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.
Solution:
We will be using the concept of two-variable linear equations to solve the given question.
(i) Let Ritu’s speed of rowing in still water and the speed of the 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 [Since, Distance = Speed × Time]
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
x + y + x - y = 10 + 2
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 stream 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 ⇒ 2p + 5q = 1/4 ⇒ 8p + 20q - 1 = 0 ....(3)
3/x + 6/y = 1/3 ⇒ 3p + 6q = 1/3 ⇒ 9p +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) [Since, Distance / Speed = Time]
If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer
100/u + 200/v = 4 + 1/6 [Since, 10 minutes = 1/6 hours]
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 ⇒ 60p + 240q = 4 ....(3)
100/u + 200/v = 25/6 ⇒ 100p + 200q = 25/6 ⇒ 600p + 1200q = 25 ....(4)
Multiplying equation (3) by 10, we obtain
600p + 2400q = 40 ....(5)
Subtracting equation (4) from (5), we obtain
600p + 2400q - (600p + 1200q) = 40 - 25
1200q = 15
q = 15/1200
q = 1/80
Substituting q = 1/80 in equation (3), we obtain
60p + 240 × 1/80 = 4
60p = 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
☛ Check: NCERT Solutions Class 10 Maths Chapter 3
Video Solution:
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.
NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.6 Question 2
Summary:
(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Hence, Ritu’s speed of rowing in still water is 6 km/h and the speed of the current is 4 km/h.
(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. Hence, the number of days taken by a woman is 18 and by a man is 36. (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. Hence, speed of the train = 60 km/h and speed of the bus = 80 km/h
☛ Related Questions:
- Solve the following pairs of equations by reducing them to a pair of linear equations: (i) 1/2x + 1/3y = 2; 1/3x + 1/2y = 13/6 (ii) 2/√x + 3/√y = 2; 4/√x - 9/√y = -1 (iii) 4/x + 3y = 14; 3/x - 4y = 23 (iv) 5/(x - 1) + 1/(y - 2) = 2; 6/(x - 1) - 3/(y - 2) = 1 (v) (7x - 2 y)/xy = 5; (8x + 7 y)/xy = 15 (vi) 6x + 3y = 6xy; 2x + 4 y = 5xy (vii) 10/(x + y) + 2/(x - y) = 4; 15/(x + y) - 5/(x - y) = - 2 (viii) 1/(3x + y) + 1/(3x - y) = 4; 1/2(3x + y) - 1/2(3x - y) = 8
- The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju are twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
- One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II] [Hint: x +100 = 2 ( y -100), y +10 = 6 (x -10)]
visual curriculum