## Chain Rule

**Points to Remember**

**1) Direct Proportion:**Any two quantities are said to be directly proportional, if on the increase of one quantity, the other quantity increases and vice-versa.

**Example:**Cost is directly proportional to number of objects

Cost ∝ Number of objects

Number of objects increases (↑) Cost (↑)

**Example:**Work done is directly proportional to number of working men

Work done ∝ Number of working men

Number of men increase (↑) Work done (↑)

**2) Indirect Proportion:**Any two quantities are said to be indirectly proportional, if on the increase of one quantity, the other quantity decreases and vice-versa.

**Example:**If speed of car is increases, then the time required to cover the distance decreases.

Speed of car (↑) Time required decreases (↓)

**Example:**Time taken to finish work increases, if number of men decrease.

Time (↑) Number of men (↓)

**Tips and Tricks**

In this chapter generally different types of numerical related to time and work, time and speed, cost and number of articles, men and work, etc. are asked.

Type 1: Indirect proportion

**Q 1. If 30 men can do a piece of work in 20 hours, then in how many hours will 12 men do it?**

a. 18 hours

b. 30 hours

c. 40 hours

d. 50 hours

View solution

Correct Option: (d)**Hint: ** As number of workers increase, the time required decreases. Hence, this is a problem related to indirect proportion.

Workers (↑),Time (↓)

Let the number of hours be x.

12 : 30 :: 20 : x

20 | = | 20 |

30 | x |

x = | 20 × 30 | = 50 |

12 |

**Q 2.**3 pumps, working 4 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work, to empty the tank in one day?

a. 7 hours

b. 8 hours

c. 6 hours

d. 5 hours

View solution

Correct Option: (c)**Hint: ** As number of pumps increase, the time required decreases and when working hours increase, fewer days are required to complete the work. Hence, this is a problem related to indirect proportion. **Given: ** 3 pumps can empty a tank in 2 days, if they are working 6 hours a day**Find: ** Number of hours a day, 3 pumps must work, to empty the tank in one day.

More pumps (↑),Less working hours (↓)

More working hours (↑),Less days (↓)

4 : x :: | Pumps are in the ratio 4 : 3 | |

Days are in the ratio 1 : 2 |

24 = 4x

x = 6

Alternate solution 1: (Trick)

Arrange all given parameters in table format.

Pumps | Days | Hours |

3 | 2 | 4 |

4 | 1 | ’A’ hrs |

3 × 2 × 4 = 4 × 1 × A

A = | 3 × 2 × 4 | = 6 hrs |

4 × 1 |

We are given that, 3 pumps, working 4 hours a day, can empty a tank in 2 days. Therefore, it means that:

3 pumps take total 8 hours to empty the tank.

**Hence, 1 pump will take 8 x 3 = 24 hours**

Remember:

**As number of pump decrease, time required increases.**

**So, if 4 pumps work, time required decreases.**

24 | = 6 hrs |

4 |

**Q 3.**A wheel that has 6 cogs is meshed with a larger wheel of 12 cogs. If the smaller wheel has made 22 revolutions, then find the number of revolutions made by the larger wheel.

a. 11

b. 13

c. 15

d. 17

View solution

Correct Option: (a)**Hint: ** As number of cogs increase, the revolutions made decrease. Hence, this is a problem related to indirect proportion.

Let the number of wheels be x.

More cogs (↑),Less revolutions (↓)**Given: ** 6 cogs meshed with wheel of 12 cogs and smaller wheel made 22 revolutions

Therefore,

12:6 :: 22:x

12 × x=6 × 22

x = | 6 × 22 | = 11 |

12 |