Question

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 the 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.

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Answer 1

Formulate the following problems as a pair of equations, and hence find their solutions:

Question:

(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.

Solution:

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

Speed of Ritu while rowing

• Upstream = (x - y) km/h
• Downstream = (x + y) km/h

According to question,

2(x + y) = 20

⇒ x + y = 10. . . . (i)

2(x - y) = 4

⇒ x - y = 2. . . . . (ii)

Adding equation (i) and (ii), we get

Putting this equation in (i), we get

y = 4

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

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Question:

(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.

Solution:

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 = 1/x

According to the question,

4(2/x + 5/y) = 1

2/x + 5/y = 1/4

3(3/x + 6/y) = 1

3/x + 6/y = 1/3

Putting 1/x = p and 1/y = q in these equations, we get,

2p + 5q = 1/4

By cross multiplication, we get

p/-20-(-18) = q/-9-(-18) = 1/144-180

p/-2 = q/-1 = 1/-36

p/-2 = -1/36 and q/-1 = 1/-36

p = 1/18 and q = 1/36

p = 1/x = 1/18 and q = 1/y = 1/36

x = 18 and y = 36

Hence, number of days taken by a woman = 18 and number of days taken by a man = 36

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Question:

(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 the 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.

Solution:

Let the speed of train and bus be u km/h and v km/h respectively.

According to the given information,

60/u + 240/v = 4 . . . . (i)

100/u + 200/v = 25/6. . . . . (ii)

Putting 1/u = p and 1/v = q in the equations, we get,

60p + 240q = 4 . . . . . (iii)

100p + 200q = 25/6

600p + 1200q = 25 . . . . (iv)

Multiplying equation (iii) by 10, we get

600p + 2400q = 40 . . . .(v)

Subtracting equation (iv) from (v), we get1200q = 15

q = 15/200 = 1/80 . . . . . (vi)

Putting equation (iii), we get

60p + 3 = 4

60p = 1

p = 1/60

p = 1/u = 1/60 and q = 1/v = 1/80

u = 60 and v = 80

Hence, speed of train = 60 km/h and speed of bus = 80 km/h.

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