maths : Exercise 3.6

Exercise 3.6

Question 1:

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

Let

and

, then the equations change as follows.

Using cross-multiplication method, we obtain

Putting

and

in the given equations, we obtain

Multiplying equation (1) by 3, we obtain
6p + 9q = 6 (3)
Adding equation (2) and (3), we obtain

Putting in equation (1), we obtain

Hence,

Substituting

in the given equations, we obtain

By cross-multiplication, we obtain

Putting

and

in the given equation, we obtain

Multiplying equation (1) by 3, we obtain

Adding (2) an (3), we obtain

Putting this value in equation (1), we obtain

Putting

and

in the given equation, we obtain

By cross-multiplication method, we obtain

Putting

and

in these equations, we obtain

By cross-multiplication method, we obtain

Hence, x = 1, y = 2

Putting

and

in the given equations, we obtain

Using cross-multiplication method, we obtain

Adding equation (3) and (4), we obtain

Substituting in equation (3), we obtain
y = 2
Hence, x = 3, y = 2

Putting

in these equations, we obtain

Adding (1) and (2), we obtain

Substituting in (2), we obtain

Adding equations (3) and (4), we obtain

Substituting in (3), we obtain

Hence, x = 1, y = 1

Question 2:

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 =

km/h
Downstream =

km/h
According to question,

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

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

By cross-multiplication, we obtain

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,