# Chain Rule Questions

**FACTS AND FORMULAE FOR CHAIN RULE QUESTIONS**

**1. Direct Proportion: **Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.

Ex. 1. Cost is directly proportional to the number of articles. (More Articles, More Cost)

Ex. 2. Work done is directly proportional to the number of men working on it (More Men, More Work)

**2. Indirect Proportion: **Two quantities are said to be indirectly proportional,if on the increase of the one, the other decreases to the same extent and vice-versa.

Ex. 1. The time taken by a car in covering a certain distance is inversely proportional to the speed of the car. (More speed, Less is the time taken to cover a distance)

Ex. 2. Time taken to finish a work is inversely proportional to the num of persons working at it. (More persons, Less is the time taken to finish a job)

A) 75 | B) 82 |

C) 100 | D) 110 |

Explanation:

Originally let there be x men.

Less men, More days (Indirect Proportion)

Therefore, (x-10) : x :: 100 :110

=> (x - 10) * 110 = x * 100 => x= 110

A) 46 | B) 47 |

C) 48 | D) 49 |

Explanation:

Let the required length be x meters

More men, More length built (Direct proportion)

Less days, Less length built (Direct Proportion)

$\left.\begin{array}{r}\begin{array}{cc}Men& 20:35\end{array}\\ \begin{array}{cc}Days& 6:3\end{array}\end{array}\right\}\vdots \vdots 56:x$

=> (20 x 6 x X)=(35 x 3 x 56)

=> x = 49

Hence, the required length is 49 m.

A) 1 | B) 3 |

C) 7 | D) 14 |

Explanation:

Let the required number days be *x*.

*Less spiders, More days (Indirect Proportion)*

*Less webs, Less days (Direct Proportion)*

$\left.\begin{array}{r}\begin{array}{cc}Spiders& 1:7\end{array}\\ \begin{array}{cc}Webs& 7:1\end{array}\end{array}\right\}\vdots \vdots 7:x$

=> 1 * 7 * *x* = 7 * 1 * 7

=> x= 7

A) 18 days | B) 21 days |

C) 24 days | D) 30 days |

Explanation:

(2 x 14) men +(7 x 14) boys = (3 x 11) men + (8 x 11) boys

=>5 men= 10 boys => 1man= 2 boys

Therefore, (2 men+ 7 boys) = (2 x 2 +7) boys = 11 boys

( 8 men + 6 boys) = (8 x 2 +6) boys = 22 boys.

Let the required number of days be x.

More boys , Less days (Indirect proportion)

More work , More days (Direct proportion)

$\left.\begin{array}{r}\begin{array}{cc}Boys& 22:11\end{array}\\ \begin{array}{cc}Work& 1:3\end{array}\end{array}\right\}\vdots \vdots 14:x$

Therefore, (22 * 1 * x) = (11 * 3 * 14)

=> x = 21

Hence, the required number of days = 21

A) 9 | B) 10 |

C) 11 | D) 12 |

Explanation:

Let the required no of working hours per day be x.

More pumps , Less working hours per day (Indirect Proportion)

Less days, More working hours per day (Indirect Proportion)

$\left.\begin{array}{r}\begin{array}{cc}Pumps& 4:3\end{array}\\ \begin{array}{cc}Days& 1:2\end{array}\end{array}\right\}\vdots \vdots 8:x$

=> (4 * 1 * x) = (3 * 2 * 8)

=> x=12

A) 10 | B) 13 |

C) 14 | D) 15 |

Explanation:

Let the required number of days be *x*.

*Less persons, More days (Indirect Proportion)*

*More working hours per day, Less days (Indirect Proportion)*

$\left.\begin{array}{r}\begin{array}{cc}Persons& 30:39\end{array}\\ \begin{array}{cc}Workinghours/day& 6:5\end{array}\end{array}\right\}\vdots \vdots 12:x$

=> 30 * 6 * *x* = 39 * 5 * 12

=> x= 13

A) 24 days | B) 28 days |

C) 34 days | D) 35 days |

Explanation:

Less Men, means more Days {Indirect Proportion}

Let the number of days be x then,

27 : 36 :: 18 : x

[Please pay attention, we have written 27 : 36 rather than 36 : 27, in indirect proportion, if you get it then chain rule is clear to you :)]

=>27x = 36 * 18

=> x = 24

So 24 days will be required to get work done by 27 men.

A) 6 days | B) 12 days |

C) 4 days | D) 3 days |

Explanation:

Let x men can do the in 12 days and the required number of days be z

More men, Less days [Indirect Proportion]

Less work, Less days [Direct Proportion ]

$\left.\begin{array}{r}\begin{array}{cccc}men& 2x& :& x\end{array}\\ \begin{array}{cccc}work& 1& :& \frac{1}{2}\end{array}\end{array}\right\}::12:z$

$\therefore \left(2x\times 1\times z\right)=\left(x\times \frac{1}{2}\times 12\right)$

$\Rightarrow z=3$