The Cost, \[$ C \$\], Of A Ham Sandwich At A Deli Varies Directly With The Number Of Sandwiches, \[$ N \$\]. If \[$ C = \$54 \$\] When \[$ N = 9 \$\], What Is The Cost Of The Sandwiches When \[$ N = 3 \$\]?A.

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Introduction

In this article, we will explore the concept of direct variation and how it can be applied to real-world problems. We will use the example of a ham sandwich at a deli to demonstrate how the cost of the sandwich varies directly with the number of sandwiches ordered.

What is Direct Variation?

Direct variation is a relationship between two variables where one variable is a constant multiple of the other variable. In other words, as one variable increases, the other variable also increases at a constant rate. This relationship can be represented mathematically as:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

The Problem

In this problem, we are given that the cost of a ham sandwich at a deli varies directly with the number of sandwiches ordered. We are also given that the cost of the sandwich is $54 when 9 sandwiches are ordered. We need to find the cost of the sandwiches when 3 sandwiches are ordered.

Step 1: Write the Equation of Variation

Since the cost of the sandwich varies directly with the number of sandwiches ordered, we can write the equation of variation as:

c = kn

where c is the cost of the sandwich, n is the number of sandwiches ordered, and k is the constant of variation.

Step 2: Find the Constant of Variation

We are given that the cost of the sandwich is $54 when 9 sandwiches are ordered. We can substitute these values into the equation of variation to find the constant of variation:

54 = k(9)

To solve for k, we can divide both sides of the equation by 9:

k = 54/9

k = 6

So, the constant of variation is 6.

Step 3: Write the Equation of Variation with the Constant of Variation

Now that we have found the constant of variation, we can write the equation of variation as:

c = 6n

Step 4: Find the Cost of the Sandwiches when 3 Sandwiches are Ordered

We can substitute n = 3 into the equation of variation to find the cost of the sandwiches when 3 sandwiches are ordered:

c = 6(3)

c = 18

So, the cost of the sandwiches when 3 sandwiches are ordered is $18.

Conclusion

In this article, we have demonstrated how to use the concept of direct variation to solve a real-world problem. We have shown how to write the equation of variation, find the constant of variation, and use the equation of variation to find the cost of the sandwiches when 3 sandwiches are ordered.

Direct Variation Formula

The direct variation formula is:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Example Problems

  1. The cost of a book varies directly with the number of books ordered. If the cost of the book is $20 when 2 books are ordered, what is the cost of the book when 5 books are ordered?
  2. The distance traveled by a car varies directly with the time traveled. If the distance traveled is 120 miles when 2 hours are traveled, what is the distance traveled when 4 hours are traveled?

Solutions

  1. Let c be the cost of the book and n be the number of books ordered. We can write the equation of variation as:

c = kn

We are given that the cost of the book is $20 when 2 books are ordered. We can substitute these values into the equation of variation to find the constant of variation:

20 = k(2)

To solve for k, we can divide both sides of the equation by 2:

k = 20/2

k = 10

So, the constant of variation is 10.

Now that we have found the constant of variation, we can write the equation of variation as:

c = 10n

We can substitute n = 5 into the equation of variation to find the cost of the book when 5 books are ordered:

c = 10(5)

c = 50

So, the cost of the book when 5 books are ordered is $50.

  1. Let d be the distance traveled and t be the time traveled. We can write the equation of variation as:

d = kt

We are given that the distance traveled is 120 miles when 2 hours are traveled. We can substitute these values into the equation of variation to find the constant of variation:

120 = k(2)

To solve for k, we can divide both sides of the equation by 2:

k = 120/2

k = 60

So, the constant of variation is 60.

Now that we have found the constant of variation, we can write the equation of variation as:

d = 60t

We can substitute t = 4 into the equation of variation to find the distance traveled when 4 hours are traveled:

d = 60(4)

d = 240

So, the distance traveled when 4 hours are traveled is 240 miles.

Direct Variation in Real-World Applications

Direct variation is a fundamental concept in mathematics that has numerous real-world applications. Some examples of direct variation in real-world applications include:

  • The cost of a product varies directly with the number of units sold.
  • The distance traveled by a car varies directly with the time traveled.
  • The cost of a service varies directly with the number of hours worked.
  • The amount of money earned varies directly with the number of hours worked.

Conclusion

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other variable. In other words, as one variable increases, the other variable also increases at a constant rate.

Q: How do I write the equation of variation?

A: To write the equation of variation, you need to identify the dependent variable (y) and the independent variable (x). Then, you can write the equation as:

y = kx

where k is the constant of variation.

Q: How do I find the constant of variation?

A: To find the constant of variation, you need to substitute the values of the dependent variable and the independent variable into the equation of variation. Then, you can solve for k.

Q: What is the difference between direct variation and inverse variation?

A: Direct variation is a relationship where one variable is a constant multiple of the other variable. Inverse variation is a relationship where one variable is a constant divided by the other variable.

Q: Can you give an example of direct variation in real-world applications?

A: Yes, here are a few examples:

  • The cost of a product varies directly with the number of units sold.
  • The distance traveled by a car varies directly with the time traveled.
  • The cost of a service varies directly with the number of hours worked.
  • The amount of money earned varies directly with the number of hours worked.

Q: How do I use the equation of variation to solve a problem?

A: To use the equation of variation to solve a problem, you need to:

  1. Write the equation of variation.
  2. Substitute the values of the dependent variable and the independent variable into the equation.
  3. Solve for the unknown variable.

Q: Can you give an example of using the equation of variation to solve a problem?

A: Yes, here is an example:

Suppose the cost of a book varies directly with the number of books ordered. If the cost of the book is $20 when 2 books are ordered, what is the cost of the book when 5 books are ordered?

To solve this problem, we can write the equation of variation as:

c = kn

We are given that the cost of the book is $20 when 2 books are ordered. We can substitute these values into the equation of variation to find the constant of variation:

20 = k(2)

To solve for k, we can divide both sides of the equation by 2:

k = 20/2

k = 10

So, the constant of variation is 10.

Now that we have found the constant of variation, we can write the equation of variation as:

c = 10n

We can substitute n = 5 into the equation of variation to find the cost of the book when 5 books are ordered:

c = 10(5)

c = 50

So, the cost of the book when 5 books are ordered is $50.

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

A: Here are a few common mistakes to avoid:

  • Not identifying the dependent variable and the independent variable correctly.
  • Not writing the equation of variation correctly.
  • Not substituting the values of the dependent variable and the independent variable into the equation correctly.
  • Not solving for the unknown variable correctly.

Q: Can you give some tips for working with direct variation?

A: Yes, here are a few tips:

  • Make sure to identify the dependent variable and the independent variable correctly.
  • Write the equation of variation correctly.
  • Substitute the values of the dependent variable and the independent variable into the equation correctly.
  • Solve for the unknown variable correctly.
  • Check your work to make sure that the equation is correct.

Conclusion

In this article, we have answered some common questions about direct variation. We have also provided examples of direct variation in real-world applications and tips for working with direct variation.