# Variation in Mathematics : 1. Direct Variation , 2. Indirect or inverse variation, 3. Joint Variation, and, 4. Partial Variation Mathematics JSS 2 Third Term Week 4

Subject : Mathematics

Class : JSS 2

Term : Third Term

Week : Week 4

Topic :

Variation in Mathematics :

1. Direct Variation ,

2. Indirect or inverse variation,

3. Joint Variation, and,

4. Partial Variation

Content

Mathematics Topic: Variation

Class: JSS 2

Today, we will be learning about different types of variations in mathematics. Variations help us understand the relationship between two or more variables.

Let’s explore the four types of variations:

1. Direct Variation:
– Direct variation occurs when two variables change in the same direction. If one variable increases, the other variable also increases, and if one variable decreases, the other variable decreases.
– We can represent direct variation using the equation y = kx, where ‘y’ and ‘x’ are the variables, and ‘k’ is a constant called the constant of variation. The value of ‘k’ remains the same throughout.
– For example, if we say the number of hours you work is directly proportional to the amount of money you earn, it means that as you work more hours, you earn more money.

2. Indirect or Inverse Variation:
– Indirect variation occurs when two variables change in opposite directions. If one variable increases, the other variable decreases, and vice versa.
– We can represent inverse variation using the equation y = k/x, where ‘y’ and ‘x’ are the variables, and ‘k’ is the constant of variation.
– For example, if we say the time taken to complete a task is inversely proportional to the number of people working on it, it means that as the number of people working on the task increases, the time taken to complete it decreases.

3. Joint Variation:
– Joint variation involves three or more variables and occurs when one variable varies directly with some factors and inversely with others.
– We can represent joint variation using the equation y = kxz, where ‘y’, ‘x’, and ‘z’ are the variables, and ‘k’ is the constant of variation.
– For example, if we say the speed of a car depends on both the distance it travels and the time it takes, we have a joint variation. As the distance increases and the time taken decreases, the speed of the car increases.

4. Partial Variation:
– Partial variation occurs when one variable depends on another variable and a constant value. It combines both direct and indirect variations.
– We can represent partial variation using the equation y = kx + c, where ‘y’ and ‘x’ are the variables, ‘k’ is the constant of variation, and ‘c’ is a constant term.
– For example, if we say the cost of buying a number of pens includes both a fixed cost and a cost per pen, we have a partial variation. The fixed cost ‘c’ remains the same, and the cost per pen ‘k’ depends on the number of pens ‘x’.

Remember, variations are a way to understand and describe how different quantities relate to each other. By understanding these variations, we can solve problems and analyze real-life situations more effectively.

[mediator_tech]

### Worked Samples

1. Direct Variation:

Example 1:

The distance traveled by a car is directly proportional to the time taken. If a car travels 100 kilometers in 2 hours, find the distance it would travel in 5 hours.

Solution:

Let ‘d’ represent the distance and ‘t’ represent the time.

We can write the direct variation equation as d = kt.

From the given information, we have:

100 = k * 2

k = 50 (constant of variation)

Now, we can use this value of ‘k’ to find the distance for 5 hours:

d = 50 * 5

d = 250 kilometers

Example 2:

The amount of money earned by a worker is directly proportional to the number of hours worked. If a worker earns \$30 for working 4 hours, find the amount earned for working 9 hours.

Solution:

Let ‘m’ represent the amount earned and ‘h’ represent the number of hours worked.

We can write the direct variation equation as m = kh.

From the given information, we have:

30 = k * 4

k = 7.5 (constant of variation)

Now, we can use this value of ‘k’ to find the amount earned for 9 hours:

m = 7.5 * 9

m = \$67.50

Example 3:

The number of items a store sells is directly proportional to the price per item. If the store sells 200 items for \$500, find the price per item for selling 400 items.

Solution:

Let ‘n’ represent the number of items and ‘p’ represent the price per item.

We can write the direct variation equation as n = kp.

From the given information, we have:

200 = k * 500

k = 0.4 (constant of variation)

Now, we can use this value of ‘k’ to find the price per item for 400 items:

p = 0.4 * 400

p = \$160

2. Indirect or Inverse Variation:

Example 1:

The time taken to complete a task is inversely proportional to the number of workers. If it takes 6 workers 4 hours to complete the task, find the time it would take for 9 workers to complete the same task.

