Question

a man can row 32km upstream and 48km downstream in 14 hours.also he can row 24km upstream and 40km downstream in 11 hours.Then what is the speed of man in still water​

Answer 1

Given :-

▪ A man can row 32 km upstream and 48 km downstream in 14 hours. Also, he can row 24 km upstream and 40 km downstream in 11 hours.

To Find :-

▪ Speed of man in still water.

Solution :-

Let the speed of man and stream be x and y respectively.

We have two cases given in the question,

Case 1

⇒ Man rows 32 km upstream and 48 km downstream in 14 hours.

To be noted that, while going upstream the speed of stream will oppose the speed of man, so speed of stream will be subtracted from the speed of man.

We know,

⇒ Time = Distance / Speed

According to case 1,

⇒ 32 / (Speed in upstream) + 48 / (speed in downstream) = 14

⇒ { 32 / (x - y) } + { 48 / (x + y) } = 14

Let 1/(x - y) = a and 1/(x + y) = b,

⇒ 32a + 48b = 14

⇒ 16a + 24b = 7 ...(1)

Case 2

⇒ Man rows 24 km upstream and 40 km downstream in 11 hours.

Similarly as in Case 1,

⇒ 24 / (Speed in upstream) + 40 / (Speed in downstream ) = 11

⇒ { 24 / (x - y) } + { 40 / (x + y) } = 11

As we assumed 1/(x - y) = a and 1/(x + y) = b, in case 1 so we substitute the value here too,

⇒ 24a + 40b = 11 ...(2)

Multiply (1) by 24 and (2) by 16 respectively,

⇒ ( 16a + 24b = 7 ) × 24

⇒ 384a + 576b = 168 ...(3)

Also,

⇒ ( 24a + 40b = 11 ) × 16

⇒ 384a + 640b = 176 ...(4)

Subtracting (3) from (4), we get

⇒ 384a + 640b - 384a - 576b = 176 - 168

⇒ 640b - 576b = 8

⇒ 64b = 8

⇒ b = 1/8

Now, Substitute [b = 1/8] in (1),

⇒ 16a + 24×1/8 = 7

⇒ 16a + 3 = 7

⇒ 16a = 4

⇒ a = 1/4

We assumed 1/(x - y) = a and 1/(x + y) = b, So

⇒ x - y = 4 ...(5)

and,

⇒ x + y = 8 ...(6)

Add (5) and (6),

⇒ x - y + x + y = 8 + 4

⇒ 2x = 12

⇒ x = 6 km/h

Further, Substituting [x = 6] in (5),

⇒ 6 - y = 4

⇒ y = 6 - 4

⇒ y = 2 km/h

Hence, The speed of man in still water is 6 km/hr.

Answer 2

Given:

A man can row 32km upstream and 48km downstream in 14 hours.also he can row 24km upstream and 40km downstream in 11 hours.

To find:

Velocity of man in still water.

Calculation:

Let Velocity of man in still water be u and Velocity of stream be v :

So, in 1st case:

$\sf{ \therefore \dfrac{32}{(u - v)} + \dfrac{48}{(u + v)} = 14 }$

In 2nd case:

$\sf{ \therefore \dfrac{24}{(u - v)} + \dfrac{40}{(u + v)} = 11 }$

Now , let's consider (u + v)= b and (u-v) = a:

So, the equations becomes:

$\sf{ \therefore \dfrac{32}{a} + \dfrac{48}{b} = 14 \: \: \: \: \: .......(1)}$

$\sf{ \therefore \dfrac{24}{a} + \dfrac{40}{b} = 11 \: \: \: \: \: \: .......(2)}$

Now, let's consider 1/a = x and 1/b = y:

So, the equations become:

$\sf \therefore \: 32x + 48y = 14 \: \: \: \: ......(3)$

$\sf \therefore \: 24x + 40y = 11 \: \: \: \: ......(4)$

Multiplying eq.(3) with 3 and eq.(4) with 4 and then Solving by elimination method , we get:

$\sf{ \therefore \: x = \dfrac{1}{4} \: and \: y = \dfrac{1}{8} }$

$\sf{ = > \: \dfrac{1}{a} = \dfrac{1}{4} \: and \: \dfrac{1}{b} = \dfrac{1}{8} }$

$\sf{ = > \: a = 4 \: and \: b = 8 }$

$\sf{ = > \: (u - v) = 4 \: and \: (u + v)= 8 }$

Now , solving these final equations:

$\sf{ \therefore \: 2u = 12}$

$\sf{ = > \: u = 6 \: km {hr}^{ - 1} }$

So, Velocity of man in still water is 6 km/hr.