A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Explanation

Let the speed of the boat in still water is x km/hr and the speed of the stream is y km/hr.
`therefore` speed of boat upstream = (x -y) km/hr
and speed of boat downstream = (x + y) km/hr
We know that, speed = `("distance")/("time")`
According to given condition,
Time taken in going 30 km upstream = `(30)/(x - y)` hr
Time taken in 44 km downstream = `(40)/(x + y)` hr
`therefore (30)/(x - y) + (44)/(x + y) = 10 " "...(1)`
Similarly in second condition.
`(40)/(x - y)+ (55)/(x + y) = 13 " ...(2)"`
Let `(1)/(x - y) = a` and `(1)/(x + y) = b`, we get
30a + 44b = 10 ...(3)
and 40a + 55b = 13 ...(4)
Multiplying equation (3) by 4 and (4) by 3, we get
12a + 176b = 40 ...(5)
120a + 165b = 39 ...(6)
linear Equations in Two Variables
Subtracting equation (6) from (5), we get
11b = 1
implies b = `(1)/(11)`
Putting b = `(1)/(11)` in equation (3), we get
`30a + 44 xx (1)/(11) = 10`
implies 30a = 6
implies `a = (1)/(5)`
implies `(1)/(x - y) = (1)/(5)` or x - y = 5 ...(7)
and `(1)/(x + y) = 11` or x + y = 11 ...(8)
Solving equations (7) and (8), we get
x = 8 and y = 3
Hence, speed of boat in still water = 8 km/hr and speed of water stream = 3 km/hr.