Question

Q1.A man goes 32km upstream and 40 km downstream in 10 hr.Again he goes 38 km upstream and 50 km downstream. then find the speed of the stream?

Answer 1

Step-by-step explanation:

*Given:A man goes 32km upstream and 40 km downstream in 10 hr.Again he goes 38 km upstream and 50 km downstream in 12 hours.

To find: Find the speed of the stream?

Solution:

Let the speed of boat is x km per hour and speed of stream is y km per hour.

Speed of upstream be (x-y)km/h

and speed of stream is (x+y)km/h

Step 1: Make equations from the given condition

Time taken to go 32 km upstream=Distance /speed

=32/x-y hours

Time taken to go 40 km downstream=Distance /speed

=40/x+y hours

Total time taken during the journey is 10 hours

Thus,

$\begin{gathered}\bold{ \frac{32}{x - y} + \frac{40}{x + y} = 10} \: \: \: ...eq1\\\end{gathered}$

Time taken to go 38 km upstream=Distance /speed

=38/x-y

Time taken to go 50 km downstream=Distance /speed

=50/x+y

Total time taken during the journey is 12 hours

Thus,

$\begin{gathered}\bold{ \frac{38}{x - y} + \frac{50}{x + y} = 12} \: \: \: ...eq2\\ \end{gathered}$

Step 2:Convert these equations in linear equations in two variables

Solve eq1 and eq2

Let

$\begin{gathered} \frac{1}{x - y} = a \\ \\ \frac{1}{x + y} = b \\ \\ \end{gathered}$

thus

$\begin{gathered}32a + 40b = 10 \\ or \\ 16a + 20b = 5 \: \: \:... eq3\\ \\ 38a + 50b = 12 \\ or \\ 19a + 25b = 6 \: \: \:...eq4 \\ \end{gathered}$

Solution of eq3 and eq4 gives value of a and b.

multiply eq3 by 25 and eq4 by 20 and subtract both

$\begin{gathered}400a + 500b = 125\\ 380a + 500b = 120 \\ - - - - - - - - - \\ 20a = 5\\ \\ a = \frac{5}{20} \\ \\ a = \frac{1}{4} \\ \end{gathered}$

put the value of a in eq3

$\begin{gathered}16 \times \frac{1}{4} + 20b = 5 \\ \\ 4+ 20b = 5 \\ \\ 20b = 1 \\ \\ b = \frac{ 1}{20} \\\\ \end{gathered}$

Step 3: Find values of x and y.

Put these values of a and b in assumption

$\begin{gathered} \frac{1}{x - y} = \frac{1}{4} \\ \\ x - y = 4 \: \: \: ...eq5\\ \\ \frac{1}{x + y} = \frac{ 1}{20} \\ \\ x + y = 20 \: \: \: ...eq6\\ \end{gathered}$

add both equations 5 and 6

$\begin{gathered}x - y = 4\\ x + y =20 \\ - - - - - - - - - \\ 2x = 24 \\ \\ x = \frac{24}{2} \\ \\ \bold{\boxed{x = 12}} \\ \end{gathered}$

put value of x in eq5

$\begin{gathered}x - y = 4 \\ 12 - y = 4 \\ \\ - y = 4 - 12 \\ \\ - y = - 8 \\ \\ \bold{\boxed{y = 8}} \\ \\ \end{gathered}$

Final answer:

Speed of stream is 8 km/hour.

*Note: Question is incomplete.It was assumed that he goes 38 km upstream and 50 km downstream in 12 hours.

Hope it helps you.

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