Question

3
s boat goes 16 km upstream & returns the
same distance downstream
in 6 hours. If it
goes 6 km upstream & 8 km downstream, it
takes 2 1/2 his
find the speed of
the boat.​

Answer 1

Correct Question

A boat goes 16 km upstream & returns the

same distance downstream

in 6 hours. If it

goes 6 km upstream & 8 km downstream, it

takes 2$\frac{1}{2}$ his

find the speed of

the boat.

✪AnSwEr

${\tt{\red{\underline{\large{Given}}}}}$

• A word problem
• Related to stream

${\sf{\green{\underline{\large{To\:find}}}}}$

• Speed of the boat

${\sf{\pink{\underline{\Large{Explanation}}}}}$

Let the speed of still water be x km/h and Speed of stream be y

Downstream = X+Y

while

Upstream = X-Y

D= speed × time

=>T=Distance/speed

According to the question

• 16km upstream
• 16km downstream
• it takes 6hour

$\rightarrow\tt\dfrac{16}{X-Y}+\dfrac{16}{X+Y}=T_1$

$\rightarrow\tt\dfrac{16}{X-Y}+\dfrac{16}{x+y}=6$_____(1)

And

• 6km Upstream
• 8km downstream
• it takes 2$\frac{1}{2}$

$\rightarrow\tt\dfrac{6}{X-Y}+\dfrac{8}{X+Y}=T_2$

$\rightarrow\tt\dfrac{6}{X-Y}+\dfrac{8}{x+y}=\frac{21}{2}$_____(2)

Solving the Equation

$\rightarrow\tt\dfrac{16}{X-Y}+\dfrac{16}{x+y}=6$_____(1)

$\rightarrow\tt\dfrac{6}{X-Y}+\dfrac{8}{x+y}=\dfrac{5}{2}$_____(2)

Taking

$\rightarrow\tt\dfrac{1}{X-Y}=a\:And\dfrac{1}{x+y}=b$

New Equations are

16a+16b =6_____(3)

6a+8b=5/2_____(4)

multiply be 2 in Equation 4

12a+16b=5______(5)

Solving equation (3) and (5)

16a+16b =6

12a+16b=5

________

4a=1

=>a=1/4

¶utting the value in Equation 5

3+16b=5

=>16b=2

=>b=1/8

HERE

$\implies\tt\dfrac{1}{X-Y}=a\:And\dfrac{1}{x+y}=b$

$\implies\tt\dfrac{1}{X-Y}=\frac{1}{4}\:And\dfrac{1}{x+y}=\frac{1}{8}$

X-Y=4

x+y=8

_____

=>2x=12

=>x=6

andy=2

Hence , the speed of still water be 6 km/h and Speed of stream be 2

Answer 2

Answer:

Let the Speed of Boat be x km/hr and Speed of Stream be y km/hr.

• Upstream = (x – y)
• Downstream = (x + y)

⠀

Given : A boat goes 16 km upstream & returns the same distance downstream in 6 hours.

$:\implies\sf \dfrac{Distance}{Upstream}+\dfrac{Distance}{Downstream}=Time\: Taken\\\\\\:\implies\sf \dfrac{16}{(x - y)}+\dfrac{16}{(x + y)}=6\\\\\\:\implies\sf 16\left( \dfrac{1}{x - y} + \dfrac{1}{x + y}\right)= 6\\\\\\:\implies\sf 8\left( \dfrac{1}{x - y} + \dfrac{1}{x + y}\right)= 3\\\\\\:\implies\sf \dfrac{8}{3}\left( \dfrac{x + y + x - y}{x - y} \right) = 1\\\\\\:\implies\sf \dfrac{8}{3}\left( \dfrac{2x}{x^2 - y^2} \right) = 1\\\\\\:\implies\sf \dfrac{8 \times 2x}{3} =x^2 - y^2\qquad...eq \:(l)$

$\rule{90}{0.8}$

Given : If it goes 6 km upstream & 8 km downstream, it takes 2½ hours.

$:\implies\sf \dfrac{Distance_1}{Upstream}+\dfrac{Distance_2}{Downstream}=Time\: Taken\\\\\\:\implies\sf \dfrac{6}{(x - y)}+\dfrac{8}{(x + y)}=2{}^{1}\!/{}_{2}\\\\\\:\implies\sf \dfrac{6x + 6y + 8x + 8y}{x^2 - y^2} = \dfrac{5}{2}$

★ Putting value from eq. ( 1)

$:\implies\sf \dfrac{(14x-2y)3}{2x \times 8} = \dfrac{5}{2}\\\\\\:\implies\sf \dfrac{(14x-2y)3}{x \times 8}=5\\\\\\:\implies\sf 42x - 6y = 40x\\\\\\:\implies\sf 42x - 40x = 6y\\\\\\:\implies\sf 2x = 6y\\\\\\:\implies\sf x = 3y\\\\\\:\implies\sf y=\dfrac{x}{3}$

$\rule{150}{1}$

☯ Using the value of y in eq. ( 1)

$\dashrightarrow\sf\:\:\dfrac{8 \times 2x}{3} =x^2 - y^2\\\\\\\dashrightarrow\sf\:\:\dfrac{8 \times 2x}{3} =x^2 - \left(\dfrac{x}{3}\right)^2\\\\\\\dashrightarrow\sf\:\: \dfrac{8 \times 2x}{3} =x^2 - \dfrac{x^2}{9}\\\\\\\dashrightarrow\sf\:\: \dfrac{8 \times 2x}{3} = \dfrac{8x^2}{9}\\\\\\\dashrightarrow\sf\:\:2 = \dfrac{x}{3}\\\\\\\dashrightarrow\sf\:\:x = 6 \:km/hr\quad\bigg\lgroup Speed \:of \: Boat\bigg\rgroup$

⠀

$\therefore\:\underline{\textsf{Hence, the speed of Boat is \textbf{6 km/hr}}}.$