Solution:

Let ‘t’ represent the time taken and ‘w’ represent the number of workers.

We can write the inverse variation equation as t = k/w.

From the given information, we have:

4 = k/6

k = 24 (constant of variation)

Now, we can use this value of ‘k’ to find the time for 9 workers:

t = 24/9

t ≈ 2.67 hours

Example 2:

The pressure of a gas is inversely proportional to its volume. If a gas has a pressure of 20 Pa when its volume is 5 cubic meters, find the pressure when the volume is 10 cubic meters.

Solution:

Let ‘p’ represent the pressure and ‘v’ represent the volume.

We can write the inverse variation equation as p = k/v.

From the given information, we have:

20 = k/5

k = 100 (constant of variation)

Now, we can use this value of ‘k’ to find the pressure for 10 cubic meters:

p = 100/10

p = 10 Pa

Example 3:

The number of days it takes to complete a project is inversely proportional to the number of workers. If it takes 15 days with 10 workers, find the number of days it would take with 5 workers.

Solution:

Let ‘d’ represent

the number of days and ‘w’ represent the number of workers.
We can write the inverse variation equation as d = k/w.
From the given information, we have:
15 = k/10
k = 150 (constant of variation)
Now, we can use this value of ‘k’ to find the number of days with 5 workers:
d = 150/5
d = 30 days

3. Joint Variation:

Example 1:
The force applied to lift an object is jointly proportional to the mass of the object and the acceleration. If a force of 100 Newtons is required to lift an object with a mass of 5 kilograms and an acceleration of 10 meters per second squared, find the force required for an object with a mass of 8 kilograms and an acceleration of 6 meters per second squared.
Solution:
Let ‘F’ represent the force, ‘m’ represent the mass, and ‘a’ represent the acceleration.
We can write the joint variation equation as F = kma.
From the given information, we have:
100 = k * 5 * 10
k = 2 (constant of variation)
Now, we can use this value of ‘k’ to find the force for the second object:
F = 2 * 8 * 6
F = 96 Newtons

Example 2:
The volume of a rectangular prism is jointly proportional to its length, width, and height. If a rectangular prism with dimensions 4 meters by 6 meters by 3 meters has a volume of 72 cubic meters, find the volume of a prism with dimensions 8 meters by 9 meters by 5 meters.
Solution:
Let ‘V’ represent the volume, ‘l’ represent the length, ‘w’ represent the width, and ‘h’ represent the height.
We can write the joint variation equation as V = klwh.
From the given information, we have:
72 = k * 4 * 6 * 3
k = 1 (constant of variation)
Now, we can use this value of ‘k’ to find the volume for the second prism:
V = 1 * 8 * 9 * 5
V = 360 cubic meters

Example 3:
The speed of sound in a medium is jointly proportional to the square root of temperature and the square root of pressure. If the speed of sound is 340 meters per second at a temperature of 25 degrees Celsius and a pressure of 1 atmosphere, find the speed of sound at a temperature of 30 degrees Celsius and a pressure of 2 atmospheres.
Solution:
Let ‘s’ represent the speed of sound, ‘t’ represent the temperature, and ‘p’ represent the pressure.
We can write the joint variation equation as s = k√(tp).
From the given information, we have:
340 = k * √(25 * 1)
k = 20 (constant of variation)
Now, we can use this value of ‘k’ to find the speed of sound for the second scenario:
s = 20 * √(30 * 2)
s ≈ 267.89 meters per second

4. Partial Variation:
(Note: The partial variation examples require an additional constant term.)

Example 1:
The total cost of buying apples includes a fixed cost of \$5 and a cost per apple of \$2. If you buy 10 apples, find the total cost.
Solution:
Let ‘C’ represent the total cost and ‘n’ represent the number of apples.
We can write the partial variation equation as C = kn + c.
From the given information, we have:
C = 2n + 5
Now, we can use this equation to find

[mediator_tech]

### Formulas For calculating different types of variation

1. Direct Variation:

The formula for direct variation is:

y = kx

where ‘y’ and ‘x’ are the variables, and ‘k’ is the constant of variation.

2. Indirect or Inverse Variation:

The formula for inverse variation is:

y = k/x

where ‘y’ and ‘x’ are the variables, and ‘k’ is the constant of variation.

