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**NCERT Solutions for Class 8 Maths Chapter 11****Direct and Inverse Proportions**Ex 11.1**NCERT Solutions for Class 8 Maths Chapter 11****Direct and Inverse Proportions**Ex 11.2

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

**Ex 11.2 Class 8 Maths Question 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)** Clearly, the more is the number of workers to do a job, the less is the time taken to complete the job.

So, it is a case of inverse proportion.**(ii)** Clearly, the more is the time taken, the more is the distance travelled in a uniform speed.

So, it is a case of direct proportion.**(iii)** Clearly, the more is the area cultivated land, the more is the crop harvested.

So, it is a case of direct proportion.**(iv)** Clearly, the more is the speed of the vehicle, the less is the time taken to cover a fixed distance.

So, it is a case of inverse proportion.**(v)** Clearly, the more is the population, the less is the area of land per person in a country.

So, it is a case of inverse proportion.

** **

**Ex 11.2 Class 8 Maths Question 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 |

The prize for each winner (in ₹) | 1,00,000 | 50,000 | - | - | - | - | - |

**Solution:**Let, the blank spaces be denoted by a, b, c, d and e.

So, we observe that 1 × 100,000 = 2 × 50,000

⇒ 1,00,000 = 1,00,000

Hence, they are inversely proportional.

2 × 50,000 = 4 × a

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

The prize for each winner (in ₹) | 1,00,000 | 50,000 | 25,000 | 20,000 | 12,500 | 10,000 | 5,000 |

**Ex 11.2 Class 8 Maths Question 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 |

The angle between a pair of consecutive spokes | 90° | 60° | - | - | - |

(i) Are the number of spokes and the angle 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:**From the above table, we observe that

4 × 90° = 6 × 60°

360° = 360°

Thus, the two quantities are in inverse proportion.

Let the blank spaces be denoted by a, b and c.

4 × 90° = 8 × a

Hence, the required table is shown below.

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

The angle between a pair of consecutive spokes | 90° | 60° | 45° | 36° | 30° |

(i) Yes, the number of spokes and the angle formed between the pairs of consecutive spokes are in inverse proportion.

(ii) If the number of spokes is 15, then

4 × 90° = 15 × x

x = (4 × 90)/15 = 24°

(iii) If the angle between two consecutive spokes is 40°, then

4 × 90° = y × 40°

y = (4 × 90)/40 = 9 spokes

Thus, the required number of spokes is 9.

**Ex 11.2 Class 8 Maths Question 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 decreased by 4?

**Solution:**

Number of children | Number of Sweets |

24 | 5 |

(24 – 4) or 20 | a |

We observe that on increasing the number of children, number of sweets got by each child will be less. So, they are in inverse proportion.

x_{1}y_{1} = x_{2}y_{2}

where x_{1} = 24, y_{1} = 5, x_{2} = 20 and y_{2} = a (let)

24 × 5 = 20 × a

a = 6

Hence, the required number of sweets is 6.

**Ex 11.2 Class 8 Maths Question 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:**If the number of animals increases, then it will take less days to last.

Then the two quantities are in inverse proportions.

Number of animals | Number of days |

20 | 6 |

(20 + 10) or 30 | p |

Let the required number of days be p.

x_{1}y_{1} = x_{2}y_{2}

where x_{1} = 20, y_{1} = 6, x_{2} = 30 and y_{2} = p (let)

20 × 6 = 30 × p

p = 4

Hence, the required number of days is 4.

**Ex 11.2 Class 8 Maths Question 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:**If the number of persons is increased, it will take less number of days to complete the job.

Thus, the two quantities are in inverse proportion.

Number of persons | Number of days |

3 | 4 |

4 | k |

Let the required number of days be k. Then,

x_{1}y_{1} = x_{2}y_{2}

3 × 4 = 4 × k

k = 3 days

Hence, the required number of days is 3.

**Ex 11.2 Class 8 Maths Question 7.**

### A batch of bottles was 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:**If the number of bottles is increased, then the required number of boxes will be decreased. Thus, the two quantities are in inverse proportion.

Number of boxes | Number of bottles in each box |

25 | 12 |

x | 20 |

Let the required number of boxes be x.

x_{1}y_{1} = x_{2}y_{2}

25 × 12 = x × 20

x = 15

Hence, the required number of boxes is 15.

**Ex 11.2 Class 8 Maths Question 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:**If the number of machines is increased, then the less number of days would be required to produce the same number of articles.

Thus, the two quantities are in inverse proportion.

Number of machines | Number of days |

42 | 63 |

x | 54 |

Let the required number of machines be x.

x_{1}y_{1} = x_{2}y_{2}

42 × 63 = x × 54

x = 49

Hence, the required number of machines is 49.

**Ex 11.2 Class 8 Maths Question 9.**

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

**Solution:**On increasing the speed, it will take less time to travel a distance.

Thus, the two quantities are in inverse proportions.

Speed (in km/h) | Time (in hours) |

60 | 2 |

80 | x |

Let the required time be x hours.

x_{1}y_{1} = x_{2}y_{2}

60 × 2 = 80 × x

x = 3/2 hours = 1^{1}/_{2} hours

Hence, the required time is 1^{1}/_{2} hours.

**Ex 11.2 Class 8 Maths Question 10.**

### Two persons could fit new windows in a house in 3 days.

(i) One of the people 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:**On increasing the number of persons, the less time will be required to complete a job.

Thus, the quantities are in inverse proportion.

Number of persons | Number of days |

2 | 3 |

(i) 1 (2 – 1) | x |

(ii) y | 1 |

(i) Let the required number of days be x.

x_{1}y_{1} = x_{2}y_{2}

2 × 3 = 1 × x

x = 6

Hence, the required number of days is 6.

(ii) Let the required number of persons be y.

x_{1}y_{1} = x_{2}y_{2}

2 × 3 = y × 1

y = 6

Hence, the required number of persons is 6.

**Ex 11.2 Class 8 Maths Question 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:**On increasing the duration of periods, the number of periods will be reduced.

Thus, the two quantities are in inverse proportion.

Number of periods | Duration of periods in minutes |

8 | 45 |

9 | x |

Let the required duration of each period be x.

x_{1}y_{1} = x_{2}y_{2}

8 × 45 = 9 × x

x = 40 minutes

Hence, the required duration of each period is 40 minutes.

**Related Links:**

**NCERT Solutions for Maths Class 9**

**NCERT Solutions for Maths Class 10**

**NCERT Solutions for Maths Class 11**