Unitary Method

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:

1. With more money , more articles can be bought.
2. With less number of men , less work can be done in same time .
3. 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:

1. With a greater speed, less time will be taken to cover the same distance .
2. 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:

1. For getting more, multiply.
2. 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:

1. Fill in the blanks:
   1. Cost of 2.25m cloth is required for 1 shirt , then cloth required for 3 shirts =______
   2. 4 men can do a work in 6days,then 6men will do the same work in ______
   3. With speed=40km/h , a certain distance is covered in 6 hours , then with speed=60km/h , the same distance is covered in ________ hours.
   4. For 8 days ,a family requires Rs320 then for 4 days , the family will require =__________
   5. 1 man does a work in 15days , 3 men will do the same work in ________
2. 6 books weigh 1 Kg 260 g . How many books will weigh 3 Kg 150 g?
3. 8 men complete a work in 6 hours . In how many hours will 12 men complete the same work ?
4. 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?
5. 400 men have provision for 23 weeks. They are joined by 60 men. How long will the provisions last?
6. If the interest on Rs 200 be Rs 25 in a certain time , what will be the interest on Rs750 for the same time?
7. If 3 dozen eggs cost Rs30 , find the cost of 3 scores of eggs .(1score=20)
8. If the fare for 48km is Rs288, what is it for 36km?
9. 9 lines of a print ,in a column of a book ,contains 36words. How many words will a column of 51 lines contain?
10. 15 masons can build a wall in 20 days. How many masons will build the wall in 12days?
11. The cost of of a cargo is Rs 7200. Find the cost of of the same cargo.
12. 50 men mow 32 hectares of land in 3 days .How many days will 15 men take to mow it?
13. 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:

1. The work done by A in 3 days
2. The work left after A has worked for 3 days
3. 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:

1. 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?
2. 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?
3. 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?
4. Mohit can complete a work in 50 days, whereas Anuj can complete the same work in 40 days. Find:
   1. Work done by Mohit in 20 days.
   2. Work left after Mohit has worked on it for 20 days.
   3. Time taken by Anuj to complete the remaining work.
5. Joseph and Peter can complete a work in 20 hours and 25 hours respectively. Find:
   1. Work done by both together in 4 hours.
   2. Work left after both worked together for 4 hours.
   3. Time taken by Peter to complete the remaining work.
6. 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:
   1. How much work can A do in 1 hour?
   2. How much work can B do in 1 hour?
   3. In how much time will the work be completed, if both work together.