3. Joint Variation:

The formula for joint variation is:

y = kxz

where ‘y’, ‘x’, and ‘z’ are the variables, and ‘k’ is the constant of variation.

4. Partial Variation:

The formula for partial variation is:

y = kx + c

where ‘y’ and ‘x’ are the variables, ‘k’ is the constant of variation, and ‘c’ is a constant term.

These formulas help us understand and express the relationships between variables in different types of variations. By using these formulas, we can solve problems and analyze real-life situations more effectively.

### Evaluation

1. The equation y = kx represents ______________ variation.

a) Direct

b) Indirect

c) Joint

2. In ______________ variation, if one variable increases, the other variable decreases, and vice versa.

a) Direct

b) Indirect

c) Joint

3. The formula y = kxz represents ______________ variation.

a) Direct

b) Indirect

c) Joint

4. In partial variation, the equation y = kx + c includes a ______________ term.

a) Variable

b) Constant

c) Exponential

5. The equation y = k/x represents ______________ variation.

a) Direct

b) Indirect

c) Joint

6. In ______________ variation, as one variable increases, the other variable also increases, and vice versa.

a) Direct

b) Indirect

c) Joint

7. The formula y = kx represents ______________ variation, where ‘k’ is the constant of variation.

a) Direct

b) Indirect

c) Joint

8. The equation y = k/x represents ______________ variation, where ‘k’ is the constant of variation.

a) Direct

b) Indirect

c) Joint

9. In ______________ variation, one variable depends on another variable and a constant value.

a) Direct

b) Indirect

c) Partial

10. The formula y = kx + c represents ______________ variation, where ‘k’ is the constant of variation and ‘c’ is a constant term.

a) Direct

b) Indirect

c) Partial

[mediator_tech]

### Homework

1. The length of a rectangle is directly proportional to its width. If a rectangle has a length of 8 cm and a width of 4 cm, what is the length of a similar rectangle with a width of 10 cm?

a) 16 cm

b) 20 cm

c) 25 cm

d) 32 cm

2. The time taken to complete a journey is inversely proportional to the speed. If a journey takes 5 hours at a speed of 40 km/h, how long will the journey take at a speed of 60 km/h?

a) 2 hours

b) 2.5 hours

c) 3 hours

d) 4 hours

3. The cost of 6 apples is directly proportional to the number of apples. If 6 apples cost \$12, how much will 10 apples cost?

a) \$10

b) \$15

c) \$18

d) \$20

4. The weight of an object is jointly proportional to its mass and the acceleration due to gravity. If the weight of an object is 50 N when the mass is 2 kg and the acceleration due to gravity is 10 m/s^2, what will be the weight of an object with a mass of 4 kg and an acceleration due to gravity of 5 m/s^2?

a) 25 N

b) 50 N

c) 100 N

d) 200 N

5. The price of a book is directly proportional to the number of pages. If a book with 200 pages costs \$20, how much will a similar book with 300 pages cost?

a) \$15

b) \$20

c) \$25

d) \$30

6. The volume of a cylinder is jointly proportional to its height and the square of its radius. If the volume of a cylinder is 100 cubic centimeters when the height is 5 cm and the radius is 2 cm, what will be the volume of a cylinder with a height of 8 cm and a radius of 3 cm?

a) 144 cubic centimeters

b) 216 cubic centimeters

c) 256 cubic centimeters

d) 288 cubic centimeters

7. The cost of 3 pens includes a fixed cost of \$5 and a cost per pen of \$2. What will be the total cost of 8 pens?

a) \$16

b) \$21

c) \$24

d) \$28

8. The speed of a car is inversely proportional to the time taken to travel a certain distance. If a car travels 100 km at a speed of 50 km/h, how long will it take to travel the same distance at a speed of 80 km/h?

a) 1 hour

b) 1.5 hours

c) 2 hours

d) 2.5 hours

9. The temperature of a gas is directly proportional to its pressure. If the temperature of a gas is 300 K when the pressure is 2 atm, what will be the temperature when the pressure is 3 atm?

a) 200 K

b) 300 K

c) 400 K

d) 600 K

10. The number of workers required to complete a task is inversely proportional to the time taken. If 6 workers can complete a task in 8 hours, how long will it take for 10 workers to complete the same task?

a) 4 hours

b) 6 hours

c) 8 hours

d) 12 h ours

### Lesson Plan Presentation

Topic: Variation in Mathematics: Direct Variation, Indirect Variation, Joint Variation, and Partial Variation

