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 an Equation

To solve this problem, we need to write an equation that represents the relationship between the cost of the sandwich and the number of sandwiches ordered. Since the cost varies directly with the number of sandwiches, we can write the equation 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 use this information to find the constant of variation, k. We can plug in the values of c and n into the equation and solve for k:

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 with the Constant of Variation

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

c = 6n

Step 4: Find the Cost of the Sandwiches when n = 3

Now that we have the equation with the constant of variation, we can use it to find the cost of the sandwiches when 3 sandwiches are ordered. We can plug in the value of n into the equation and solve for c:

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 direct variation to solve a real-world problem. We have shown how to write an equation that represents the relationship between the cost of a ham sandwich and the number of sandwiches ordered, and how to use the equation to find the cost of the sandwiches when a different number of 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

Here are a few example problems that demonstrate how to use direct variation:

  • If the cost of a book varies directly with the number of pages, and the cost of a book with 200 pages is $20, what is the cost of a book with 400 pages?
  • If the weight of a box varies directly with the number of items in the box, and the weight of a box with 10 items is 20 pounds, what is the weight of a box with 20 items?
  • If the cost of a shirt varies directly with the number of buttons on the shirt, and the cost of a shirt with 5 buttons is $15, what is the cost of a shirt with 10 buttons?

Solutions

Here are the solutions to the example problems:

  • If the cost of a book varies directly with the number of pages, and the cost of a book with 200 pages is $20, what is the cost of a book with 400 pages?

To solve this problem, we need to write an equation that represents the relationship between the cost of the book and the number of pages. Since the cost varies directly with the number of pages, we can write the equation as:

c = kp

where c is the cost of the book, p is the number of pages, and k is the constant of variation.

We are given that the cost of a book with 200 pages is $20. We can use this information to find the constant of variation, k. We can plug in the values of c and p into the equation and solve for k:

20 = k(200)

To solve for k, we can divide both sides of the equation by 200:

k = 20/200

k = 0.1

So, the constant of variation is 0.1.

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

c = 0.1p

We can use this equation to find the cost of a book with 400 pages. We can plug in the value of p into the equation and solve for c:

c = 0.1(400)

c = 40

So, the cost of a book with 400 pages is $40.

  • If the weight of a box varies directly with the number of items in the box, and the weight of a box with 10 items is 20 pounds, what is the weight of a box with 20 items?

To solve this problem, we need to write an equation that represents the relationship between the weight of the box and the number of items in the box. Since the weight varies directly with the number of items, we can write the equation as:

w = ki

where w is the weight of the box, i is the number of items, and k is the constant of variation.

We are given that the weight of a box with 10 items is 20 pounds. We can use this information to find the constant of variation, k. We can plug in the values of w and i into the equation and solve for k:

20 = k(10)

To solve for k, we can divide both sides of the equation by 10:

k = 20/10

k = 2

So, the constant of variation is 2.

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

w = 2i

We can use this equation to find the weight of a box with 20 items. We can plug in the value of i into the equation and solve for w:

w = 2(20)

w = 40

So, the weight of a box with 20 items is 40 pounds.

  • If the cost of a shirt varies directly with the number of buttons on the shirt, and the cost of a shirt with 5 buttons is $15, what is the cost of a shirt with 10 buttons?

To solve this problem, we need to write an equation that represents the relationship between the cost of the shirt and the number of buttons on the shirt. Since the cost varies directly with the number of buttons, we can write the equation as:

c = kb

where c is the cost of the shirt, b is the number of buttons, and k is the constant of variation.

We are given that the cost of a shirt with 5 buttons is $15. We can use this information to find the constant of variation, k. We can plug in the values of c and b into the equation and solve for k:

15 = k(5)

To solve for k, we can divide both sides of the equation by 5:

k = 15/5

k = 3

So, the constant of variation is 3.

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

c = 3b

We can use this equation to find the cost of a shirt with 10 buttons. We can plug in the value of b into the equation and solve for c:

c = 3(10)

c = 30

So, the cost of a shirt with 10 buttons is $30.

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 know if a problem is a direct variation problem?

A: To determine if a problem is a direct variation problem, look for the following characteristics:

  • The problem involves a relationship between two variables.
  • The relationship is proportional, meaning that as one variable increases, the other variable also increases at a constant rate.
  • The problem can be represented mathematically using the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Q: How do I write an equation for a direct variation problem?

A: To write an equation for a direct variation problem, follow these steps:

  1. Identify the dependent variable (y) and the independent variable (x).
  2. Determine the constant of variation (k) by using the given information.
  3. Write the equation in the form y = kx.

Q: How do I find the constant of variation (k)?

A: To find the constant of variation (k), follow these steps:

  1. Use the given information to write an equation in the form y = kx.
  2. Plug in the values of y and x into the equation.
  3. Solve for k by dividing both sides of the equation by x.

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

A: Direct variation and inverse variation are two types of proportional relationships. The main difference between them is the direction of the relationship.

  • Direct variation: As one variable increases, the other variable also increases at a constant rate.
  • Inverse variation: As one variable increases, the other variable decreases at a constant rate.

Q: Can I have a negative constant of variation (k)?

A: Yes, you can have a negative constant of variation (k). This means that as one variable increases, the other variable decreases at a constant rate.

Q: Can I have a zero constant of variation (k)?

A: No, you cannot have a zero constant of variation (k). This would mean that the relationship between the two variables is not proportional.

Q: Can I have a fractional constant of variation (k)?

A: Yes, you can have a fractional constant of variation (k). This means that the relationship between the two variables is proportional, but the constant of variation is a fraction.

Q: Can I have a negative fractional constant of variation (k)?

A: Yes, you can have a negative fractional constant of variation (k). This means that the relationship between the two variables is proportional, but the constant of variation is a negative fraction.

Q: How do I graph a direct variation problem?

A: To graph a direct variation problem, follow these steps:

  1. Plot the points (x, y) on a coordinate plane.
  2. Draw a line through the points to represent the relationship between the two variables.
  3. Label the axes with the variables x and y.

Q: Can I have a direct variation problem with a non-linear relationship?

A: No, a direct variation problem must have a linear relationship between the two variables. If the relationship is non-linear, it is not a direct variation problem.

Q: Can I have a direct variation problem with a non-constant of variation (k)?

A: No, a direct variation problem must have a constant of variation (k). If the constant of variation is not constant, it is not a direct variation problem.

Conclusion

In this article, we have answered some common questions about direct variation. We have discussed the characteristics of direct variation problems, how to write an equation for a direct variation problem, and how to find the constant of variation (k). We have also discussed the differences between direct variation and inverse variation, and how to graph a direct variation problem.