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If you are in search of **NCERT solutions for Class 8 Maths Chapter 11 Direct and Inverse Proportions**, then you are at the right place. When it comes to **Mathematics**, it is a subject that holds value for students across various academic streams, be it science, biology, or commerce and having a very good understanding of basic math concepts will help you in achieving your goal and make it much more easier.

Here we provide detailed **NCERT solutions for Class 8 Maths** of all the chapters, **exercise wise** with **updated syllabus by NCERT**. These NCERT solutions for Class 8 Maths have been carefully crafted to provide detailed explanations, ensuring that students can grasp the concepts effectively. We also provide extra practice questions for the same, so feel free to bookmark **ncertforclass8.com** to boost your preparation.

## NCERT Solutions for Class 8 Maths Chapter 11 **Direct and Inverse Proportions** Ex 11.2

**Class 8 Maths Chapter** **11 ****Direct and Inverse Proportions** **Exercise 11.2**

**Direct and Inverse Proportions**

**Exercise 11.2**

**1. Which of the following are in inverse proportion?**

(i) The number of workers on a job and the time to complete the job.

(ii) The time taken for a journey and the distance travelled in a uniform speed.

(iii) Area of cultivated land and the crop harvested.

(iv) The time taken for a fixed journey and the speed of the vehicle.

(v) The population of a country and the area of land per person.

Solution-

(i) The number of workers and the time to complete the job is in inverse proportion because less workers will take more time to complete a job, and more workers will take less time to complete the same job.

(ii) Direct proportion.

(iii) Direct proportion because more are of cultivated land will yield more crops.

(iv) Inverse proportion because if time is less, speed is more.

(v) It is an inverse proportion. If the population of a country increases, the area of land per person decreases.

**2. In a Television game show, the prize money of ` 1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?**

Number of winners | 1 | 2 | 4 | 5 | 8 | 10 | 20 |

Prize for each winner (in ₹) | 1,00,000 | 50,000 | … | … | … | … | … |

Solution-

According to the question, the number of winners and prize money are in inverse proportion because winners are increasing, and prize money is decreasing.

`\[When\ the\ number\ of\ winners\ is\ 4,\ each\ winner\ will\ get=\frac{100000}{4}=₹\ 25,000\]`

`\[When\ the\ number\ of\ winners\ is\ 5,\ each\ winner\ will\ get=\frac{100000}{5}=₹\ 20,000\]`

`\[When\ the\ number\ of\ winners\ is\ 8,\ each\ winner\ will\ get=\frac{100000}{8}=₹\ 12,500\]`

`\[When\ the\ number\ of\ winners\ is\ 10,\ each\ winner\ will\ get=\frac{100000}{10}=₹\ 10,500\]`

`\[When\ the\ number\ of\ winners\ is\ 20,\ each\ winner\ will\ get=\frac{100000}{20}=₹\ 5,000\]`

**3. Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table-**

Number of spokes | 4 | 6 | 8 | 10 | 12 |

Angle between a pair of Consecutive spokes | 90° | 60° | … | … | … |

**(i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion?(ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes.(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is 40°?**

Solution-

According to the question, the number of spokes is increasing and the angle between a pair of consecutive spokes is decreasing.

So, it is an inverse proportion and the angle at the centre of a circle is 360°

`\[When\ the\ number\ of\ spokes\ is\ 8,\ then\ the\ angle\ between\ a\ pair\ of\ con\sec utive\ spokes=\frac{360}{8}=\ 45°\]`

`\[When\ the\ number\ of\ spokes\ is\ 10,\ then\ the\ angle\ between\ a\ pair\ of\ con\sec utive\ spokes=\frac{360}{10}=\ 36°\]`

`\[When\ the\ number\ of\ spokes\ is\ 12,\ then\ the\ angle\ between\ a\ pair\ of\ con\sec utive\ spokes=\frac{360}{12}=\ 30°\]`

Number of spokes | 4 | 6 | 8 | 10 | 12 |

Angle between a pairof Consecutive spokes | 90° | 60° | 45° | 36° | 30° |

(i) Yes, the number of spokes and the angles formed between a pair of consecutive spokes is in inverse proportion.

(ii) When the number of spokes is 15, then the angle between a pair of consecutive spokes = 360/15= 24°

(iii) The number of spokes would be needed = 360/40 = 9

**4. If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?**

Solution-

Each child gets = 5 sweets

24 children will get 24×5 = 120 sweets.

