Direct Proportion
- Two quantities are said to be in direct proportion when they increase or decrease together at the same rate.
- If one quantity is doubled, the other quantity is also doubled.
Example:
- The cost of petrol and the number of liters purchased.
Inverse Proportion
- Two quantities are said to be in inverse proportion when an increase in one quantity leads to a decrease in the other quantity, and vice versa.
Example:
- The speed of a car and the time taken to cover a fixed distance.
Key Concepts:
- Constant of proportionality: The constant ratio in direct proportion or the constant product in inverse proportion.
- Identifying direct and inverse proportions: Understanding the relationship between quantities.
- Solving problems: Applying the concepts of direct and inverse proportions to real-life situations.
By understanding these concepts, you can effectively solve problems involving quantities that change in relation to each other.
Exercise 11.1
1. 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 proportions to the parking time.
Ans :
Let’s calculate the ratio of parking time to parking charges for each case:
- For 4 hours: 4/60 = 1/15
- For 8 hours: 8/100 = 2/25
- For 12 hours: 12/140 = 3/35
- For 24 hours: 24/180 = 2/15
The ratios are not equal. This means that the parking charges are not directly proportional to the parking time.
Therefore, the parking charges are not in direct proportion to the parking time.
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.
Ans :
| Parts of red pigment | 1 | 4 | 7 | 12 | 20 |
| Parts of base | 8 | 32 | 56 | 96 | 160 |
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?
Ans :
We can set up a proportion:
- 1 part of red pigment / 75 mL of base = x parts of red pigment / 1800 mL of base
Cross-multiplying, we get:
- 75 * x = 1 * 1800
Solving for x:
- x = 1800 / 75
- x = 24
Therefore, 24 parts of red pigment should be mixed with 1800 mL of base.
4. A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Ans :
The machine will fill 700 bottles in five hours.
Here’s how we can calculate it:
- The machine fills 840 bottles in 6 hours.
- So, in 1 hour, it fills 840 / 6 = 140 bottles.
- Therefore, in 5 hours, it will fill 140 * 5 = 700 bottles.
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?
Ans :
Calculate the actual length of the bacteria:
Actual length
= Enlarged length / Magnification factor
= 5 cm / 50,000
= 0.0001 cm
Calculate the enlarged length of the bacteria if the photograph is enlarged 20,000 times:
Enlarged length
= Actual length * Magnification factor
= 0.0001 cm * 20,000
= 2 cm
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 :
- Height of model ship’s mast / Height of actual ship’s mast = Length of model ship / Length of actual ship
Substituting the given values:
- 9 cm / 12 m = x cm / 28 m
Converting meters to centimeters:
- 9 cm / 1200 cm = x cm / 2800 cm
Cross-multiplying:
- 1200x = 9 * 2800
Solving for x:
- x = (9 * 2800) / 1200
- x = 21 cm
Therefore, the length of the model ship is 21 cm.
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?
Ans :
i) For 5 kg of sugar:
- If 2 kg of sugar contains 9 * 10^6 crystals,
- Then, 1 kg of sugar contains (9 * 10^6) / 2 crystals
- Therefore, 5 kg of sugar contains (9 * 10^6 / 2) * 5 crystals = 4.5 * 10^7 crystals
ii) For 1.2 kg of sugar:
- If 2 kg of sugar contains 9 * 10^6 crystals,
- Then, 1 kg of sugar contains (9 * 10^6) / 2 crystals
- Therefore, 1.2 kg of sugar contains ((9 * 10^6) / 2) * 1.2 crystals = 5.4 * 10^6 crystals
8. Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road of 72 km. What would be her distance covered in the map?
Ans :
Let the distance covered on the map be x cm.
We can set up a proportion:
- 1 cm / 18 km
- = x cm / 72 km
Cross-multiplying, we get:
- 18x = 72
- x = 72 / 18
- x = 4 cm
Therefore, the distance covered on the map would be 4 cm.
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 5 m long.
Ans :
Given:
- Height of first pole = 5 m 60 cm = 560 cm
- Shadow of first pole = 3 m 20 cm = 320 cm
i) Length of shadow for a 10 m 50 cm high pole:
- Let the length of the shadow be x cm.
- Setting up a proportion:
- Height of first pole / Shadow of first pole = Height of second pole / Shadow of second pole
- 560 cm / 320 cm = 1050 cm / x cm
- x = (1050 * 320) / 560
- x = 600 cm or 6 m
Therefore, the length of the shadow cast by the 10 m 50 cm high pole is 6 m.
ii) Height of the pole for a 5 m long shadow:
- Let the height of the pole be y cm.
- Setting up a proportion:
- Height of first pole / Shadow of first pole = Height of second pole / Shadow of second pole
- 560 cm / 320 cm = y cm / 500 cm
- y = (560 * 500) / 320
- y = 875 cm or 8 m 75 cm
Therefore, the height of the pole which casts a shadow of 5 m is 8 m 75 cm.
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 :
Given:
- Time for which we need to find the distance = 5 hours = 5 * 60 minutes = 300 minutes
Let’s set up a proportion:
- Distance traveled in 25 minutes / Time taken = Distance traveled in 300 minutes / Time taken
- 14 km / 25 minutes = Distance traveled / 300 minutes
Solving for the unknown distance:
- Distance traveled = (14 km * 300 minutes) / 25 minutes
- Distance traveled = 168 km
Exercise 11.2
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 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.
