Q2 of 30 Page 67

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.

(i) 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............(1)


2(x-y) = 4


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


Adding equation (1) and (2),


we obtain


2x = 12 = x = 6


Putting this in equation (1),


we 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 =


Work done by a man in 1 day =


According to the question,






Putting in these equations,


we obtain



= 8p + 20q = 1



= 9p + 18q = 1


By cross-multiplication, we obtain







x = 18, y = 36


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


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


According to the given information,




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


More from this chapter

All 30 →
4

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.


(ii) A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is added to its denominator. Find the fraction.


(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?


(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?


(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

 1

Solve the following pairs of equations by reducing them to a pair of linear equations:



















(i)




(ii)




(iii)




(iv)




(v)




(vi)




(vii)




(viii)



 1

The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differs by 30 years. Find the ages of Ani and Biju.

 2

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)]