A) In The Following Tables, The Two Variables $x$ And $y$ Are Inversely Proportional. In Each Case:(i) Find The Values Of $a$ And $b$. (ii) Draw A Graph Of $y$ Versus $x$, Using An Appropriate

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Inverse Proportionality: Understanding the Relationship Between Variables

In mathematics, proportionality is a fundamental concept that describes the relationship between two variables. When two variables are inversely proportional, it means that as one variable increases, the other decreases, and vice versa. This relationship is often represented by the equation y = k/x, where k is a constant. In this article, we will explore the concept of inverse proportionality, and we will use tables to find the values of a and b in each case. We will also draw a graph of y versus x to visualize the relationship between the variables.

What is Inverse Proportionality?

Inverse proportionality is a relationship between two variables where the product of the two variables is constant. This means that as one variable increases, the other decreases, and vice versa. The equation for inverse proportionality is y = k/x, where k is a constant. This equation can be rewritten as xy = k, which shows that the product of the two variables is constant.

Finding the Values of a and b

To find the values of a and b, we need to use the equation y = k/x. We are given two tables with the values of x and y. We can use these tables to find the values of a and b.

Table 1

x y
2 3
4 1.5
6 1

Table 2

x y
3 2
6 1
9 0.67

Finding the Values of a and b in Table 1

To find the values of a and b in Table 1, we can use the equation y = k/x. We can start by finding the value of k. We can do this by multiplying the values of x and y in the table.

x y xy
2 3 6
4 1.5 6
6 1 6

As we can see, the product of x and y is constant, which means that the relationship between the variables is inverse proportionality. We can now find the value of k by dividing the product of x and y by x.

x y xy k
2 3 6 3
4 1.5 6 1.5
6 1 6 1

The value of k is 3, which means that the equation for inverse proportionality is y = 3/x. We can now find the values of a and b by substituting the values of x and y into the equation.

x y y = 3/x
2 3 1.5
4 1.5 0.75
6 1 0.5

As we can see, the values of a and b are 1.5, 0.75, and 0.5.

Finding the Values of a and b in Table 2

To find the values of a and b in Table 2, we can use the equation y = k/x. We can start by finding the value of k. We can do this by multiplying the values of x and y in the table.

x y xy
3 2 6
6 1 6
9 0.67 6

As we can see, the product of x and y is constant, which means that the relationship between the variables is inverse proportionality. We can now find the value of k by dividing the product of x and y by x.

x y xy k
3 2 6 2
6 1 6 1
9 0.67 6 0.67

The value of k is 2, which means that the equation for inverse proportionality is y = 2/x. We can now find the values of a and b by substituting the values of x and y into the equation.

x y y = 2/x
3 2 0.67
6 1 0.33
9 0.67 0.22

As we can see, the values of a and b are 0.67, 0.33, and 0.22.

Drawing a Graph of y versus x

To visualize the relationship between the variables, we can draw a graph of y versus x. We can use the values of x and y from the tables to plot the points on the graph.

Graph 1

x y
2 3
4 1.5
6 1

As we can see, the graph shows an inverse relationship between the variables. When x increases, y decreases, and vice versa.

Graph 2

x y
3 2
6 1
9 0.67

As we can see, the graph shows an inverse relationship between the variables. When x increases, y decreases, and vice versa.

Conclusion

In this article, we have explored the concept of inverse proportionality and used tables to find the values of a and b in each case. We have also drawn a graph of y versus x to visualize the relationship between the variables. As we can see, the relationship between the variables is inverse proportionality, where the product of the two variables is constant. We can use this concept to model real-world relationships and make predictions about the behavior of the variables.

References

Further Reading

Inverse proportionality is a fundamental concept in mathematics that describes the relationship between two variables. In this article, we will answer some of the most frequently asked questions about inverse proportionality, covering topics such as the definition, examples, and applications of this concept.

Q: What is inverse proportionality?

A: Inverse proportionality is a relationship between two variables where the product of the two variables is constant. This means that as one variable increases, the other decreases, and vice versa.

Q: What is the equation for inverse proportionality?

A: The equation for inverse proportionality is y = k/x, where k is a constant.

Q: What is the significance of the constant k in the equation for inverse proportionality?

A: The constant k represents the product of the two variables. It is a measure of the strength of the inverse relationship between the variables.

Q: Can you provide an example of inverse proportionality in real life?

A: Yes, a classic example of inverse proportionality is the relationship between the number of people at a party and the amount of food available. As the number of people at the party increases, the amount of food available decreases, and vice versa.

Q: How do you determine if two variables are inversely proportional?

A: To determine if two variables are inversely proportional, you can use the following steps:

  1. Plot the data on a graph.
  2. Check if the graph shows a straight line with a negative slope.
  3. If the graph shows a straight line with a negative slope, then the variables are inversely proportional.

Q: What are some common applications of inverse proportionality?

A: Inverse proportionality has many applications in real life, including:

  1. Economics: Inverse proportionality is used to model the relationship between the price of a good and the quantity demanded.
  2. Physics: Inverse proportionality is used to model the relationship between the force of gravity and the distance between two objects.
  3. Engineering: Inverse proportionality is used to model the relationship between the speed of a vehicle and the distance traveled.

Q: Can you provide some examples of inverse proportionality in different fields?

A: Yes, here are some examples of inverse proportionality in different fields:

  1. Economics: The relationship between the price of a good and the quantity demanded is an example of inverse proportionality. As the price of a good increases, the quantity demanded decreases, and vice versa.
  2. Physics: The relationship between the force of gravity and the distance between two objects is an example of inverse proportionality. As the distance between two objects increases, the force of gravity between them decreases, and vice versa.
  3. Engineering: The relationship between the speed of a vehicle and the distance traveled is an example of inverse proportionality. As the speed of a vehicle increases, the distance traveled decreases, and vice versa.

Q: How do you solve problems involving inverse proportionality?

A: To solve problems involving inverse proportionality, you can use the following steps:

  1. Write down the equation for inverse proportionality.
  2. Plug in the values of the variables into the equation.
  3. Solve for the unknown variable.

Q: What are some common mistakes to avoid when working with inverse proportionality?

A: Some common mistakes to avoid when working with inverse proportionality include:

  1. Confusing inverse proportionality with direct proportionality.
  2. Failing to check if the variables are inversely proportional before using the equation.
  3. Not considering the units of the variables when using the equation.

Conclusion

Inverse proportionality is a fundamental concept in mathematics that describes the relationship between two variables. By understanding the definition, examples, and applications of inverse proportionality, you can solve problems involving this concept and make predictions about the behavior of the variables. Remember to avoid common mistakes and use the equation for inverse proportionality correctly to get accurate results.

References

Further Reading