Direct And Inverse Proportions
Class 8th Mathematics CBSE Solution
- Following are the car parking charges near a railway station upto ll 4 & 60 8 &…
- A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of…
- In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how…
- A machine in a soft drink factory fills 840 bottles in six hours. How many…
- A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as…
- In a model of a ship, the mast is 9 cm high, while the mast of the actual ship…
- Suppose 2 kg of sugar contains 9 10^6 crystals. How many sugar crystals are…
- Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a…
- A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same…
- A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how…
- Which of the following are in inverse proportion? (i) The number of workers on…
- In a Television game show, the prize money of Rs 1,00,000 is to be divided…
- Rehman is making a wheel using spokes. He wants to fix equal spokes in such a…
- If a box of sweets is divided among 24 children, they will get 5 sweets each.…
- A farmer has enough food to feed 20 animals in his cattle for 6 days. How long…
- A contractor estimates that 3 persons could rewire Jasminders house in 4 days.…
- A batch of bottles were packed in 25 boxes with 12 bottles in each box. If the…
- A factory requires 42 machines to produce a given number of articles in 63…
- A car takes 2 hours to reach a destination by travelling at the speed of 60…
- Two persons could fit new windows in a house in 3 days. (i) One of the persons…
- A school has 8 periods a day each of 45 minutes duration. How long would each…
Exercises 13.1
Question 1.Following are the car parking charges near a railway station upto
Check if the parking charges are in direct proportion to the parking time.
Answer:
For the data to be in direct proportion, the ratio of the charged money and the time should remain equal.
The ratio of parking charges to their respective number of hours may be calculated as
= 15
As, each ratio is not same therefore, the parking charges to their respective number of hours are not in direct proportion.
Question 2.
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.
Answer:
A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base
x1 = If 1 part of red pigment is mixed with 8 part of base, 4 parts of red pigment is mixed with 4 x 8 parts of base. Therefore,X1 = 4 × 8 = 32
x2 = If 1 part of red pigment is mixed with 8 part of base, 7 parts of red pigment is mixed with 7 x 8 parts of base. Therefore,X2 = 7 × 8 = 56
x3 = If 1 part of red pigment is mixed with 8 part of base, 12 parts of red pigment is mixed with 12 x 8 parts of base. Therefore,X3 = 12 × 8 = 96
x4 = If 1 part of red pigment is mixed with 8 part of base, 20 parts of red pigment is mixed with 20 x 8 parts of base. Therefore,X4 = 20 × 8 = 160
Question 3.
In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?
Answer:
The table of the given information is as follows:
The parts of the red pigment and parts of base are in direct proportion,
Therefore,
⇒ x = 24
Thus, 24 parts of the red pigment mixed with 1800 ml of base.
Method 2:
Given: 1 part of red pigment makes 75 ml
1 ml will be made from 1/75 parts
Therefore,
1800 ml will be made from 1/75 × 1800 = 24 parts
Question 4.
A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Answer:
The given information in the form of table is as follows:
Let x be the no of bottles filled in 5 hours.
The no of bottles and no of hours are in direct proportion as the larger the number of bottles the longer it take to fill them and vice-versa.
Therefore,
6x = 5 × 840
→x = 700
Thus, 700 bottles will be filled in 5 hours.
Question 5.
A photograph of a bacteria enlarged 50,000 times attains a length of 5 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?
Answer:
Let the actual length of the bacteria be x cm and enlarged length be y cm when it is enlarged 20,000 times.
The given information in the form of table is as follows
The above information is in direct proportion.
Therefore,
x = 10-4 cm
Hence the actual length of bacteria is 10-4 cm
y = 2 cm
Hence, the enlarged length of bacteria when it is enlarged 20,000 times is 2cm
Question 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?
Answer:
Let the length of the mast of modal ship be x when the length of the actual ship be 28 cm
The given information in the form of table is as follows:
The above information is in direct information:
9/12 = x/28
Cross-multiplying, we get,9 × 28 = 12 × x
→x = 21 cm
Question 7.
Suppose 2 kg of sugar contains 9 × 106 crystals. How many sugar crystals are there in
(i) 5 kg of sugar?
(ii) 1.2 kg of sugar?
Answer:
(i) Let x be the sugar crystals in 5kg of sugar
The above information in the form of table is as follows:
The above information are in direct proportion:
Number of Crystals ∝ Weight of Sugar (The more the weight, the more the number of crystals and vice versa)
(ii) Let x be the sugar crystals in 1.2kg of sugar
The above information in the form of table is as follows:
The above information is in direct proportion:
Therefore,
2/(9 × 106) = 1.2/x
→ x = 5.4 × 106 crystals
Question 8.
Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road for 72 km. What would be her distance covered in the map?
Answer:
Let the distance measured on map be x cm
The above information in the form of table is as follows:
The above information is in direct proportion because distance covered in cm will increase or decrease with distance covered in km.
Therefore,
1 × 18 = x × 72
x = 4 cm
Method 2:
1 cm on map = 18 km on road
1 km on road = 1/18 cm on map
72 km on road = 1/18 × 72 cm on map = 4 cm on map.
Question 9.
