# Chain Rule

**The technique of chain rule is applicable in all the cases where two or more than two components are given.**

Out of these components, one component would be having its one part missing and the other given part of the same component is taken as base and it is compared individually with all the other components one by one and the following two methods are applied. The technique is as follows:

1. If the missing part of the same component should be greater than the given part, then numerator is kept greater than the denominator.

2. Secondly if the missing part of the same component is smaller than the given part, then numerator is kept smaller than the denominator.

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# Concept 1: Inverse Relationship

**When doing sums on time and work remember that inverse relationship applies. Students often make a mistake by using direct relationship.**

**Example :**

If 5 men do a piece of work in 1 day, in how many days will 10 men do the same work?

The tendency here is to use direct relationship and arrive at the answer of 2 days, since the quantity on the left has doubled.

However, the answer should be 1⁄2 day, since more men will take lesser time.

Students should learn to avoid this common mistake.

**Illustration 1:**

24 men can do a piece of work in 40 days. 20 men will do the same work in how many days?

Earlier 24 men were working and now 20 men are working i.e. lesser number of men are working.

Less men, more days and more men, less days (The relation is ⇑⇓ & ⇓⇑).

In this case there are *lesser *men, so *more *days are required as per the rule 1.

Above the numerator should be greater than the denominator i.e. 40 × 24/20 = 48 days.

**Illustration 2:**

20 women working 12 hrs a day can do a piece of work in 36 days. 12 women working 16 hrs a day will take how many days to complete that work?

Earlier 12 hrs, now 16 hrs i.e. more hrs per day means lesser days (rule 2 ⇑⇓ & ⇓⇑)

Earlier 20 women, now 12 women i.e. lesser persons means more days (rule 1 ⇑⇓ & ⇓⇑).

It takes 36 days earlier. Now = 36 × 12/16 × 20/12 = 45 days

# Concept 2: Using Three Variables

**Illustration 3:**

18 men or 27 women can do a piece of work in 40 days. 22 men and 21 women will do the same work in how many days?

Important: Take care that in the first part it is ‘or’ then in the second part it is ‘and’.
From the first part 18 men = 27 women ⇒ 2 men = 3 women.

In the second part 22 men and 21 women.
Now 21 women would be equal to lesser men (since 3 women = 2 men) hence 21 women = 21 × 2/3 = 14 men. Now add the 22 men also. So we have total men = 22 + 14 = 36.
Chain rule can now be applied:
Earlier there were 18 men and now there are 36 men.
More men mean lesser number of days (rule 2 ⇑⇓ & ⇓⇑) ⇒ 40 × 18/36 = 20 days.

**Illustration 4**:

A group of 600 men has provision for 39 days. After 12 days a reinforcement of 300 men arrives. The food will last for how many days more?

Originally 600 men have provision for 39 days.

Since 12 days have passed, this means that 600 men have eaten their share of 12 days.

The remaining provisions will last for 39 –12 = 27 days.

Now when 300 men arrive. After their joining, the total strength becomes 600 + 300 = 900.

Chain rule can now be applied: 600 men have provision for 27 days, 900 will have for how many days?

MORE MEN LESSER DAYS (⇑⇓ rule 2): Hence 27 × 600/900 = 18 days.

**Illustration 5:**

A contractor employed 120 men to complete a strip of road in 200 days. After 80 days he found that only 1⁄4th of the strip is complete. How many additional men should be employed to complete the work on time?

We make a table as follows:

More work, more men are required (rule 1).
More days remain, lesser men are required (rule 2).
Total men required = 120 × 3⁄4 × 4/1 × 80/120 = 240 men. (Note how the fractions are arrived at, following the above rules) Since 120 men are already there, so 240 –120 = 120 additional men are required.

**Illustration 6**:

A ship was to be unloaded in 20 days. A contractor appointed 280 persons. After 6 days only 1⁄4th of the work was done. Find the number of additional men required to finish the job on time.

More work, more men are required (rule 1).
More days remain, lesser men are required (rule 2).
Hence total men required = 280 × 3⁄4 × 4/1 × 6/14 = 360 men
As 280 men are already there, additional men required = 360 – 280 = 80 men.

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**Illustration 7**:

If 12 men can build a wall 100 Mts. long, 3 Mts. high and 5 Mts. thick in 25 days, in how many days will 20 men build a wall 60 m × 4 m × 2.5 Mts.

First calculate the total volume of the wall which is 100 × 3 × 5 = 1500 cubic Mts.

Volume of the second wall 60 × 4 × 2.5 = 600 cubic Mts. More number of men, lesser days (rule 2).
Lesser volume of the wall, lesser days (rule 1).
Hence 25 × 12/20 × 600/1500 = 6 days