Chapter 3: Pair of Linear Equations in Two Variables

Q
Pair of linear equations in two variables CBSE NCERT Solutions

Question:

Formulate the following problems as a part 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.

Solutions:

(i) Let the speed of rowing in still water is x km/h

Let the speed of current is y km/h.

So, the speed of rowing downstream is (x + y) km/h.

And, the speed of rowing upstream is (x − y) km/h.

According to given conditions,

$\frac{20}{x+y} = 2$  and   $\frac{4}{x-y} = 2$

2x + 2y = 20 and 2x − 2y = 4

+ y = 10 … (1) and x – y = 2 … (2)

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

2x = 12

= 6

Putting x = 6 in equation (1), we get

6 + y = 10

= 10 – 6 = 4

Therefore, the speed of rowing in still water is 6 km/h and the speed of the current is 4 km/h.

(ii) Let the time taken by 1 woman alone to finish the work in x days

and the time is taken by 1 man alone to finish the work in y days.

So, 1 woman’s 1-day work = $\left ( \frac{1}{x} \right )^{th}$part of the work.

And, 1 man’s 1-day work =  $\left ( \frac{1}{y} \right )^{th}$part of the work.

So, 2 women’s 1-day work = $\left ( \frac{2}{x} \right )^{th}$part of the work

And, 5 men’s 1-day work = $\left ( \frac{5}{y} \right )^{th}$ part of the work.

Therefore, 2 women and 5 men’s 1-day work = $\left (\frac{2}{x} + \frac{5}{y} \right )^{th}$part of the work… (1)

It is given that 2 women and 5 men complete work in 4 days

It means that in 1 day, they will be completing  $\left ( \frac{1}{4} \right )^{th}$part of the work … (2)

We can say that equation (1) is the same as (2).

$\Rightarrow \frac{2}{x} + \frac{5}{y} = \frac{1}{4}$          … (3)

Similarly,     $\frac{3}{x} + \frac{6}{y} = \frac{1}{3}$      … (4)

Let 1/x = p and 1/y = q

Putting 1/x = p and 1/y = q  in equation (3) and equation (4), we get

$2p + 5q = \frac{1}{4}$    and    $3p + 6q = \frac{1}{3}$

8p + 20q = 1 … (5)   and    9p + 18q = 1 … (6)

Multiplying equation (5) by 9 and equation (6) by 8, we get

72p + 180q = 9 … (7)

72p + 144q = 8 … (8)

Subtracting (8) from equation (7), we get

36q = 1

=   1/36

Putting the value of q = 1/36  in equation (6), we get

$9p + 18 \left ( \frac{1}{36} \right ) = 1$

9p = 1/2

p =    1/18

Putting values of p = 1/18  and q = 1/36  in  respectively, we get x = 18 and y = 36

Therefore, 1 woman completes work in 18 days.

And, 1 man completes work in 36 days.

(iii) Let the speed of the train is x km/h and the speed of the bus is y km/h

According to given conditions,

$\frac{60}{x} + \frac{240}{y} = 4$   and  $\frac{100}{x} + \frac{200}{y} = 4 + \frac{10}{60}$

Let   1/x = p and 1/y = q

Putting this in the above equations, we get

60p + 240q = 4 … (1)

And 100p + 200q =  $\frac{25}{6}$ … (2)

Multiplying equation (1) by 5 and equation (2) by 3, we get

300p + 1200q = 20 … (3)

300p + 600q =   $\frac{25}{2}$  … (4)

Subtracting equation (4) from equation (3), we get

600q = 20 − $\frac{25}{2}$ = 7.5

$\frac{7.5}{600}$

Putting the value of q in (2), we get

$100p + 200 \left ( \frac{7.5}{600} \right ) = \frac{25}{6}$

$\Rightarrow 100p + 2.5 = \frac{25}{6}$

$\Rightarrow 100p = \frac{25}{6} - 2.5$

$\Rightarrow 100p = \frac{25}{6} - \frac{25}{10}$

$\Rightarrow 100p = \frac{25}{6} - \frac{5}{2}$

$\Rightarrow 100p = \frac{25-15}{6}$

$\Rightarrow 100p = \frac{10}{6}$

$\frac{10}{600}$

But  1/x = p and  1/y = q

Therefore, x = 600/10 = 60 km/h and y = 600/7.5 = 80 km/h.

Therefore, the speed of the train is 60 km/h and the speed of the bus is 80 km/h.

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