Formulate the following problems as a pair of equations, and hence find their solutions: (i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. (ii) 2 wome

Explanation

(i) Let the speed of Ritu in still water = x km/hr
and speed of current = y km/h
Then, speed downstream = (x + y) km/h
speed upstream = (x - y) km/h
Time = `("Distance")/("Speed")`
`(20)/(x + y) = 2` and `(4)/(x - y) = 2`
implies x + y = 10 ...(1) and x - y = 2 ...(2)
On adding equations (1) and (2), we get
2x = 12 implies x = 6
On putting x = 6 in equation (1) we get, 6 + y = 10 implies y = 4
`therefore` Speed of Ritu in still water = 6 km/h
and speed of current = 4 km/h
(ii) Let 1 women finish the work in x days and let 1 man finish the work in y days.
1 day work of 1 women = `(1)/(x)`
1 day work of 1 man = `(1)/(y)`
1 day work of 2 women and 5 men = `(2)/(x) + (5)/(y) = (5x + 2y)/(xy)`
The number of days required for complete work = `(xy)/(5x + 2y)`
We are given that, `(xy)/(5x + 2y) = 4`
Similarly, in second case
`(xy)/(6x + 3y) = 3` (given)
Then `(5x + 2y)/(xy) = (1)/(4)` and `(6x + 3y)/(xy) = (1)/(3)`
implies `(20)/(y) + (8)/(x) = 1` and `(18)/(y) + (9)/(x) = 1`
Let `(1)/(x)` = u and `(1)/(y) = v`
`therefore 20 v + 8 u = 1 ...(1)`
and `18 v + 9 u = 1 ...(2)`
On multiplying equation (1) by 9 and (2) by 8, then subtracting later from first, we get
180v - 144v = 9 - 8
implies 36v = 1
implies `v = (1)/(36)`
On substituting v = `(1)/(36)` in equation (2), we get
`18 xx (1)/(36) + 9u = 1`
implies`9u = 1 - (1)/(2) implies 9u = (1)/(2) implies u = (1)/(18)`
Now, u = `(1)/(18)` and v = `(1)/(36)`
implies `(1)/(x) = (1)/(18)` and `(1)/(y) = (1)/(36)`
implies x = 18 and y = 36
Hence, one woman alone finishes the work in 18 days and one man alone finishes the work in 36 days.
(iii) Let the speed of the train = x km/h and the speed of the bus = y km/h
In first case, Roohi travels 60 km by train and 240 km by bus in 4 h.
Thus, `(60)/(x) + (240)/(y) = 4`
implies `(15)/(x) + (60)/(y) = 1 " ...(1)"`
Similarly, in second case, she travels 100 km by train and 200 km by bus in `4(1)/(6)h`
`therefore " "(100)/(x) + (200)/(y) = 4 + (1)/(6) " "(because 10 min = (1)/(6) h)`
implies `(100)/(x) + (200)/(y) = (25)/(6)`
implies `(24)/(x) + (48)/(y) = 1`
Let `(1)/(x) = u` and `(1)/(y) = v`,
15u + 60v = 1 ...(3)
and 24u + 48v = 1 ...(4)
On multiplying equation (3) by 24 and (4) by 15, then subtracting from equation (4), we get
24(15u + 60v) - 15(24u + 48v) = 24 - 15
implies 1440v - 720v = 9
implies 720v = 9
implies `v = (9)/(720) = (1)/(80)`
Substituting v = `(1)/(8)` in equation (3), we get
`15u + 60 xx (1)/(80) = 1`
implies 15u = 1 - `(3)/(4) = (1)/(4)`
implies `u = (1)/(60)`
Now, `u = (1)/(60)` and `v = (1)/(80)`
`(1)/(x) = (1)/(60)` and `(1)/(y) = (1)/(80)`
implies x = 60 and y = 80
Hence, the speed of the train = 60 km/h and the speed of the bus = 80 km/h.