**Chain Rule Solved Examples**

**Study following chain rule problems for a deeper understanding of chain rule:**

**1. If 40 men working 16 hrs a day can do a piece of work in 48 days, then 48 men working 10 hrs a day can do the same piece of work in how many days?**

**Answer :**

Here the component days is having its one part missing and the given part of this i.e. 48 days, will be taken as base and will be compared individually with the other components.

When 40 men are working, it takes 48 days and when 48 men will work, they will take lesser days (more men, lesser days). So, numerator should be lesser.

When work is done for 16hrs- it takes 48 days, when it is done for 10hrs, more days will be taken (less hours, more days). So, numerator should be more. The answer would be 48 × (40/48) × (16/10) = 64 days.Hence, answer is 64 days.

** 2. A group of 900 men has provisions for 70 days. After 34 days, a reinforcement of 300 men comes. The food will last for how many days more?**

**Answer :**

Originally there are 900 men having provisions for 70 days. After 34 days, 900 men would have eaten their share of 34 days. The remaining provision is for 70 –34 = 36 days. Now, when 300 men came, the total strength becomes 900 + 300 = 1200.

900 men have provision for 36 days, 1200 will have for how many days? (More men, lesser days). So, the answer would be 36 × 900/1200 = 27 days.Hence, answer is 27 days.

**3. A contractor employed 300 men to complete a strip of highway in 300 days. After 100 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?**

**Answer :**

men | work | days |

300 | 1/4 | 100 |

? | ¾ (remaining work) | 200 (remaining days) |

More work is there, more men are required (rule 2).

More days are remaining; fewer men are required (rule 1).

Total men required = 300 × (3/4) × (4/1) × (100/200) = 450

Now, 300 men are already there, so 450 – 300 = 150 additional men are required.Hence, answer is 150 men.

**4. If 12 men or 16 women can do a piece of work in 12 days, in how many days can the work be done by 9 men and 12 women working together?**

**Answer :**

12m = 16w ⇒ 3m = 4w

⇒ 3 men are equal to 4 women i.e. lesser men are equal to more women.

9 men will be equal to 9×4/3 = 12 women.

Second part of the problem is 9 men and 12 women.

As calculated, 9 men are equal to 12 women and there are 12 women themselves.

So, the total number of women is 12 + 12 = 24.

Apply the chain rule as

Women | Days |

16 | 12 |

24 | ? |

More women, lesser days i.e. numerator should be smaller ⇒12× 16/24 = 8 days.Hence, answer is 8 days.

**5. Some men promised to do a job in 40 days. 12 of them were absent and the remaining men did the job in 60 days. What is the original number of men?**

**Answer :**

Let there be m men in the beginning, who promised to do the job in 40 days.

But 12 became absent, means the remaining (m – 12) men did the job in 60 days.

The work is same ⇒40m = 60(m – 12) ⇒40m = 60m – 720 ⇒ m = 36.Hence, answer is 36 men.

**6.Lenin Square was built by 2000 men in 25 days. In how many days can 2500 men do the work if their working hours per day are reduced in the ratio 2: 5?**

**Answer : **

Let the number of working hours per day be t hours. Therefore, Lenin Square was built in 2000 × 25 × t man hours = 50,000t man hours. If 2500 men work 2t/5 hours a day, they can build it in (2000 × 25 × t) / (2500 × (2t/5)) = 50 days. Hence, answer is 50 days.

7. **A group of people can do a work in 30 days. With 10 of them being absent, the work is done in 40 days. How many people are there in the original group?**

**Answer :**

Let there be x number of people who do the work in 30 days. When there are x – 10 people, they do the work in 40 days. (x / (x-10)) = (40/30) (Note that it is not (40/30) , as the number of people working and the number of days it takes to finish the work are inversely proportional).So, 30x = 40x – 400 ⇒x = 40. Hence, answer is 40 days.

**8. If 10 men or 12 women take 40 days to complete a piece of work, how long will 15 men and 6 women take to do the same work?**

**Answer :**

Productivity of 10 men = Productivity of 12 women

⇒ Productivity of 1 man = Productivity of (12/10) women.

∴Productivity of (15 men + 6 women)

= Productivity of (15 + 6 × (10/12) ) men = Productivity of 20 men

If 10 men take 40 days, then 20 men will take (10/20) × 40 days = 20 days.

(Again note the inverse variance relationship used in computing the number of days). Hence, answer is 20 days.

**9. If 10 programmers, working 16 hours a day, can finish the coding of a project in 25 days, then how many days will be taken by 12 software engineers, working 5 hours a day to finish the coding of the project? It is known that 3 software engineers code as much as 2 programmers in the same time.**

**Answer :**

Numbers | Hrs | Days | |

Programmer | 10 | 16 | 25 |

Engineers | 12 | 5 | x |

Also, 3E = 2P i.e. 12 E = 8P ⇒ 10 P × 16 × 25 = 12 E × 5 × x.

⇒10 P × 16 × 25 = 8P × 5 × x. Number of days = x = 100 days. Hence, answer is 100 days.

**10. A certain group of men were assigned to do a piece of work in 50 days. But 6 of them became absent and it took 56 days to complete the work. Find the original number of men in the group.**

**Answer :**

Let the original number of men = x.

They were to complete the job in 50 days but as 6 of them became absent from the 1

^{st}day. ∴ Men left = x – 6.

They will complete the work in 56 days. ∴ M1D1 = M2 D2

x × 50 = (x – 6) × 56 ⇒ x = 56. Hence, answer is 56 men

**11. 300 persons working 18 hours a day complete 1/2th of the work in 20 days. The number of additional persons, working 16 hours a day, reqd. to complete the remaining work in 10 days is**

**Answer :**

Total number of persons required

=300 × (20/10) × (18/16) × (2/1) × (1/2) = 675

∴ Additional person required = 675 – 300 = 375. Hence, answer is 375 persons

**12. A contractor employed 30 men to do a piece of work in 38 days. After 25 days, he employed 5 men more and the work was finished one day earlier. How many days he would have been behind, if he had not employed additional men?**

**Answer :**

After 25 days, 35 men complete the work in 12 days. Thus, 35 men can finish remaining work in 12 days.

Therefore,30 men can do it in(12 x 15)/30 = 14 days, which is 1 day behind.

Hence answer is option 1.