Important Formulas - Chain Rule
1. Direct Proportion:
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)
Understanding chain rule:
Chain Rule can be applied in questions where two or more than two elements are given. Each element has two figures except one element that has one part missing. Chain rule is used to find out this missing part of an element by subsequent comparison. The other given part of the same element is taken as base and is compared separately with all the other elements.
Example:
If you have a question having following
elements: ‘men’, ‘days’ and ‘hours’. Let us say
‘hours’ are missing then you will compare ‘hours’
with all the other elements independently.
Here
independently means, comparing ‘hours’ with
‘men’ ignoring the element- ‘days’. Similarly while
comparing ‘hours’ and ‘days’ you will ignore the
element- ‘men’.
Chain rule Principle:
After the comparison, the following two
principles are followed:
1.If the missing part is greater than the given part,
then the numerator (n) is kept greater than the
denominator (d) i.e. n/d>1, where n & d are the
given parts of other element.
2.If the missing part is smaller than the given part, then the numerator (n) is kept smaller than the denominator (d) i.e. n/d<1, where n & d are the given parts of other element.
Question 1:
12 examiners (men) work 16 hours a day to check 24000 answer sheets in 18 days. Now, 24 examiners would work how many hours per day to check 36000 answer sheets in 36 days?
Solution:
| Examiner | Hours | Answer Sheets | Days |
|---|---|---|---|
| 12 | 16 | 24000 | 18 |
| 24 | ? | 36000 | 36 |
As we have to calculate the hours in this case, the base would be hours.
Comparisons:
Element = Examiners: If there are 24 examiners now and there were 12 earlier, they need to work lesser hours per day.
Element = Answer sheets: If there are more answer sheets to be checked now, they need to work more hours per day.
Element = Days: If there are more days available, then they need to work lesser hours per day.
Applying chain rule: 16 × (12/24) × (36000/24000) × (18/36) = 6 hours.
In the same illustration if hours were given and answer sheets were missing, then also the method would have been same. Let us solve the same illustration in that manner as well.
Another Method:
12 examiners work 16 hours a day to check 24000 answer sheets in 18 days. Now, 24 examiners would check how many answer sheets working 6 hours a day in 36 days?
Solution :
| Examiner | Hours | Answer Sheets | Days |
|---|---|---|---|
| 12 | 16 | 24000 | 18 |
| 24 | ? | 36000 | 36 |
Following the same steps :
As we have to calculate the answer sheets in
this case, the base would be answer sheets.
If there are 24 examiners now and there were 12
earlier, they will check more answer sheets.
If they work for lesser hours per day, they will
check lesser answer sheets.
If there are more days available, then they will
check more answer sheets.
24000 × (24/12) × (6/16) × (36/18) = 36000.
Question 2:
A certain number of men can complete a piece of work in 180 days. If there are 30 men less, it will take 20 days more for the work to be completed. How many men were there originally?
Solution:
Let there be x men originally.
They were to complete the work in 180 days but as
the number of persons is reduced to x – 30.
? Work takes 20 more days.
So the equation is
180x = (x – 30)200
? 20x = 6000
? x =
300.