Variation-Chain rule
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When one instance of a problem is given & a question is asked on another instance, it involves a rule called chain rule.
A very simple example
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4 carpenters make 20 chairs in 5 days. How many chairs will 8 carpenters make in 10 days?
4 carpenters make 20 chairs in 5 days. So 8 carpenters will make double the number of chairs in 5 days (same time as earlier) i.e. 20 × (8/4) = 40 chairs
So, 8 carpenters will make in 10 days= 40 × (10/5) = 80 chairs ==> Final Answer
Easy, isn't it. But don't underestimate these type of questions. When a long chain rule is applied, it becomes complicated to solve the question as it will involve many steps by this approach.
A General Approach to be followed
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To ease out the step by step process, there is a general approach. It is easier than above method & applicable for any number of steps involved, called Direct Proportionality rule.
Direct proportionality rule
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E.g: Eight men working 10 hrs everyday can completely build a wall of length 180 m, breadth 4 m and height 20 m in 20 days. In how many days can 12 men working 8 hrs a day build a wall of length 300 m, breadth 6 m and height 12 m?
Since we have to find the number of days, so we will use the proportion relative to the number of days.
In General,
==> for less days to be involved, we use proportion making less than 1.
==> for more days to be involved, use proportion making greater than 1.
- 8-->12 men means less days will be taken to do the work. So we use 8/12.
- 10-->8 hrs per day means more days will be taken. So we use 10/8 this time.
- 180-->300 m means more length-->more days. Thus we use 300/180
- 4-->6 m means more breadth-->more days. Thus we use 6/4.
- 20-->12 means less height-->less days. Thus we use 12/20.
In case of other problems, we will define proportions in terms of the quantity asked in the problem.
I hope this concept is clear to everyone & in general could be used to any problem involving chain variation.
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