In Each Table, \[$ Y \$\] Is Directly Proportional To \[$ X \$\]. Find The Constant Of Proportionality And Then Complete The Table.Table 1:$\[ \begin{tabular}{|c|c|c|c|} \hline \( X \) & 20 & 40 & ? \\ \hline \( Y \) & 4 & ? & 16

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Introduction

In mathematics, direct proportionality is a fundamental concept that describes the relationship between two variables. When we say that one variable is directly proportional to another, it means that as the value of one variable increases, the value of the other variable also increases at a constant rate. In this article, we will explore the concept of direct proportionality and use it to find the constant of proportionality and complete a given table.

What is Direct Proportionality?

Direct proportionality is a relationship between two variables, x and y, where y is directly proportional to x. This can be represented mathematically as:

y ∝ x

or

y = kx

where k is the constant of proportionality.

Understanding the Constant of Proportionality

The constant of proportionality, k, is a value that represents the rate at which y changes with respect to x. It is a measure of how much y increases when x increases by a certain amount. In other words, k is the factor by which y is multiplied to get the value of y for a given value of x.

Finding the Constant of Proportionality

To find the constant of proportionality, we can use the given values of x and y in the table. Let's use the first two rows of the table to find the value of k.

x y
20 4
40 ?

We know that y is directly proportional to x, so we can write:

y = kx

Substituting the values of x and y from the first row, we get:

4 = k(20)

To find the value of k, we can divide both sides of the equation by 20:

k = 4/20 k = 0.2

Now that we have found the value of k, we can use it to find the value of y for x = 40.

y = kx y = 0.2(40) y = 8

So, the value of y for x = 40 is 8.

Completing the Table

Now that we have found the value of k, we can use it to complete the table.

x y
20 4
40 8
? 16

We know that y is directly proportional to x, so we can write:

y = kx

Substituting the values of x and y from the last row, we get:

16 = k(?)

To find the value of x, we can divide both sides of the equation by k:

x = 16/k x = 16/0.2 x = 80

So, the value of x for y = 16 is 80.

Conclusion

In this article, we have explored the concept of direct proportionality and used it to find the constant of proportionality and complete a given table. We have seen how direct proportionality can be represented mathematically and how the constant of proportionality can be used to find the value of y for a given value of x. We have also seen how to use the constant of proportionality to complete a table.

Table 1: Direct Proportionality

x y
20 4
40 8
80 16

Discussion

Direct proportionality is a fundamental concept in mathematics that describes the relationship between two variables. It is used in a wide range of applications, including physics, engineering, and economics. In this article, we have seen how direct proportionality can be used to find the constant of proportionality and complete a table.

Key Takeaways

  • Direct proportionality is a relationship between two variables, x and y, where y is directly proportional to x.
  • The constant of proportionality, k, is a value that represents the rate at which y changes with respect to x.
  • The constant of proportionality can be used to find the value of y for a given value of x.
  • Direct proportionality can be used to complete a table.

References

Glossary

  • Direct Proportionality: A relationship between two variables, x and y, where y is directly proportional to x.
  • Constant of Proportionality: A value that represents the rate at which y changes with respect to x.
  • Proportionality Constant: A value that represents the rate at which y changes with respect to x.
    Direct Proportionality Q&A =============================

Frequently Asked Questions

Q: What is direct proportionality?

A: Direct proportionality is a relationship between two variables, x and y, where y is directly proportional to x. This means that as the value of x increases, the value of y also increases at a constant rate.

Q: How is direct proportionality represented mathematically?

A: Direct proportionality can be represented mathematically as:

y ∝ x

or

y = kx

where k is the constant of proportionality.

Q: What is the constant of proportionality?

A: The constant of proportionality, k, is a value that represents the rate at which y changes with respect to x. It is a measure of how much y increases when x increases by a certain amount.

Q: How do I find the constant of proportionality?

A: To find the constant of proportionality, you can use the given values of x and y in a table. Let's use the first two rows of the table to find the value of k.

x y
20 4
40 ?

We know that y is directly proportional to x, so we can write:

y = kx

Substituting the values of x and y from the first row, we get:

4 = k(20)

To find the value of k, we can divide both sides of the equation by 20:

k = 4/20 k = 0.2

Q: How do I use the constant of proportionality to complete a table?

A: To use the constant of proportionality to complete a table, you can substitute the values of x and y from the last row into the equation:

y = kx

Substituting the values of x and y from the last row, we get:

16 = k(?)

To find the value of x, we can divide both sides of the equation by k:

x = 16/k x = 16/0.2 x = 80

So, the value of x for y = 16 is 80.

Q: What are some real-world applications of direct proportionality?

A: Direct proportionality has many real-world applications, including:

  • Physics: The relationship between the force applied to an object and the distance it travels.
  • Engineering: The relationship between the speed of a vehicle and the distance it travels.
  • Economics: The relationship between the price of a product and the quantity demanded.

Q: How do I determine if a relationship is directly proportional?

A: To determine if a relationship is directly proportional, you can use the following steps:

  1. Plot the data on a graph.
  2. Check if the graph is a straight line.
  3. If the graph is a straight line, then the relationship is directly proportional.

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

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

  • Assuming that a relationship is directly proportional when it is not.
  • Failing to check if the relationship is directly proportional before using it.
  • Using the wrong value for the constant of proportionality.

Q: How do I use direct proportionality to solve problems?

A: To use direct proportionality to solve problems, you can follow these steps:

  1. Identify the variables involved in the problem.
  2. Determine if the relationship between the variables is directly proportional.
  3. Use the constant of proportionality to find the value of one variable.
  4. Use the value of one variable to find the value of the other variable.

Conclusion

Direct proportionality is a fundamental concept in mathematics that describes the relationship between two variables. It has many real-world applications and can be used to solve a wide range of problems. By understanding direct proportionality and how to use it, you can become a more confident and proficient problem-solver.

Glossary

  • Direct Proportionality: A relationship between two variables, x and y, where y is directly proportional to x.
  • Constant of Proportionality: A value that represents the rate at which y changes with respect to x.
  • Proportionality Constant: A value that represents the rate at which y changes with respect to x.
  • Directly Proportional: A relationship between two variables, x and y, where y is directly proportional to x.

References