A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time
(i) the length of the shadow cast by another pole 10 m 50 cm high
(ii) the height of a pole which casts a shadow 5m long.
Answer:
(i) let the length of the shadow of other pole be x m
The given information in the form of table is as follows:
The above information is in direct proportion.
5.6x = 33.60
x = 33.6/5.6
→x = 6m
(ii) let the length of other pole be x m
The given information in the form of table is as follows:
The above information is in direct proportion.
3.2x = 5 × 5.6
3.2x = 28
x = 28/3.2
→x = 8.75 m
Question 10.
A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?
Answer:
Let the distance covered by truck in 5 hours be x km
The given information in the form of table is as follows:
Given: In 25 minutes, the truck travels 14km
Now, in 5 hours the truck will travel more distance.
Therefore, above information is in direct proportion.
Therefore,
14/25 = x/300
25x = 14 × 300
→x = 168 km
Exercises 13.2
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.
Answer:
(i) Yes, These are in inverse proportion because if there are more workers, then it will take lesser time to complete that job.
(ii) No, these are not in inverse proportion because in more time, we may cover more distance with a uniform speed. It is a direct Proportion.
(iii) No, these are not in inverse proportion because in more area, more quantity of crop may be harvested. It is a direct Proportion.
(iv) Yes, These are in inverse proportion because with more speed, we may complete a certain distance in a lesser time.
(v) Yes, These are in inverse proportion because if the population in increasing, then area of the land per person will be decreasing accordingly.
Question 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 numberof winners?
Answer:
A table of the given information is as follows:
From the table, we obtain:
1 ×100,000 = 2 × 50,000 = 100,000
Thus, the number of winners and amount given to an individual winner is inversely proportional.
Therefore,
1 ×100,000 = x1 × 4
→x1 = 25,000
1 × 100,000 = x2 × 5
→x2 = 20,000
1 × 100,000 = x3 × 8
→x3 = 12500
1 × 100,000 = x4 × 10
→x4 = 10,000
1 × 100,000 = x5 × 20
→x5 = 5,000
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.
(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) The table of the required information is as follows:
4 × 90° = 6 × 60° = 360°
Therefore the number of spokes and the angles formed between the pairs of consecutive spokes is in inverse proportion.
(ii) 4 × 90° = 15 × angle
Angle = 24°
the angle between a pair of consecutive spokes on a wheel with 15 spokes is 24°
(iii) 4 × 90° = 40° × spokes
→spokes = 9
9 spokes would be needed, if the angle between a pair of consecutive spokes is 40°
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 reduced by 4?
Answer:
According to the question:
a box of sweets is divided among 24 children, they will get 5 sweets each
total no of sweets = 24 × 5 = 120
the number of the children is reduced by 4.
Now, total no of children = 24 – 4 = 20
Sweets get by each children = 
= 6
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?
Answer:
A farmer has enough food to feed 20 animals in his cattle for 6 days.
More the number of animals, the lesser will be the number of days food lasts.
Therefore, the number of animals and number of days are inversely proportional to each other.
Hence,
20 × 6 = 30 × x
→x = 4 days
Now, food will last in 4 days.
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?
Answer:
If the number of persons be more, it will take less time to complete the job
Therefore,
The number of persons and time required to complete the work are inversely proportional to each other.
Hence,
3 × 4 = 4 × x { let x be the number of days required to complete the work by 4 persons}
→x = 3 days
Question 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?
Answer:
A batch of bottles were packed in 25 boxes with 12 bottles in each box.
1 box contains 12 bottles, so 25 boxes will contain = 25 x 12 bottlesTotal no of bottles = 25 × 12 = 300
If the same batch is packed using 20 bottles in each box
Now the total number of bottles = 300And each box contains 20 bottles.
Then, no of boxes = 300/20 = 15
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 in54 days?
Answer:
The more the no of machines, the less days will be required to produce a given no of articles.
Therefore, the no of machines and no of days are in inverse proportion.
Hence,
42 × 63 = 54 × x { x be the no of machines required to produce a given no of articles in 54 days}
→x = 49
Question 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?
Answer:
Distance = speed × time
Therefore, speed and time are inversely proportional
This means that,
Speed × Time = Constant
2 × 60 = 80 × t { t be the time required to cover a given distance at 80 km/hr}
→ t = 1.5 hr
Question 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?
Answer:
(i) If the number of persons be more, it will take less time to complete the job. Now, One of the persons fell ill before the work started.
Persons left = 2 – 1 = 1
Therefore,
The number of persons and time required to complete the work are inversely proportional to each other.
Therefore,
2 × 3 = 1 × t ( t be the time required to fit the windows by 1 person}
→t = 6 days
(ii) If the number of persons be more, it will take less time to complete the job. Now, One of the persons fell ill before the work started.
Therefore,
The number of persons and time required to complete the work are inversely proportional to each other
Let n be the numbers of persons required to complete the work in 1 day.
2 × 3 = 1 × n
→ n = 6 persons
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 tobe the same?
Answer:
A school has 8 periods a day each of 45 minutes duration.
Total time = 8 × 45 minutes
if the school has 9 periods a day.
Then duration of period = (8 × 45)/ 9
= 40 minutes