Q. 24.2( 285 Votes )

# 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 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

Answer :

(i) Let the speed of Ritu in still water and the speed of stream be *x* km/h and *y* km/h respectively.

While rowing upstream, Ritu's speed slows down and the speed will be her speed minus speed of stream and while rowing downstream her speed will increase and will be equal to sum of her speed and speed of stream. Therefore,

The speed of Ritu while rowing

Upstream = (x - y) km/h

Downstream = (x + y) km/h

According to the question:

Ritu can row downstream 20 km in 2 hours, anddistance = speed x time

⇒ 2 (x+y) = 20

⇒ x+y = 10............(1)

also,

Ritu can row upstream 4 km in 2 hours

⇒ 2 (x - y) = 4

⇒ x-y = 2 ............(2)

Adding equation (1) and (2),

we obtain

⇒ x + y + x - y = 10 + 2⇒ 2 x = 12

⇒ x = 6

Putting this in equation (1),

6 + y = 10we obtain *y* = 4

Hence, Ritu's speed in still water is 6 km/h and the speed of the current is 4 km/h.

(ii)Let the number of days taken by a woman and a man be *x* and *y* respectively.

Therefore, work done by a woman in 1 day =

and work done by a man in 1 day =

According to the question,

2 women and 5 men take 4 days to complete the worki.e. they take 4 days to complete one work

⇒

⇒

Also,

3 women and 6 men take 3 days to complete the work i.e.

they take 3 days to complete one work

⇒

⇒

Putting in these equations,

we obtain

⇒ 8p + 20q = 1

and

⇒ 9p + 18q = 1

By cross-multiplication, we obtain

x = 18, y = 36

Hence, number of days taken by a woman, x = 18

Number of days taken by a man, y = 36

(iii) Let the speed of train and bus be *u* km/h and *v* km/h respectively.

According to the given information,

It takes her 4 hours if she travels 60 km by bus and rest ( i.e. 240 km) by trainAs

we have,

and also, If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer i.e. 4 hours and 10 minutes

Also,

1 hour = 60 minutes

and

we have,

Putting in these equations, we obtain

60p+240q = 4 ......(3)

100p+200q =

600p+1200q = 25 ....(4)

Multiplying equation (3) by 10, we obtain

600p+2400q = 40 ....(5)

Subtracting equation (4) from (5), we obtain

1200q = 15

..... (6)

Substituting in equation (3), we obtain

60p+3 = 4

60p = 1

u = 60km/h and v = 80km/h

Hence, speed of train = 60 km/h

Speed of bus = 80 km/h

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