The method in which the value of a unit quantity is first calculated to get the value of any quantity is called the unitary method. In unitary method we have two types of variations :

### 1. Direct variation:

**Condition:** Increase in one quantity causes increase in the other and decrease in one quantity causes decrease in the other .

**Examples:**

- With more money , more articles can be bought.
- With less number of men , less work can be done in same time .
- With a greater speed a larger distance can be covered in the same time.

### 2. Inverse variation:

**Condition:** Increase in one quantity causes decrease in the other and decrease in one quantity causes increase in the other.

**Examples:**

- With a greater speed, less time will be taken to cover the same distance .
- With less number of men , more days are required for the same work .

## Direct Variation Examples:

**A man earns Rs 400 in 10 days . How much will he earn in 28 days ?**

**Solution:**

Since in 10 days , the man earns = Rs 400

Therefore in 1 day , he will earn =

Therefore in 28 days , he will earn =

**0.75 metre of cloth costs Rs 45. What will be the cost of 0.6 metre of same cloth?**

**Solution:**

Given: The cost of 0.75 m cloth =Rs 45

Therefore the cost of 1m cloth =

And, the cost of 0.6 m cloth =

## Inverse Variation Examples:

**4 men can do a piece of work in 5 days. How many men will do it in 4 days?**

**Solution :**

Since in 5days , the work is done by 4 men

Therefore in 1 day , the work will be done by

In 4 days , the work will be done by

**With a speed of 60 km/h , it takes 4 hours to cover a certain distance. What should be the speed, if the same journey is to be completed in 3 hours?**

**Solution:**

Since to cover a certain distance in 4hours , speed required =60km/h

Therefore to cover the same distance in 1hour , speed required =

And, to cover the same distance in 3 hours ,speed required =

### Remember:

- For getting more, multiply.
- For getting less, divide.

**Example: A watch gains 42 sec in 3 days and 8 hours. How long will it take to gain 2 min 6 sec?**

**Solution:**

3 days+8 hours = and 2 min 6 sec=

Given: The watch gains 42 sec in 3 days 8 hours

Then the watch gains 1 sec in

And , it will gain 126 sec in .

Therefore time taken by watch to gain 2 min 6 sec=240 hours =10 days.

**Example: If 135 kg of corn feeds 45 horses for 8 days, how long will the same quantity of corn feed 24 horses?**

**Solution:**

Here, in both the cases , the quantity of corn remains the same .

Given : For 45 horses , the given quantity of corn is sufficient for 8 days .

For 1 horse , the same quantity of corn will be sufficient for

For 24 horses, it will be sufficient for

## Exercise:

- Fill in the blanks:
- Cost of 2.25m cloth is required for 1 shirt , then cloth required for 3 shirts =______
- 4 men can do a work in 6days,then 6men will do the same work in ______
- With speed=40km/h , a certain distance is covered in 6 hours , then with speed=60km/h , the same distance is covered in ________ hours.
- For 8 days ,a family requires Rs320 then for 4 days , the family will require =__________
- 1 man does a work in 15days , 3 men will do the same work in ________

- 6 books weigh 1 Kg 260 g . How many books will weigh 3 Kg 150 g?
- 8 men complete a work in 6 hours . In how many hours will 12 men complete the same work ?
- If a 25 cm long candle burns for 45 minutes , how long will another candle of the same material and same thickness but 5 cm longer than the previous one, burn?
- 400 men have provision for 23 weeks. They are joined by 60 men. How long will the provisions last?
- If the interest on Rs 200 be Rs 25 in a certain time , what will be the interest on Rs750 for the same time?
- If 3 dozen eggs cost Rs30 , find the cost of 3 scores of eggs .(1score=20)
- If the fare for 48km is Rs288, what is it for 36km?
- 9 lines of a print ,in a column of a book ,contains 36words. How many words will a column of 51 lines contain?
- 15 masons can build a wall in 20 days. How many masons will build the wall in 12days?
- The cost of of a cargo is Rs 7200. Find the cost of of the same cargo.
- 50 men mow 32 hectares of land in 3 days .How many days will 15 men take to mow it?
- A garrison of 1200 men has provisions for 15 days.How long will the provisions last if the garrison be increased by 600 men?

## TIME AND WORK:

For solving problems on time and work , make the following notes:

**Note 1: **

If a man does a work in 50 days , his one day’s work =

Conversely, If one day’s work of a man is , he will complete the work in 50 days.

Thus, one day’s work = (No. of days required to complete the work)

And, number of days required to complete a work = (one day’s work)

**Note 2:** Time required to do a certain work=

**Example: Ajay and vijay can paint a hall in 6days. Ajay alone can paint it in 8 days. In how many days can Vijay alone paint it ?**

**Solution:**

Ajay and Vijay, in 1 day, can paint of the hall and Ajay alone, in 1 day, can paint of the hall.

Vijay alone, in 1 day, can paint of the hall, i.e., of the hall

Vijay alone, can paint the hall in 24 days.

**Example: A tap fills a cistern in 4 hours . Another tap emties the full cistern in 6 hours . How long will it take to fill the cistern, if the cistern is empty and both the taps are open?**

**Solution:**

one tap, in 1 hour, fills of the cistern

And, the second tap, in 1 hour, empties of the cistern

Therefore both the taps together in 1 hour will fill of the cistern, i.e., of the cistern.

the cistern will be full in 12 hours.

**Example: A and B can do a piece of work in 15 days and 20 days respectively.** Find:

**The work done by A in 3 days****The work left after A has worked for 3 days****The number of days that B will take to complete the remaining work.**

**Solution:**

1. A’s one day’s work=

work done by A in 3 days=

2. Work left after A has worked for 3 days=

3. Remaining work= and B’s one day’s work=

number of days taken by B to complete the remaining work

**Example 10: A completes a piece of work in 4 days and B completes it in 6 days. How long will it take to complete the same work, if they both work on it, together?**

**Solution:**

Given: A does the work in 4 days

A’s one day’s work=

B does the work in 6 days

B’s one day’s work=

This implies (A+B)’s one day’s work=

## Exercise 2:

- A can do a pirce of work in 4days and B can do the same work in 5 days. Find how much work can be done by them working together in: i) 1 day ii) 2 days. Also find how much work will be left ,after they have worked together for 2 days?
- A, B and C can do a piece of work in 12, 15 and 20 days respectively. How long will they take to do it working together?
- Two taps fill a cistern in 10 hours and 8 hours respectively . A third tap can empty it in 15hours .How long will it take to fill the cistern, if all of them are opened together?
- Mohit can complete a work in 50 days, whereas Anuj can complete the same work in 40 days. Find:
- Work done by Mohit in 20 days.
- Work left after Mohit has worked on it for 20 days.
- Time taken by Anuj to complete the remaining work.

- Joseph and Peter can complete a work in 20 hours and 25 hours respectively. Find:
- Work done by both together in 4 hours.
- Work left after both worked together for 4 hours.
- Time taken by Peter to complete the remaining work.

- A is able to complete of a certain work in 10 hours and B is able to complete of the same work in 12 hours. Find:
- How much work can A do in 1 hour?
- How much work can B do in 1 hour?
- In how much time will the work be completed, if both work together.

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