**Direct and Inverse Proportions**

**NCERT Exercise and Other Q & A***I*f there is an increase (↑) [decrease (↓)] in one quantity produces a proportionate decrease (↓) [increase (↑)] in another quantity, then we say that the two quantities are in

*inverse variation*or

*inverse proportion*.

If two quantities vary inversely, their product is a constant.

e.g. if and y are two quantities which are in inverse proportion then,

x ×

*y = constant*

or x

_{1}× y

_{1}= x

_{2}× y

_{2}

__NCERT Exercise 13.2__

**Q1: 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.**

Answer:

(i) With an increase in number of workers, the job will complete in lesser time. ∴ The number of workers on a job and the time to complete the job are in inverse proportion

*.*

(ii) At constant (uniform) speed, more time means more distance will be covered. ∴ The time taken for a journey and the distance travelled in a uniform speed are in direct proportion.

(iii) An increase (↑) in the cultivated area means an increase (↑) in the crop harvested. ∴ Area of cultivated land and the crop harvested are in direct proportion.

(iv) An increase (↑) in speed of the vehicle means the distance will be covered in lesser (↓) time. ∴ The time taken for a fixed journey and the speed of the vehicle. are in inverse proportion.

(v) With an increase (↑) in population of a country, the area of land per person will decrease (↓). ∴ these two quantities are in inverse proportion.

**Q2: In a Television game show, the prize money of Rs 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 Rs) | 1,00,000 | 50,000 | p_{1} | p_{2} | p_{3} | p_{4} | p_{5} |

Answer: More (↑) the winners, lesser (↓) will be the prize money for each winner. The number of winners and the prize money for each winner are inversely proportional to each other.

⇒ 1 × 100000 = 4 × p

_{1}

⇒ p

_{1}= 100000 / 4 = 25,000

⇒ 1 × 100000 = 5 × p

_{2}

⇒ p

_{2}= 100000 / 5 = 20,000

⇒ 1 × 100000 = 8 × p

_{3}

⇒ p

_{3}= 100000 / 8 = 12,500

⇒ 1 × 100000 = 10× p

_{4}

⇒ p

_{4}= 100000 / 10 = 10,000

⇒ 1 × 100000 = 20× p

_{5}

⇒ p

_{5}= 100000 / 20 = 5,000

Thus the required values are:

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

Prize for each winners (in Rs) | 1,00,000 | 50,000 | 25,000 | 20,000 | 12,500 | 10,000 | 5,000 |

**Q3: 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° | v_{1} | v_{2} | v_{3} |

**(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°?**

Answer:

(i) An increase (↑) in number of spokes means a corresponding decrease (↓) in the angle between a pair of consecutive spokes. Thus the number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion.

⇒ 4 × 90° = 8 × v

_{1}

⇒ v

_{1}= 4 × 90°/8 = 45°

⇒ 4 × 90° = 10 × v

_{2}

⇒ v

_{2}= 4 × 90°/10 = 36°

⇒ 4 × 90° = 8 × v

_{3}

⇒ v

_{3}= 4 × 90°/12 = 30°

Thus the required table is

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

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

(ii) Let the angle between a pair of consecutive spokes on a wheel with 15 spokes = v

_{4}

⇒ 4 × 90° = 15 × v

_{4}

⇒ v

_{1}= 4 × 90°/15 = 24° ..... (answer)

(iii) Let the number for spokes needed making with each pair angle of 40°= x

⇒ 4 × 90° = x × 40°

⇒ x = 4 × 90°/ 40° = 9 ..... (answer)

Q4: 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?

Answer: Number of Children after reducing by 4 = 24 - 4 = 20.

If there is decrease (↓) in number of students, each student will get more (↑) sweets. Thus the two quantities are in inverse proportion.

Number of students | 24 | 20 |

Number of sweets to each student | 5 | x |

⇒ 24 × 5 = 20 × x

⇒ x = 24 × 5/20 = 6

Thus each student will get 6 sweets.

**Q5: 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?**

Answer: If there is increase (↑) in number of animals, food will last in lesser (↓) number of days. Thus the two quantities vary inversely.

Animals to feed | 20 | 30(=20+10) |

Number of days food last | 6 | x |

⇒ 20 × 6 = 30 × x

⇒x = 20 × 6 / 30 = 4

Thus the food will last for 4 days.

**Q6: 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?**

Answer: More (↑) number of people hired means lesser (↓) time it will take finish the job. Thus, number of persons and the days to finish the job are in inverse proportion.

Number of persons | 3 | 4 |

Number of days to finish the job | 4 | x |

⇒ 3 × 4 = 4 × x

⇒ x = 3 × 4 / 4 = 3 days.

Thus number of days to complete the job is 3 days.

**Q7: 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?**

Answer: More (↑) bottles in the box means, lesser (↓) number of boxes are required. Thus, number of boxes and number of bottles per box are in inverse proportion.

Number of boxes | 25 | 20 |

Number of bottles per box | 12 | x |

⇒ 25 × 12 = 20 × x

⇒ x = 25 × 12 / 20 = 15

Thus 15 boxes are required, if number of bottles per box is 20.

**Q8: 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?**

Answer: An increase (↑) in number of machines means lesser number of days are required to produce required number of articles. Thus the two quantities are in inverse variation.

Number of machines | 42 | x |

Number of days to produce articles | 63 | 54 |

⇒ 42 × 63 = x × 54

⇒ x = 42 × 63/ 54 = 49

Thus to produce the same number of articles in 54 days, machines required are 49.

**Q9: 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?**

Answer: An increase (↑) in speed of the car means, lesser (↓) time is required to cover the same distance.Thus speed of car and the time required to cover the fixed distance are in inverse proportion.

Speed of car (km/h) | 60 | 80 |

Time taken (hrs) | 2 | t |

⇒ 60 × 2 = 80 × t

⇒ t = 60 × 2 / 80 = 1.5 hrs

Thus car travelling with speed of 80 km/h will take time of 1 hour and 30 minutes.

**Q10: 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?**

Answer: Less (↓) number of persons means, more (↑) time to fit the windows. Thus the two quantities are in inverse proportion.

Number of persons working | 2 | 1(=2-1) | n |

Days required to fit windows | 3 | d | 1 |

(i) Number of persons working after one person fell ill = 2 - 1 = 1

⇒ 2 × 3 = 1 × d

⇒ d = 2 × 3 / 1 = 6

Thus 6 days are required if only one person is working.

(ii) Days required to fit the windows = 1

⇒ 2 × 3 = n × 1

⇒ n = 2 × 3 / 1 = 6

Thus 6 persons are required to complete the work in 1 day.

**Q11: 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?**

Answer: For a fixed school hours, an increase (↑) in duration of a period means a decrease (↓) in number of periods. Thus the two variables are in inverse proportion.

Duration of each period(mins) | 45 | d |

Number of periods | 8 | 9 |

⇒ 45 × 8 = d × 9

⇒ d = 45 × 8/ 9 = 40

Thus the duration of each period will be 40 minutes.

__Miscellaneous Questions__

**Q12: In a village, 120 men had provision for food for 210 days. After 10 days, 40 men left the village. How long the remaining food last?**

Answer: After 10 days, 40 men left.

⇒ After 10 days, for 120 men, food will last = 210 - 10 = 200 days.

⇒ Remaining men at the village = 120 - 40 = 80

Less men, more food. Thus the two quantities are in inverse variation.

For 120 men, food will last for = 200 days

For 1 man, food will last for = 200 × 120 (inverse proportion)

For 80 men, food will last for = 200 × 120 / 80 = 300

Thus for 80 men, food will last for 300 days.

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