Also total number of sweets = 120

If the number of children is reduced by 4, then children left = 24-4 = 20

`\[Now,\ each\ child\ will\ get\ sweets=\frac{120}{20}=6\ sweets\]`

**5. A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?**

Solution-

Let the number of days be x.

Animals | 20 | 30 |

Days | 6 | x |

Total number of animals = 20+10 = 30

Here, the number of animals and the number of days are in inverse proportion.

`\[∴\ \frac{20}{30}=\ \frac{x}{6}\]`

`\[⟹30\times\ x=20\times6\]`

`\[⟹x=\frac{20\times6}{30}=4\ days.\]`

Hence, the food will last for four days.

**6. A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should they take to complete the job?**

Solution-

Let the time taken to complete the job be x.

Persons | 3 | 4 |

Days | 4 | x |

Here, the number of persons and the number of days are in inverse proportion.

`\[∴\ \frac{3}{4}=\frac{x}{4}\]`

`\[⟹4\times x=3\times4\]`

`\[⟹x=\frac{3\times4}{4}=\ 3\ days.\]`

Hence, 4 persons will complete the job in 3 days.

**7. A batch of bottles were packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would be filled?**

Solution-

Let the number of boxes be x.

Number of boxes in each box | 12 | 20 |

Boxes | 25 | x |

Here, the number of bottles and the number of boxes are in inverse proportion.

`\[∴\ \frac{12}{20}=\frac{x}{25}\]`

`\[⟹20\times x\ =\ 12\times25\]`

`\[⟹x=\frac{12\times25}{20}=15\ boxes\]`

Hence, 15 boxes would be filled.

**8. A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?**

Solution-

Let the number of machines required be x.

Days | 63 | 54 |

Machines | 42 | x |

Here, the number of machines and the number of days are in inverse proportion.

`\[∴\ \frac{63}{54}=\frac{x}{42}\]`

`\[⟹54\times x=\ 63\times42\]`

`\[⟹\ x=\frac{63\times42}{54}=49\ machines\]`

Hence, 49 machines would be required.

**9. A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h?**

Solution-

Let the number of hours be x.

Speed (in Km/hour) | 60 | 80 |

Time (in Hours) | 2 | x |

Here, the speed of the car and time are in inverse proportion.

`\[∴\ \frac{60}{80}=\frac{x}{2}\]`

`\[⟹\ 80\times x=60\times2\]`

`\[⟹\ x=\ \frac{60\times2}{80}=\ \frac{3}{2}=\ 1\ \frac{1}{2}\ hours\]`

`\[Hence\ the\ car\ will\ take\ \ 1\frac{1}{2}\ hours\ to\ reach\ the\ destination.\]`

**10. Two persons could fit new windows in a house in 3 days.(i) One of the persons fell ill before the work started. How long would the job take now?(ii) How many persons would be needed to fit the windows in one day?**

Solution-

i) Let the number of days be x.

Persons | 2 | 1 |

Days | 3 | x |

Here, the number of persons and the number of days are in inverse proportion.

`\[∴\ \frac{2}{1}=\frac{x}{3}\]`

`\[⟹\ \ 1\times x=2\times3\]`

`\[⟹\ \ x=\frac{2\times3}{1}=6\ days\]`

(ii) Let the number of persons be x.

Persons | 2 | x |

Days | 3 | 1 |

Here, the number of persons and the number of days are in inverse proportion.

`\[∴\ \ \frac{2}{x}=\frac{1}{3}\]`

`\[⟹\ 1\times x=2\times3\]`

`\[⟹\ x=\frac{2\times3}{1}=\ 6\ persons.\]`

**11. A school has 8 periods a day each of 45 minutes duration. How long would each period be, if the school has 9 periods a day, assuming the number of school hours to be the same?**

Solution-

Let the duration of each period be x.

Period | 8 | 9 |

Duration of period (in minutes) | 45 | x |

Here, the number of periods and the duration of periods are in inverse proportion.

`\[∴\ \ \frac{8}{9}=\frac{x}{45}\]`

`\[⟹\ 9\times x=8\times45\]`

`\[⟹\ x=\frac{8\times45}{9}=40\ \min utes\]`

Hence, the duration of each period would be 40 minutes.

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