Duration: 1 hour

Learning Objectives:

1. Understand the concept of variation in mathematics.

2. Differentiate between direct, indirect, joint, and partial variation.

3. Apply the appropriate formulas and equations to solve problems related to variation.

4. Analyze real-life situations and identify the type of variation involved.

Embedded Core Skills:

1. Critical thinking

2. Problem-solving

3. Mathematical reasoning

4. Analytical skills

5. Communication skills

Learning Materials:

– Whiteboard or blackboard

– Markers or chalk

– Worksheets with variation problems

– Examples and illustrations of direct, indirect, joint, and partial variation

– Calculator (if necessary)

Presentation:

Introduction (5 minutes):

– Greet the students and briefly explain the topic of variation in mathematics.

– Discuss the importance of understanding variation in solving real-life problems.

– Present the learning objectives and explain how the lesson will be structured.

[mediator_tech]

Teacher’s Activities:

1. Direct Variation (10 minutes):

– Define direct variation and provide examples.

– Introduce the formula y = kx for direct variation.

– Explain the concept of the constant of variation (k).

– Present examples and guide students through solving problems related to direct variation.

– Engage students in a class discussion to ensure understanding.

2. Indirect or Inverse Variation (10 minutes):

– Define indirect or inverse variation and provide examples.

– Introduce the formula y = k/x for inverse variation.

– Explain the concept of the constant of variation (k).

– Present examples and guide students through solving problems related to inverse variation.

– Encourage students to discuss and share their solutions with the class.

3. Joint Variation (10 minutes):

– Define joint variation and provide examples.

– Introduce the formula y = kxz for joint variation.

– Explain the concept of the constant of variation (k).

– Present examples and guide students through solving problems related to joint variation.

– Encourage students to work collaboratively in pairs or groups to solve joint variation problems.

4. Partial Variation (10 minutes):

– Define partial variation and provide examples.

– Introduce the formula y = kx + c for partial variation.

– Explain the concept of the constant of variation (k) and the constant term (c).

– Present examples and guide students through solving problems related to partial variation.

– Encourage students to apply critical thinking and analyze the given information.

Learners’ Activities:

1. Direct Variation (10 minutes):

– Students solve direct variation problems individually or in pairs.

– Students participate in class discussions, sharing their solutions and strategies.

– Students ask questions to clarify any confusion.

2. Indirect or Inverse Variation (10 minutes):

– Students solve inverse variation problems individually or in pairs.

– Students present and explain their solutions to the class.

– Students engage in peer discussions to deepen their understanding.

3. Joint Variation (10 minutes):

– Students work collaboratively in pairs or groups to solve joint variation problems.

– Students discuss their approaches and strategies with their peers.

– Students present their solutions and explanations to the class.

4. Partial Variation (10 minutes):

– Students solve partial variation problems independently or in pairs.

– Students analyze the given information and apply the appropriate formula.

– Students discuss their findings and solutions with their classmates.

Assessment:

1. Formative Assessment (15 minutes):

– Distribute worksheets with variation problems.

– Monitor students’ progress and provide guidance when needed.

– Collect and review the completed worksheets to assess understanding.

Ten Evaluation Questions:

1. What is direct variation?

2. Give an example of direct variation in real life.

3. What is the formula for direct variation?

4. Define inverse variation.

5. Provide an example of inverse variation.

6. What is the formula for inverse variation?

7. Explain joint variation.

8. Give an example of joint variation.

9. What is the formula for joint variation?

10. Describe partial variation and provide an example.

Conclusion (5 minutes):

– Summarize the key points discussed in the lesson, emphasizing the different types of variation.

– Reinforce the importance of understanding variation in solving mathematical problems.

– Encourage students to practice solving variation problems and apply the concepts learned.

– Address any remaining questions or concerns from the students.

Note: The evaluation questions can be adjusted or modified based on the specific content covered during the lesson.

[mediator_tech]

### Assignment

1. The equation y = __________ represents direct variation, where ‘y’ and ‘x’ are the variables.

2. In indirect or inverse variation, as one variable increases, the other variable __________.

3. The formula y = __________ represents joint variation, where ‘y’, ‘x’, and ‘z’ are the variables.

4. In partial variation, the equation y = kx + __________ includes a constant term.

5. The equation y = __________ represents inverse variation, where ‘y’ and ‘x’ are the variables.

6. In direct variation, if one variable increases by a factor of ‘a’, the other variable increases by a factor of __________.

