# Direct and Inverse Proportions - Class 8 - Mathematics

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## Examples - Direct and Inverse Proportions | NCERT | Mathematics | Class 8

The cost of 5 metres of a particular quality of cloth is ₹210 . Tabulate the cost of 2, 4, 10 and 13 metres of cloth of the same type. Explain steps too.

First, let's find the cost per metre of the cloth. Since 5 metres of the cloth costs ₹210, the cost per metre can be found by dividing ₹210 by 5.

$$ \text{Cost per metre} = \frac{\text{Total cost for 5 metres}}{\text{Number of metres}} = \frac{210}{5} $$

After calculating the cost per metre, we can determine the cost for 2 metres, 4 metres, 10 metres, and 13 metres of the cloth by multiplying the cost per metre by the respective number of metres.

Let's perform these calculations:

Calculate the cost per metre.

Calculate the cost for each requested amount of cloth (2, 4, 10, 13 metres).

This can be expressed mathematically as:

$$ \text{Cost for } n \text{ metres} = \text{Cost per metre} \times n $$

Where "$n$" will be replaced with 2, 4, 10, and 13 for each calculation. After calculating, here are the results:

**Cost per metre:**₹42**Cost for 2 metres:**₹84**Cost for 4 metres:**₹168**Cost for 10 metres:**₹420**Cost for 13 metres:**₹546

Therefore, the tabulation of costs for the respective lengths of the cloth is as follows:

Metres of Cloth | Cost (₹) |
---|---|

2 | 84 |

4 | 168 |

10 | 420 |

13 | 546 |

This tabulation was derived by first finding the unit cost of the cloth per metre, which is ₹42. This unit cost was then multiplied by the respective number of metres (2, 4, 10, and 13) to find the total cost for each specified length of cloth.

An electric pole, 14 metres high, casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions.

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Sign up nowIf the weight of 12 sheets of thick paper is 40 grams, how many sheets of the same paper would weigh $2 \frac{1}{2}$ kilograms?

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Sign up nowA train is moving at a uniform speed of $75 \mathrm{~km} / \mathrm{hour}$.

(i) How far will it travel in 20 minutes?

(ii) Find the time required to cover a distance of $250 \mathrm{~km}$.

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Sign up nowThe scale of a map is given as 1:30000000. Two cities are $4 \mathrm{~cm}$ apart on the map. Find the actual distance between them.

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6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?

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There are 100 students in a hostel. Food provision for them is for 20 days. How long will these provisions last, if 25 more students join the group?

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If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?

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## Exercise 11.1 - Direct and Inverse Proportions | NCERT | Mathematics | Class 8

Following are the car parking charges near a railway station up to

4 hours | $₹ 60$ |

8 hours | $₹ 100$ |

12 hours | $₹ 140$ |

24 hours | $₹ 180$ |

Check if the parking charges are in direct proportion to the parking time.

To check if the parking charges are in direct proportion to the parking time, we can compare the ratios of charges to time for each of the different parking durations provided. In a direct proportion, these ratios should be constant.

The ratios for the given charges and times are:

For 4 hours at ₹60: $$\frac{₹60}{4\text{ hours}} = ₹15\text{ per hour}$$

For 8 hours at ₹100: $$\frac{₹100}{8\text{ hours}} = ₹12.50\text{ per hour}$$

For 12 hours at ₹140: $$\frac{₹140}{12\text{ hours}} = ₹11.67\text{ per hour}$$

For 24 hours at ₹180: $$\frac{₹180}{24\text{ hours}} = ₹7.50\text{ per hour}$$

Since the charge per hour is not constant across different durations, the parking charges are not in direct proportion to the parking time.

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

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

Parts of base | 8 |

if 1 part of a red pigment requires $75 \mathrm{~mL}$ of base, how much red pigment should we mix with $1800 \mathrm{~mL}$ of base?

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A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?

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A photograph of a bacteria enlarged 50,000 times attains a length of $5 \mathrm{~cm}$ as shown in the diagram. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?

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In a model of a ship, the mast is $9 \mathrm{~cm}$ high, while the mast of the actual ship is $12 \mathrm{mhigh}$. If the length of the ship is $28 \mathrm{~m}$, how long is the model ship?

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Suppose $2 \mathrm{~kg}$ of sugar contains $9 \times 10^{6}$ crystals. How many sugar crystals are there in (i) $5 \mathrm{~kg}$ of sugar? (ii) $1.2 \mathrm{~kg}$ of sugar?

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Rashmi has a road map with a scale of $1 \mathrm{~cm}$ representing $18 \mathrm{~km}$. She drives on a road for $72 \mathrm{~km}$. What would be her distance covered in the map?

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A $5 \mathrm{~m} 60 \mathrm{~cm}$ high vertical pole casts a shadow $3 \mathrm{~m} 20 \mathrm{~cm}$ long. Find at the same time (i) the length of the shadow cast by another pole $10 \mathrm{~m} 50 \mathrm{~cm}$ high (ii) the height of a pole which casts a shadow $5 \mathrm{~m}$ long.

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A loaded truck travels $14 \mathrm{~km}$ in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?

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## Exercise 11.2 - Direct and Inverse Proportions | NCERT | Mathematics | Class 8

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.

Let's analyze each option to determine whether they are in inverse proportion or not:

**The number of workers on a job and the time to complete the job.**This is in inverse proportion. If more workers are assigned to a job, the time taken to complete the job decreases.

**The time taken for a journey and the distance travelled in a uniform speed.**This is not in inverse proportion; it is in direct proportion. If the speed is constant, a longer distance will take more time.

**Area of cultivated land and the crop harvested.**This is not in inverse proportion; it is typically in direct proportion. Generally, a larger area of cultivated land will yield a larger harvest.

**The time taken for a fixed journey and the speed of the vehicle.**This is in inverse proportion. If the speed of the vehicle increases, the time taken to complete a fixed journey decreases.

**The population of a country and the area of land per person.**This is in inverse proportion. If the population increases, assuming the total land area remains constant, the area of land available per person decreases.

Thus, options (i), (iv), and (v) are in inverse proportion.

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 ₹) | $1,00,000$ | $50,000$ |

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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 | $90^{\circ}$ | $60^{\circ}$ |

(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 $40^{\circ}$ ?

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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 ?

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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?

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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?

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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?

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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?

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

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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?

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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?