Chain Rule

1.Direct Proportion:
Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.
For example: Cost is directly proportional to the number of articles.
(More Articles, More Cost)

2.Indirect Proportion:
Two quantities are said to be indirectly proportional, if on the increase of the one, the orther decreases to the same extent and vice-versa.
For example:
The time taken by a car is covering a certain distance is
inversely proportional to the speed of the car. (More speed, Less is
the time taken to cover a distance.)

Note: In solving problems by chain rule, we compare every item with the term to be found out.

Chain Rule Formula and Terms

Example:1
In a dairy farm, 40 cows eat 40 bags of husk in 40 days. In how many days one cow will eat one bag of husk?
Solution:-
40 cows eat 40 bags of husk in 40 days
1 cow 1 bag of husk in 40 days

Example:2
36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work?
Solution:-
Given that;
Persons and no of days are indirect proportional to each other
Which means if persons increased then the number of days will reduce to do a job and vice versa
36 men in 18 days
27 men in ? days
36(9 x4)=18(6x3)
27(9x3)=24(6x4)
27 men can finish it in 24 days

Example:3
3 pumps, working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
Solution:-
Given that;
Number of pipes and Number of days are indirect proportional to each other Which means if no of pipes are increased to fill/empty a tank then the number of days required to fill/empty the tank will reduce
3 pumps working 2 days (8 hour a day)
So 3 pumps working 16 hour to empty
Our question is
4 pumps working 1 day ( ? hour a day)
3(1x3)=16(4x4)
4(1x4)=12(4x3)
4 pumps can empty the tank in 1 day by working( 12 hour a day)

Example:4
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?
1R
Solution:-
Persons and no of days are indirect proportional to each other
Which means if persons increased then the number of days will reduce to do a job and vice versa
39 persons 12 days(5 hour a day)
So, 39 persons are working 60 hours each
Our question is 30 person ? days (6 hour a day)
39(13x3)=60(10x6)
30(10x3)=78(13x6)
So 30 persons are working 78 hours each
30 person can finish that work in 13 days (78/6=13, 6 hours a day)

Example:5
If a quarter kg of potato costs 60 paise, how many paise will 200 gm cost?
Solution:-
Given that
Kilogram and cost are direct proportional to each other
This means if the number of kgs of purchase is increased and then cost for that purchase also will increase
Quarter kg =250 gm
250 g = 60 paise
200 g=? paise
250(5x50)= 60(5x12)
200(4x50)=48(4x12)
200 g potato is 48 paisa

Example:6
4 persons can do 4 works in 4 days. In how many such works can be done by 8 men in 8 days?
Solution:-
Given that;
4 men 4 works in 4 days
8 men ? works in 8 days
1 men 1 work in 4 days
1 man 2 works in 8 days
8 man (8x2) 16 works in 8 days