*NCERT Solutions of Chapter: Direct and Inverse Proportions. NCERT Solutions along with worksheets and notes for Class 8.*

**(2)** **A mixture of paint is prepared by mixing 1 part of the red pigments with 8 part of base. In the following table, find the parts of the base that need to be added.**

Parts of red pigment |
1 |
4 |
7 |
12 |
20 |

Parts of base |
8 |
… |
… |
…. |
…. |

**Ans-**

Let parts of red pigment is x and parts of base is y.

Parts of red pigment (x) | 1(x_{1}) |
4(x_{2}) |
7(x_{3}) |
12(x_{4}) |
20(x_{5}) |

Parts of base (y) | 8(y_{1}) |
(y_{2}) |
(y_{3}) |
(y_{4}) |
(y_{5}) |

**Given**

Parts of red pigment x_{1} = 1

Parts of base y_{1} = 8

**(6)In a model of a ship, the mast is 9 cm high, while the mast of the actual, ship is 12 m high. If the length of the ship is 28 m, how long is the model ship?**

**Ans-**

Let the length of the ship is y metres.

9 cm = .09 m [1m = 100 cm]

Put all the information in the table given below:-

Length of the ship | 28 m (x_{1}) |
x (x_{2}) |

Length of the mast | 12 m (y_{1)} |
.09 m(y_{2}) |

As the length of the mast of the ship increases, the length of the ship also increases. It is a case of direct proportion

**(10)** **A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?**

**Ans-**

5 hours = 5 60 minutes [1 hour = 60 minutes]

5 hours = 300 minutes

Let the distance covered by truck in 300 minutes is x km. Put all the information in the table given below:-

Distance covered by the truck (km) | 14(x_{1}) |
x (x_{2}) |

Time taken by truck to cover distance (minutes) | 25 (y_{1}) |
300 (y_{2}) |

**Exercise 13.2**

**(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 traveled in a uniform, speed.**

**(iii) Area of cultivated land 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.**

**Ans-**

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

The above relationship is in **inverse proportion.**

Increasing number of the workers on a job will complete that work in less time.

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

The above relationship is **not in inverse proportion.**

As distance increases time taken to cover that distance will also increase.

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

The above relationship is **not in inverse proportion.**

As area of cultivated land increases crop harvested will also increase.

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

The above relationship is in **inverse proportion.**

As speed of the vehicle increases for fixed journey the time taken decreases.

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

The above relationship is in **inverse proportion.**

As the population of a country increase the land per person decreases.

** (2) 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?**

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

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

**Ans-** Let prize for each winner Be Rs y. Put all the information in the table given below:-

Numbers of winners (x) | 1 (x_{1}) |
2(x_{2}) |
4(x_{3}) |
5(x_{4}) |
8(x_{5}) |
10(x_{6}) |
20(x_{7}) |

Prize for each winner (y)(Rs) | 1,00,000
(y |
50,000
(y |
(y_{3}) |
(y_{4}) |
(y_{5}) |
(y_{6}) |
(y_{7}) |

**(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^{0} |
60^{0} |
… |
… |
… |

**(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 whel with 15 spokes.**

**(iii) How many spjes would be needed, if the amgle between a pair of consecutive spokes is 40 ^{0}?**

**Ans-**

Let the number of spokes is x and angle between a pair of consecutive spokes be y^{0}.

Number of spokes (x) | 4(x_{1}) |
6(x_{2}) |
8(x_{3}) |
10(x_{4}) |
12(x_{5}) |

Angle between a pair of consecutive spokes(y) | 90^{0}(y_{1}) |
60^{0}(y_{2}) |
(y_{3}) |
(y_{4}) |
(y_{5}) |

As the number of spokes increases the angle between a pair of consecutive spokes decreases. It is a case of inverse proportion.

Therefore,

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

**(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 jb take now?**

**(ii) How many persons would be needed to fit the windows in one day?**

**Ans-**

**(i)** Let the job will be complete in y days.

Number of persons | 2(x_{1}) |
1(x_{2}) |

Number of days required to complete the work | 3(y_{1}) |
y(y_{2}) |

**(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?**

**Ans-**

Let the duration of the period be y hours.

Number of periods | 8(x_{1}) |
45(x_{2}) |

Duration of the periods (minutes) | 9(y_{1}) |
y(y_{2}) |

**Helping Topics**

**Direct and Inverse Proportions**