Chain Rule

 Chain Rule :-

1. Direct Proportion

                               Two quantities are said to be directly proportional,

                               if on the increase or decrease of one, the other increases or decreases

                               the same extent.

Examples 

              (a) Cost of the goods is directly proportional to the number of goods.

                   (More goods, More cost)

              (b) Amount of work done is directly proportional to the number of persons

                   who did the work.     (More persons, More Work)

2. Indirect Proportion (inverse proportion)

                          Two quantities are said to be indirectly proportional

                          (inversely proportional) if on the increase of one, the other

                           decreases to the same extent and vice-versa.

Examples

             (a) Number of days needed to complete a work is indirectly proportional

                (inversely proportional) with the number of persons who does the work

                (More Persons, Less Days needed)

             (b) The time taken to travel a distance is indirectly proportional

                (inversely proportional) with the speed in which one is travelling 

                (More Speed, Less Time)

Note :-In solving questions by chain rule, we compare every item with the term to be find out.

Example 1.If  the price of 12 toys is Rs. 270, what will be the price of 10 toys? 

Explanation:
                  More Toys,More price            (Direct Proportion )

                 12 : 10 ::270 : x
                 12✖x = 270✖10

                  x =(270✖10)/12 =225.

Example 2.If 36 men can do a piece of work in 25 days, then in how many days will 15 men do it? 

Explanation:
                  More men,Less days            (Indirect Proportion )

                 15 : 36 ::25 : x
                 15✖x = 36✖25

                  x =(36✖25)/15 =60
Example 3.39 persons can repair a road in 12 days, working 5 hours a day.
                  In how many days will 30 persons, working 6 hours a day, complete the work?

Explanation:

           Let the required number of days be x.

            Less Persons, More days                          (Indirect Proportion )

             More working hour per day, Less days   (Indirect Proportion )

Persons	30	:	39		:: 12 : x
Working hours/day	6	:	5

              30 ✕ 6 ✕ x = 39 ✕ 5 ✕ 12

x =	(39 ✕ 5 ✕ 12)
	(30 ✕6)

                          x = 13.

Example 4.If 30 men can build a wall 56 m long in 4 days, what length of a similar
                  wall can be built by 25 men in 3 days?

Explanation:

                 
      Let the required length be x meter.
      More Persons, More length built                          (Direct Proportion )
      Less days, Less length built                                  (Direct Proportion )
     

Persons	30	:	25		:: 56 : x
days	4	:	3

              30 ✕ 4 ✕ x = 25 ✕ 3 ✕ 56

x =	(25 ✕ 3 ✕ 56)
	(30 ✕4)

                          x = 35 meters.

Example 5.A soft drink machine fills 980 bottles of 250 g in 7 hours.
                   How many bottles of 300 g can be filled in 6 hours?

Explanation:

                 Let the required number of bottles be x.

                 More hours, More fill bottles            (Direct Proportion )

                 More g in Bottles , Less fill bottles   (Indirect Proportion )

                

Hours	7	:	6		:: 980 : x
g	300	:	250

          300 ✕ 7 ✕ x = 250✕ 6 ✕ 980

x =	(250 ✕ 6✕ 980)
	(300 ✕ 7)

          x = 700.

Example 6. If 6 persons can do a piece of work in 10 days working 7 hours a day.
                   How many days will it take to complete a piece of work twice 14 persons
                   working together for 8 hours a day

Explanation:
                     Let the required number of days be x.
                    More work, More days                      (Direct Proportion )
                    More boys, Less days                        (Indirect Proportion )
                     More hours per ay, Less days            (Indirect Proportion )

           

1✕14✕8✕x = 2✕6✕7✕10

                                                       

x =	(2 ✕ 6✕ 7✕10)
	(1 ✕14✕8)

                                                           x = 15/2 days