# 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 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 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

**Solution:**

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

Ritu’s speed of rowing;

Upstream = ( x - y ) km / h

Downstream = ( x + y ) km / h

According to question,

Ritu can row downstream 20 km in 2 hours,

2( x + y) = 20

x + y = 10 ....(1)

Ritu can row upstream 4 km in 2 hours,

2( x - y) = 4

x - y = 2 ....(2)

Adding equation (1) and (2), we obtain

2x = 12

x = 6

Putting x = 6 in equation (1), we obtain

6 + y = 10

y = 4

Hence, Ritu’s speed of rowing 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 to finish the work be x and y respectively.

Therefore, work done by a woman in 1 day = 1/x and work done by a man in 1 day = 1/y

According to the question,

2 women and 5 men can together finish an embroidery work in 4 days;

2/x + 5/y = 1/4 ....(1)

3 women and 6 men can finish it in 3 days

3/x + 6/y = 1/3 ....(2)

Substituting 1/x = p and 1/y = q in equations (1) and (2), we obtain

2/x + 5/y = 1/4 ⇒ 2 p + 5q = 1/4 ⇒ 8 p + 20q - 1 = 0 ....(3)

3/x + 6/y = 1/3 ⇒ 3 p + 6q = 1/3 ⇒ 9 p +18q - 1 = 0 ....(4)

By cross-multiplication, we obtain

p/-20 - (-18) = q/-9 - (-8) = 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

Therefore, p = 1/x = 1/18

⇒ x = 18

and, q = 1/y = 1/36

⇒ y = 36

Hence, the number of days taken by a woman is 18 and by a man is 36.

(iii) Let the speed of train and bus be u km/h and v km/h respectively. According to the given information, Roohi travels 300 km and takes 4 hours if she travels 60 km by train and the remaining by bus

60/u + 240/v = 4 ....(1)

If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer

100/u + 200/v = 25/6 ....(2)

Substituting 1/u = p and 1/v = q in equations (1) and (2), we obtain

60/u + 240/v = 4 ⇒ 60 p + 240q = 4 ....(3)

100/u + 200/v = 25/6 ⇒ 100 p + 200q = 25/6 ⇒ 600 p +1200q = 25 ....(4)

Multiplying equation (3) by 10, we obtain

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

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

1200q = 15

q = 15/1200

q = 1/80

Substituting q = 1/80 in equation (3), we obtain

60 p + 240 × 1/80 = 4

60 p = 4 - 3

p = 1/60

Therefore, p = 1/u = 1/60

⇒ u = 60

and, q = 1/v = 1/80

⇒ v = 80

Hence,

speed of the train = 60 km / h

And speed of the bus = 80 km / h

**Video Solution:**

### NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.6 Question 2:

(i) Hence, Ritu’s speed of rowing in still water is 6 km/h and the speed of the current is 4 km/h.

(ii) Hence, the number of days taken by a woman is 18 and by a man is 36.

(iii) Hence, speed of the train = 60 km / h and speed of the bus = 80 km / h