Question

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 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 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.

Hint: We are going to solve this by using the idea of upstream and downstream speeds.Upstream refers to the direction towards the source of the river, against the direction of flow. Downstream describes the direction towards the mouth of the river, in which the current flows.

Let her speed in still water be x and speed of stream be y.
Now, According to question and using $time = \frac{{dis\tan ce}}{{speed}}$
$\Rightarrow \frac{{20}}{{x + y}} = 2$ and $\frac{4}{{x - y}} = 2$
$\Rightarrow x + y = 10;x - y = 2$
On solving we get $x = 6$ and $y = 4$
Speed of Ritu in still water is 6 km/hr and that of a stream is 4 km/hr.

Hint: We are going to solve this problem by using the idea of time and work concept.
Let the one woman do x unit work in one day and man do y unit.
Now according to question,
$\Rightarrow 2 \cdot x \cdot 4 + 5 \cdot y \cdot 4 = 1$\Rightarrow 8x + 20y = 1-- (1) \Rightarrow 3 \cdot x \cdot 3 + 6 \cdot y \cdot 3 = 1$ \Rightarrow 9x + 18y = 1$ -- (2)
Multiply equation (1) by 9 and (2) by 8 then we have
$\Rightarrow 72x + 180y = 9$ and $\Rightarrow 72x + 144y = 8$
On solving these two we have
$y = \frac{1}{{36}}$ that is one man can finish that work in 36 days
$x = \frac{1}{{18}}$ That is one woman can finish that work in 18 days.

Hint: we need to have a basic idea on the topic time, speed and distance to solve this problem.
Let the speed of the bus be x and that of the train be y.
Now, according to question and using $time = \frac{{dis\tan ce}}{{speed}}$
$\Rightarrow \frac{{60}}{y} + \frac{{240}}{x} = 4 \Rightarrow \frac{{15}}{y} + \frac{{60}}{x} = 1$ --- (i)
$\Rightarrow \frac{{100}}{y} + \frac{{200}}{x} = \frac{{25}}{6} \Rightarrow \frac{{24}}{y} + \frac{{48}}{x} = 1$ -- (ii)
From (i)
$\Rightarrow \frac{1}{x} = \frac{1}{{60}}\left( {1 - \frac{{15}}{y}} \right)$ Putting this in equation (ii)
$\Rightarrow \frac{{24}}{y} + \frac{{48}}{{60}}\left( {1 - \frac{{15}}{y}} \right) = 1$
$\Rightarrow \frac{{24}}{y} + \frac{4}{5} - \frac{{12}}{y} = 1$
$\Rightarrow \therefore y = 60km/hr$
Then
Note:
(i) A boat is said to go downstream if it is moving along with the direction of the stream. The overall speed of the boat in this case is called downstream speed. A boat is said to go upstream if it is moving opposite to the direction of the stream. The overall speed of the boat in this case is called upstream speed. In downstream the boat speed is more compared to upstream because water speed will add to the normal boat speed.
(ii) More work means we need more time required to complete the work or we have to increase the number of workers to complete the work at the same time. Time and people are inversely proportional that means if we have more people we can complete the work in less time. Work and time are directly proportional that means we need more time to complete the more work. Rate of work*time=Work done.
(iii) We have Time=Distance/Speed, we need to understand the relation between time, speed and time and distance and distance, speed.