A boat can travel 78 km upstream and back in a total of 32 hours. It can travel 15 km upstream and 52 km downstream in a total of 9 hours. How much distance will the boat cover in 12 hours in still water?

Correct option is B

Given:

A boat travels 78 km upstream and back (78 km downstream) in 32 hours.

The boat travels 15 km upstream and 52 km downstream in 9 hours.

Solution:

Speed of the boat in still water = ( b ) km/h

Speed of the stream = ( s ) km/h

Upstream speed:( b - s ) km/h

Downstream speed:( b + s ) km/h

Time to travel 78 km upstream + time to travel 78 km downstream = 32 hours

​$$\frac{78}{b - s} + \frac{78}{b + s}$$​ = 32 ________(1)

Time to travel 15 km upstream + time to travel 52 km downstream = 9 hours

​$$\frac{15}{b - s} + \frac{52}{b + s}$$​ = 9 _________(2)

​$$\frac{1}{b - s} = \text{x}\quad \text{and} \frac{1}{b + s}$$​ = y

78x + 78y = 32 _______(1a)

15x + 52y = 9________ (2a)

78x + 78y = 32$$\implies x + y = \frac{32}{78} = \frac{16}{39}$$​ _______(3)

x =$$\frac{16}{39}$$​ – y

15$$\left(\frac{16}{39} - y\right)$$​+ 52y = 9

​$$\frac{240}{39} - 15y + 52y = 9 \implies \frac{240}{39}$$​ + 37y = 9

37y = 9 -$$\frac{240}{39} = \frac{351 - 240}{39} = \frac{111}{39} = \frac{37}{13}$$​​

y =$$\frac{37}{13 \times 37} = \frac{1}{13}$$​​

x =$$\frac{16}{39} - \frac{1}{13} = \frac{16 - 3}{39} = \frac{13}{39} = \frac{1}{3}$$​​

b - s =$$\frac{1}{x}$$​ = 3 ________(4)

b + s =$$\frac{1}{y}$$​ = 13 _______(5)

Add equations (4) and (5):

2b =$$16 \implies b$$​ = 8 km/h

Subtract equation (4) from equation (5):

2s =$$10 \implies s$$​ = 5 km/h

Distance covered in 12 hours in still water:**

Distance = Speed$$\times$$​Time = 8$$\times$$​ 12 = 96 km