7. The formula y = kxz represents __________ variation, where ‘y’, ‘x’, and ‘z’ are the variables.

8. In partial variation, if ‘x’ increases by a certain amount, ‘y’ increases by the product of that amount and the constant of variation ‘k’, plus the constant term __________.

9. The formula y = __________ represents direct variation, where ‘y’ and ‘x’ are the variables, and ‘k’ is the constant of variation.

10. In indirect or inverse variation, as one variable increases, the other variable __________ by the product of the constant of variation ‘k’ and the original value of the other variable.

1. The equation y = kx represents direct variation, where ‘y’ and ‘x’ are the variables.
2. In indirect or inverse variation, as one variable increases, the other variable decreases.
3. The formula y = kxz represents joint variation, where ‘y’, ‘x’, and ‘z’ are the variables.
4. In partial variation, the equation y = kx + c includes a constant term.
5. The equation y = k/x represents inverse variation, where ‘y’ and ‘x’ are the variables.
6. In direct variation, if one variable increases by a factor of ‘a’, the other variable increases by a factor of ‘a’.
7. The formula y = kxz represents joint variation, where ‘y’, ‘x’, and ‘z’ are the variables.
8. In partial variation, if ‘x’ increases by a certain amount, ‘y’ increases by the product of that amount and the constant of variation ‘k’, plus the constant term ‘c’.
9. The formula y = kx represents direct variation, where ‘y’ and ‘x’ are the variables, and ‘k’ is the constant of variation.
10. In indirect or inverse variation, as one variable increases, the other variable decreases by the product of the constant of variation ‘k’ and the original value of the other variable.

Well done!

[mediator_tech]

1. The length of a rectangle is directly proportional to its width. If the width is 5 cm and the length is 15 cm, what will be the length of a similar rectangle with a width of 8 cm?

a) 10 cm

b) 12 cm

c) 20 cm

d) 24 cm

2. The time taken to complete a task is inversely proportional to the number of workers. If it takes 6 workers 4 hours to complete the task, how many hours will it take for 12 workers to complete the same task?

a) 1 hour

b) 2 hours

c) 3 hours

d) 6 hours

3. The temperature of a gas is inversely proportional to its volume. If the temperature is 300 K when the volume is 2 liters, what will be the temperature when the volume is 5 liters?

a) 75 K

b) 120 K

c) 150 K

d) 750 K

4. The cost of buying pens includes a fixed cost of \$5 and a cost per pen of \$2. If you buy 8 pens, what will be the total cost?

a) \$11

b) \$16

c) \$21

d) \$26

5. The speed of a car is directly proportional to the distance traveled. If a car travels 200 km at a speed of 50 km/h, how far will it travel at a speed of 80 km/h?

a) 160 km

b) 250 km

c) 320 km

d) 400 km

6. The pressure of a gas is inversely proportional to its volume. If the pressure is 4 atmospheres when the volume is 10 liters, what will be the pressure when the volume is 5 liters?

a) 2 atmospheres

b) 4 atmospheres

c) 8 atmospheres

d) 20 atmospheres

7. The area of a square is directly proportional to the square of its side length. If the side length of a square is 4 cm, what will be the area of a square with a side length of 6 cm?

a) 12 cm^2

b) 18 cm^2

c) 24 cm^2

d) 36 cm^2

8. The volume of a rectangular prism is directly proportional to its height. If the volume is 60 cubic units when the height is 5 units, what will be the volume when the height is 8 units?

a) 24 cubic units

b) 40 cubic units

c) 75 cubic units

d) 120 cubic units

9. The number of workers required to complete a task is jointly proportional to the time taken and the difficulty level. If it takes 4 workers 3 hours to complete a task with difficulty level 2, how many workers will be needed to complete a task with difficulty level 3 in 6 hours?

a) 2 workers

b) 4 workers

c) 6 workers

d) 8 workers

10. The total cost of buying a certain number of items includes a fixed cost of \$10 and a cost per item of \$3. If the total cost is \$25, how many items were bought?

a) 5 items

b) 7 items

c) 8 items

d) 10 items

[mediator_tech]