Question

. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers
40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in
still water and speed of stream

Answer 1

Answer:

$\bigstar{\bold{Speed\:of\:stream=2\:km/hr}}$

$\bigstar{\bold{Speed\:of\:boat=10\:km/hr}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• Boat covers 32 kn upstream and 36 km downstream in 7 hours
• Boat covers 40 km upstream and 48 km downstream in 9 hours

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Speed of the boat in still water
• Speed of the stream

$\Large{\underline{\underline{\bf{Solution:}}}}$

➻ Let the speed of boat in still water be x km/hr

➻ Let the speed of the stream be y km/hr

➻ Speed while travelling upstream = x - y km/hr

➻ Speed while travelling downstream = x + y km/hr

➻ We know that

     Time = Distance/Speed

➻ Hence by first case,

    $\sf{\dfrac{32}{x-y} +\dfrac{36}{x+y}=7----(1)}$

➻ Now by second case

    $\sf{\dfrac{40}{x-y}+\dfrac{48}{x-y}=9----(2)}$

➻ Now let 1/x - y = p, 1/x + y = q

➻ Hence equation 1 and 2 changes to,

    32p + 36q = 7----(3)

    40p + 48q = 9----(4)

➻ Multiply equation 3 by 5 and equationn 4 by 4

    160p + 180 q =35----(5)

    160p + 192q = 36----(6)

➻ Solving equation 5 and 6 by elimination method,

               -12q = -1

                    q = 1/12

➻ Substitute the value of q in equation 3

    32p + 36 × 1/12 = 7

    32p + 3 = 7

    32p = 4

         p = 4/32

         p = 1/8

➻ Now we know that p =1/x-y, 1/x - y = 1/8

    q = 1/x + y, 1/x + y = 1/12

➻ Hence,

    x - y = 8

    x = 8 + y ----(7)

    x + y = 12

➻ Substitute equation 7 in the above equation,

    8 + y + y = 12

    2y = 4

      y = 2

➻ Hence speed of stream is 2 km/hr

    $\boxed{\bold{Speed\:of\:stream=2\:km/hr}}$

➻ Now substitute the value of y in equation 7

    x = 8 + 2

    x = 10

➻ Hence the speed of the boat in still water is 10 km/hr

    $\boxed{\bold{Speed\:of\:boat=10\:km/hr}}$

$\Large{\underline{\underline{\bf{Notes:}}}}$

➻ A linear equation in two variable can be solved by

• Substitution method
• Elimination method
• Cross multiplication method

Answer 2

Answer :-

▶ Speed of the stream = 2 Km/hr

▶ Speed of the boat = 10 Km/hr

________________________________

★ Concept :-

This question is based on Linear Equations in two variables, where we turn the value of one variable into other to find the value of both the variables.

• Distance = Speed × Time

_______________________________

★ Solution :-

✏Let the speed of the stream be 'x' Km/hr

✏ Let the speed of the boat be 'y' Km/hr.

So,

» Total speed of boat while travelling upstream = (y - x) Km / hr

» Total speed of boat while travelling downstream = (x + y) Km/hr

______________________________

Then, according to the question,

• Case I -

✒ {32 / (y - x)} + {36 / (x + y)} = 7 ..(i)

• Case II -

✒ {40 / (y - x)} + {48 / (x + y)} = 9 .. (ii)

Now let,

=> 1 / (y - x) = p

=> 1 / (x + y) = q

So, from equation (i) , we get,

▶ 32p + 36q = 7 ... (iii)

From equation (ii), we get,

▶ 40p + 48q = 9 ...(iv)

________________________________

Let us solve this equation by elimination method, then,

Multiplying equation (iii) by 5 , we get,

=> 160p + 180q = 35 ... (v)

Multiplying equation (iv) by 4 , we get,

=> 160p + 192q = 36 ... (vi)

Now subtracting equation (v) from equation (vi), we get,

✒ 160p + 192q - (160p + 180q) = 36 - 35

✒ 160p + 192q - 160p - 180q = 1

✒ 12q = 1

✒ q = 1/12

• So, q = 1/12

By applying the value of q in equation (iii), we get,

=> 32p + 36 × (1/12) = 7

=> 32p + 3 = 7

=> 32p = 7 - 3

=> 32p = 4

=> p = 4/32 = 1/8

• So, p = 1/8

________________________________

We know that,

=> 1 / (x - y) = p

=> 1 / (x - y) = 1/8

=> x - y = 8

=> x = 8 + y ... (vii)

Also,

=> 1 / (x + y) = q

=> 1 / (x + y) = 1 / 12

=> x + y = 12 ... (viii)

From equation (vii) and (viii) , we get,

✏ (8 + y) + y = 12

✏ 8 + 2y = 12

✏ 2y = 12 - 8 = 4

✏ y = 4/2 = 2

Hence, y = 2.

By applying the value of y in equation (vii), we get,

=> x = 8 + y = 8 + 2 = 10

Hence, x = 2.

________________________________

» Speed of the stream = x = 2 Km/hr

» Speed of the boat = y = 10 Km/hr

___________________________

★ More to know :-

• If we go on drawing the graph of these linear equations, we see that these graphs intersect each other at a point. This coordinate is the solution of these equations which give us a unique solution and this are consistent.

• You would have come across the graph of linear equations which have parallel lines. This shows that those linear equations have no solution and thus are inconsistent.

• Also you might have noticed that sometimes the lines of two given linear equations are coincident or overlapping. This shows that those linear equations have infinitely many solutions and thuse are consistent.