Question

A boat travel 4 km in Downstream and 3 km in Upstream in same time. If this Boat can travel 48 km Downstream and Upstream in total of 14 hours. Then find the Speed of Boat in Still water and Speed of Stream.​

Answer 1

AnswEr :

Let the Speed of Boat in Still water be x Km/hr and, the Speed of Stream be y Km/hr.

$\bold{Let's\: Assume} \begin{cases}\sf{Upstream = (x - y)\:Km/hr} \\ \sf{Downstream = (x + y)\:Km/hr}\end{cases}$

• According to the Question Now :

$\longrightarrow\tt Time_{Downstream} = Time_{Upstream}$

$\longrightarrow\tt \dfrac{Distance}{Downstream\:Speed} = \dfrac{Distance}{Upstream\:Speed}$

$\longrightarrow\tt \dfrac{4}{(x + y)} = \dfrac{3}{(x - y)}$

$\longrightarrow\tt 4(x - y) = 3(x + y)$

$\longrightarrow\tt 4x - 4y = 3x + 3y$

$\longrightarrow\tt 4x - 3x = 3y + 4y$

$\longrightarrow\red{\tt x = 7y} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ \: \: \: \: \:}{} \frak{eq.(i)}$

$\rule{300}{1}$

• Second Part of the Question Now :

$\implies \tt Time_{Downstream} + Time_{Upstream} = Total\:Time\:Taken$

$\implies \tt \dfrac{Distance}{Downstream\:Speed} + \dfrac{Distance}{Upstream\:Speed} = Total\:Time\:Taken$

$\implies \tt \dfrac{48}{(x + y)}+\dfrac{48}{(x - y)} = 14$

$\implies \tt \dfrac{48}{(7y + y)}+\dfrac{48}{(7y - y)} = 14$

$\implies \tt \dfrac{48}{8y}+\dfrac{48}{6y} = 14$

$\implies \tt 48\bigg(\dfrac{1}{8y}+\dfrac{1}{6y}\bigg)= 14$

$\implies \tt \dfrac{3 + 4}{24y} = \cancel\dfrac{14}{48}$

$\implies \tt \dfrac{7}{24y} = \dfrac{7}{24}$

$\implies \tt \cancel\dfrac{7\times24}{24\times7} = y$

$\implies\large\boxed{\tt y =1\:Km/hr}$

$\rule{300}{1}$

• Putting the value of y in eq.( I )

$\longrightarrow\tt x = 7y$

$\longrightarrow\tt x = 7(1)$

$\longrightarrow\large\boxed{\tt x = 7\:Km/hr}$

⠀

∴ Speed of Boat in Still water is 7 Km/hr and, Speed of Stream is 1 Km/hr.

$\rule{300}{2}$

$\star\:\Large\mathfrak{\underline{Shortcut\:Trick :}}$

As Fixed Distance is Given i.e. 48 Km, and Total Time Taken is Given i.e. 14 hr. Speed of Boat in Still water be x Km/hr and, Speed of Stream be y Km/hr

Simply Split that Distance in the form of (a × b) and, Time in (a + b). Where a will be Downstream and b will be Upstream.

◗ 48 = (8 × 6) = (a × b)

◗ 14 = (8 + 6) = (a + b)

Downstream = 8 Km/hr = (x + y) Km/hr

Upstream = 6 Km/hr = (x - y) Km/hr

$\rule{300}{1}$

⇒ x + y = 8

⇒ x – y = 6

______________

⇒ x + x = 8 + 6

⇒ 2x = 14

⇒ x = 7 Km/hr

$\rule{300}{1}$

• Using the Value of x in Any of Speed :

⇒ x + y = 8

⇒ 7 + y = 8

⇒ y = 8 - 7

⇒ y = 1 Km/hr

⠀

∴ Speed of Boat in Still water is 7 Km/hr and, Speed of Stream is 1 Km/hr.

Answer 2

$\bold{\Huge{\underline{\boxed{\rm{\red{ANSWER\::}}}}}}$

Given:

A boat travel 4km in downstream and 3km in upstream in same time. If this boat can travel 48km downstream and upstream in total of 14 hours.

$\bold{\Large{\underline{\sf{\green{To\:find\::}}}}}$

The speed of boat in still water & speed of stream.

$\bold{\Large{\underline{\sf{\purple{Explanation\::}}}}}$

• $\bold{\huge{\underline{\sf{\pink{First\:Case\::}}}}}$

A boat travel 4km in downstream & 3km in upstream in the same time.

Let the speed of boat in still water= R km/hr.

Let the speed of stream= M km/hr.

• Speed in upstream= (R-M)km/hr.
• Speed in downstream= (R+M)km/hr

According to the question:

We know that formula of the time: $\bold{Time=\:\frac{Distance}{Speed} }$

→ $\bold{\frac{4}{(R+M)} =\frac{3}{(R-M)} }$

We suppose here,

• $\bold{\frac{1}{R+M} =a}$
• $\bold{\frac{1}{R-M} =b}$

Therefore,

→ 4a = 3b

→ a = $\bold{\frac{3b}{4} }$..............................(1)

• $\bold{\huge{\underline{\sf{\pink{Second\:case\::}}}}}$

If this boat can travel 48km downstream & also upstream in total time of 14 hours.

→ $\bold{\frac{48}{R+M} +\frac{48}{R-M} =14}$

→ 48a + 48b = 14

Putting the value of a in above the equation,we get;

→ $\bold{48(\frac{3b}{4} )+48b=14}$

→ $\bold{(\frac{144b}{4} )+48b=14}$

→ 144b + 192b = 56

→ 336b = 56

→ b = $\bold{\cancel{\frac{56}{336} }}$

→ b = $\bold{\frac{1}{6} }$

Putting the value of b in equation (1), we get;

→ a = $\bold{\frac{3(\frac{1}{6} )}{4} }$

→ a = $\bold{\frac{\cancel{\frac{3}{6}} }{4} }$

→ a = $\bold{\frac{\frac{1}{2}}{4} }$

→ a = 2×4

→ a = 8

Now,

→ $\bold{\frac{1}{R+M} =8}$........................(2)

→ $\bold{\frac{1}{R-M} =\frac{1}{6} }$........................(3)

• From equation (2), we get;

⇒ R+M= 8

⇒ R = 8-M...........................(4)

Putting the value of R in equation (3), we get;

⇒ 8-M - M = 6

⇒ 8 -2M = 6

⇒ -2M = 6 -8

⇒ -2M = -2

⇒ M = $\bold{\cancel{\frac{-2}{-2} }}$

⇒ M = 1 km/hrs

Putting the value of M in equation (4), we get;

⇒ R = 8 -1

⇒ R = 7 km/hrs

Thus,

Speed of Boat in Still water is 7 Km/hr & Speed of Stream is 1 Km/hr.