Ans :
(i) This is an inverse proportion. More workers will finish the job faster.
(ii) This is not an inverse proportion. If you travel for a longer time at a constant speed, you will cover a greater distance.
(iii) This is generally not an inverse proportion. More land usually yields more crop.
(iv) This is an inverse proportion. Higher speed means less time for the same journey.
(v) This is an inverse proportion. More population on the same land means less land per person.
Therefore, options (i), (iv), and (v) are examples of inverse proportion.
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 | – | – | – | – | – |
Ans :
| Number of winners | Prize for each winner (in ₹) |
| 1 | 1,00,000 |
| 2 | 50,000 |
| 4 | 25,000 |
| 5 | 20,000 |
| 8 | 12,500 |
| 10 | 10,000 |
| 20 | 5,000 |
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°?
Ans :
| Number of spokes | Angle between a pair of consecutive spokes |
| 4 | 90° |
| 6 | 60° |
| 8 | 45° |
| 10 | 36° |
| 12 | 30° |
i)
Yes, the number of spokes and the angle formed between the pairs of consecutive spokes are inversely proportional. As the number of spokes increases, the angle between them decreases, and vice versa.
ii)
Angle between spokes = 360° / Number of spokes = 360° / 15 = 24°
iii)
Number of spokes = 360° / Angle between spokes = 360° / 40° = 9 spokes
Therefore, 9 spokes would be needed if the angle between a pair of consecutive spokes is 40°.
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?
Ans :
- Find the total number of sweets:
- Total sweets = Number of children * Sweets per child = 24 * 5 = 120 sweets
- Find the new number of children:
- New number of children = Original number of children – Decrease = 24 – 4 = 20 children
- Find the new number of sweets per child:
- Sweets per child = Total sweets / Number of children = 120 sweets / 20 children = 6 sweets/child
Therefore, each child would get 6 sweets if the number of children is decreased by 4.
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?
Ans :
Given:
- Number of animals = 20
- Number of days = 6
- Increase in animals = 10
Calculations:
- Total number of animals after increase: 20 + 10 = 30 animals
- Let the number of days the food lasts for 30 animals be x.
- We can set up an inverse proportion:
- (Number of animals1 * Number of days1) = (Number of animals2 * Number of days2)
- 20 * 6 = 30 * x
- Solving for x:
- x = (20 * 6) / 30
- x = 4
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?
Ans :
- Given:
- Number of people initially = 3
- Days to complete the work = 4
- Increased number of people = 4
- Let the number of days to complete the work with 4 people be x.
- Setting up an inverse proportion:
- (Number of people 1 * Number of days 1) = (Number of people 2 * Number of days 2)
- 3 * 4 = 4 * x
- Solving for x:
- x = (3 * 4) / 4
- x = 3
Therefore, it would take 3 days for 4 people to rewire Jasminder’s house.
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?
Ans :
To find:
- The number of boxes filled when packing with 20 bottles in each box.
Solution:
Calculate the total number of bottles:
Total number of bottles = Number of boxes * Number of bottles per box
= 25 boxes * 12 bottles/box
= 300 bottles
Calculate the number of boxes filled when packing with 20 bottles in each box:
Number of boxes = Total number of bottles / Number of bottles per box
= 300 bottles / 20 bottles/box
= 15 boxes
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?
Ans :
Given:
- Number of machines = 42
- Number of days = 63
- New number of days = 54
Setting up an inverse proportion:
- (Number of machines 1 * Number of days 1) = (Number of machines 2 * Number of days 2)
- 42 * 63 = x * 54
Solving for x:
- x = (42 * 63) / 54
- x = 49
9. A car takes 2 hours to reach a destination by traveling at a speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h?
Ans :
Given:
- Initial speed = 60 km/h
- Initial time = 2 hours
- Final speed = 80 km/h
Let the final time be x hours.
Setting up an inverse proportion:
- (Initial speed * Initial time) = (Final speed * Final time)
- 60 * 2 = 80 * x
Solving for x:
- x = (60 * 2) / 80
- x = 1.5 hours
Therefore, it will take 1.5 hours for the car to reach the destination when traveling at a speed of 80 km/h.
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?
Ans :
(i) Time taken by one person
- Given: 2 persons can complete the job in 3 days.
- To find: Time taken by 1 person to complete the job.
Let the time taken by 1 person be x days.
We can set up an inverse proportion:
- Number of persons * Number of days = constant
- 2 * 3 = 1 * x
- x = 6
(ii) Number of persons required to fit the windows in one day
- We know that 1 person can complete the job in 6 days.
- So, to complete the job in 1 day, we need 6 times the number of people.
- Number of persons required = 1 * 6 = 6 persons
Therefore, 6 persons would be needed to fit the windows in one day.
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 :
Given:
- Initial number of periods = 8
- Duration of each period = 45 minutes
- Final number of periods = 9
Let the duration of each period with 9 periods be x minutes.
Setting up an inverse proportion:
- (Initial number of periods * Duration of each period) = (Final number of periods * Duration of each period)
- 8 * 45 = 9 * x
Solving for x:
- x = (8 * 45) / 9
- x = 40 minutes
Therefore, each period would be 40 minutes long if there are 9 periods a day.
