Question

A boat goes 25km upstream and 33km downstream in 8hrs .It can go 40km upstream and 77km downstream in 15hrs .find the speed of the stream and that of boat in still water.

Answer 1

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♦ʟᴇᴛ sᴘᴇᴇᴅ ᴏғ ᴛʜᴇ ʙᴏᴀᴛ ɪɴ sᴛɪʟʟ ᴡᴀᴛᴇʀ = x ᴋᴍ/ʜ 
♦sᴘᴇᴇᴅ ᴏғ sᴛʀᴇᴀᴍ = ʏ ᴋᴍ/ʜ
♦sᴘᴇᴇᴅ ᴏғ ᴛʜᴇ ʙᴏᴀᴛ ᴜᴘsᴛʀᴇᴀᴍ = (x-ʏ) ᴋᴍ
♦sᴘᴇᴇᴅ ᴏғ ᴛʜᴇ ʙᴏᴀᴛ ᴅᴏᴡɴsᴛʀᴇᴀᴍ =(x+ʏ) ᴋᴍ
♦sɪɴᴄᴇ ᴛɪᴍᴇ ᴛᴀᴋᴇɴ ʙʏ ᴛʜᴇ ʙᴏᴀᴛ ɪɴ 25 ᴋᴍ ᴜᴘsᴛʀᴇᴀᴍ ᴀɴᴅ 33ᴋᴍ ᴅᴏᴡɴsᴛʀᴇᴀᴍ ɪs 8ʜʀs,

(25/x-ʏ) + (33/x+ʏ) = 8➖➖➖➖❕1❕

♦ᴛɪᴍᴇ ᴛᴀᴋᴇɴ ʙʏ ᴛʜᴇ ʙᴏᴀᴛ ɪɴ 40 ᴋᴍ ᴜᴘsᴛʀᴇᴀᴍ ᴀɴᴅ 77ᴋᴍ ᴅᴏᴡɴsᴛʀᴇᴀᴍ ɪs 15ʜʀs,

(40/x-ʏ) + (77/x+ʏ) = 15➖➖➖➖❕2❕

♦ʟᴇᴛ x-ʏ = 1/ᴀ ᴀɴᴅ x+ʏ = 1/ʙ, ᴛʜᴇɴ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴs 1 ᴀɴᴅ 2 ʙᴇᴄᴏᴍᴇ
25ᴀ + 33ʙ = 8➖➖➖➖❕3 ❕
40ᴀ + 77ʙ = 15➖➖➖➖❕4❕

♦ᴍᴜʟᴛɪᴘʟʏɪɴɢ 16 ᴡɪᴛʜ ❕3❕ᴀɴᴅ 10 ᴡɪᴛʜ ❕4❕,ᴡᴇ ɢᴇᴛ;
400ᴀ + 528ʙ = 128➖➖➖➖❕5❕
400ᴀ + 770ʙ = 150➖➖➖➖❕6❕

♦sᴜʙᴛʀᴀᴄᴛɪɴɢ❕ 5❕ ғʀᴏᴍ❕ 6❕
242ʙ = 22
=> ʙ = 22/242 = 1/11

♦sᴜʙsᴛɪᴛᴜᴛɪɴɢ ʙ = 1/11 ɪɴ ❕3❕
25ᴀ + 33x 1/11 = 8
=>25ᴀ = 8-3 =5
=> ᴀ = 5/25 =1/5

sᴏ,   x-ʏ = 5 ➖➖➖➖❕7❕
 ᴀɴᴅ x+ʏ = 11 ➖➖➖➖❕8❕

♦ ᴀᴅᴅɪɴɢ❕ 7❕ ᴀɴᴅ ❕8❕
2x = 16
=> x = 8

♦sᴜʙsᴛɪᴛᴜᴛɪɴɢ x=8 ɪɴ❕8❕
8 + ʏ =11
=>ʏ = 3

Hence,the speed of the boat in still water is 8 km/h and the speed of the stream is 3 km/h.

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Answer 2

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

Speed of stream = y km/h

Speed of the boat upstream = (x-y) km

And downstream = (x+y) km

Since time taken by the boat in upstream = 25 km and downstream = 33 km is 8hrs.

$\dfrac{25}{x - y}$ + $\dfrac{33}{x + y}$ = $8$ ....... (1)

Also,time taken by the boat in 40 km upstream and 77km downstream is 15hrs,

$\dfrac{40}{x - y}$ + $\dfrac{77}{x + y}$= $15$ ......... (2)

Let 

x-y = $\dfrac{1}{a}$ and x+y = $\dfrac{1}{b}$

Then the eq. (1) and (2) is

25a + 33b = 8 ....... (3)
40a + 77b = 15 ....... (4)

Multiplying 16 by (3) and 10 by (4) then we get

400a + 528b = 128 ...... (5)
400a + 770b = 150 ..... (6)

Subtract eq. (5) from (6)

400a + 528b = 128 
400a + 770b = 150 [Change the signs]
_______________
000a - 242b = - 22
______________

$242b\:=\:22$

$b$ = $\dfrac{22}{242}$

$b$ = $\dfrac{1}{11}$

Put value of b in (3)

$25a\:+\:33$ × $\dfrac{1}{11}$ = $8$

$25a$ + $\dfrac{33}{11}$ = $8$

$25a$ = $8\:-\:3$

$25a\:=\:5$

$a$ = $\dfrac{5}{25}$

$a$ = $\dfrac{1}{5}$

So,

$x\:-\:y$ = $\dfrac{1}{a}$

$x\:-\:y$ = $5$ ...... (7)

$x\:+\:y$ = $\dfrac{1}{b}$

$x\:+\:y$ = $11$ ...... (8)

Solve (7) and (8) by elimination method

x - y = 5
x + y = 11
_________
2x + 0 = 16
_________

2x = 16

$\textbf{x = 8}$

Put value of x in (7)

8 - y = 5

- y = 5 - 8

- y = - 3

$\textbf{y = 3}$

So,

$\textbf{Speed of boat in still water}$

$\huge\textbf{= 8 km/hr.}$

$\textbf{Speed of water in stream water}$

$\huge\textbf{= 3 km/hr.}$

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