ hours

Distance upstream = 40 km

Distance downstream = 30 km

Speed of the stream = $$7 \frac{1}{2}$$​ km/h = $$\frac{15}{2}$$​ km/h

Solution:

Let the speed of the boat in still water be x km/h.

Upstream speed = x - $$\frac{15}{2}$$​​

Downstream speed = x + $$\frac{15}{2}$$​​

Now,

$$\frac{40}{x - \frac{15}{2}} + \frac{30}{x + \frac{15}{2}} = \frac{11}{3}$$​​

​$$\frac{40}{\frac{2x - 15}{2}} + \frac{30}{\frac{2x + 15}{2}} = \frac{11}{3}$$​​

​$$\frac{80}{2x - 15} + \frac{60}{2x + 15} = \frac{11}{3}$$​​

​$$\frac{80(2x + 15) + 60(2x - 15)}{(2x - 15)(2x + 15)} = \frac{11}{3}$$​​

​$$\frac{160x + 1200 + 120x - 900}{(2x)^2 - (15)^2} = \frac{11}{3}$$​​

​$$\frac{280x + 300}{4x^2 - 225} = \frac{11}{3}$$​​

11(4x² - 225) = 3(280x + 300)

44x²​ - 2475 = 840x + 900

44x² - 840x - 3375 = 0 

44x²- 990x + 150x - 3375 = 0 

22x(2x - 45) + 75(2x - 45) = 0 

(22x + 75)(2x - 45) = 0 

Speed can't be negative, so

x = $$\frac{45}{2}\ km/ h$$  

Now,  time to cover 90 km

Time = $$\frac{90}{45/2} = 4 \, \text{hours}$$​​

The boat will take 4 hours to travel 90 km in still water.