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.

**Solution:**

Let speed of rowing in still water km/h

Let speed of current km/h

So, speed of rowing downstream km/h

And, speed of rowing upstream km/h

According to given conditions, we have

and

and

** (1) ** and ** (2)**

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

Putting in **(1)**, we get

Therefore, speed of rowing in still water km/h

Speed of current 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. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

**Solution:**

Let time taken by 1 woman alone to finish the work days

Let time taken by 1 man alone to finish the work days

So, 1 woman’s 1 day work part of the work

And, 1 man’s 1 day work part of the work

So, 2 women’s 1 day work part of the work

And, 5 men’s 1 day work part of the work

Therefore, 2 women and 5 men’s 1 day work 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 part of the work. **(2)**

Clearly, we can see that **(1)** = **(2)**

**(3)**

Similarly, 3 women’s 1 day work part of the work

And, 6 men’s 1 day work part of the work

Therefore, 3 women and 6 men’s 1 day work part of the work **(4)**

It is given that 3 women and 6 men complete work in = 3 days

It means that in 1 day, they will be completing part of the work. **(5)**

Clearly, we can see that **(4)** = **(5)**

**(6)**

Let and

Putting this in **(3)** and **(6)**, we get

and

**(7)** and ** (8)**

Multiplying** (7)** by 9 and **(8)** by 8, we get

** (9)**

**(10)**

Substracting **(10)** from **(9)**, we get

Putting this in **(8)**, we get

Putting values of **p and q** in and , we get

and

Therefore, 1 woman completes work in **days**

And, 1 man completes work in **days**

**(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:**

Let speed of train km/h

Let speed of bus km/h

According to given condition, we have

and

Let and

Putting this in the above equations, we get

** (1)** and **(2)**

Multiplying **(1)** by 5 and **(2)** by 3 we get

** (4)**

**(5)**

Subtracting **(5)** from **(4)**, we get

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

We know that and

Therefore, km/h and km/h

Therefore, speed of train km/h

And, speed of bus km/h