If The Cost \$c$ Of 9 Ham Sandwiches Is \$54, And The Cost Varies Directly With The Number Of Sandwiches $n$, What Is The Cost Of 3 Sandwiches?

Jun 19, 2025 by ADMIN 144 views

The Cost of Ham Sandwiches A Direct Variation Problem

#Introduction In the realm of mathematics, we often encounter scenarios where quantities are related to each other in a predictable manner. One such relationship is direct variation, where two quantities increase or decrease proportionally. In this article, we will delve into a classic problem involving direct variation, focusing on the cost of ham sandwiches at a deli. We will explore how the cost varies with the number of sandwiches purchased and, using the principles of direct variation, determine the cost for a specific number of sandwiches.

Understanding Direct Variation

Direct variation is a fundamental concept in mathematics that describes a relationship between two variables where one variable is a constant multiple of the other. In simpler terms, it means that as one quantity increases, the other quantity increases proportionally, and as one quantity decreases, the other quantity decreases proportionally. This relationship can be expressed mathematically as:

y = kx

Where:

y is the dependent variable
x is the independent variable
k is the constant of variation

The constant of variation, k, represents the ratio between the two variables and remains constant throughout the relationship. It essentially determines the proportionality between the variables. For example, if k is 2, it means that for every unit increase in x, y increases by 2 units.

Key Characteristics of Direct Variation

To further solidify our understanding of direct variation, let's highlight some of its key characteristics:

Linear Relationship: The graph of a direct variation equation is always a straight line passing through the origin (0, 0). This indicates a consistent and proportional relationship between the variables.
Constant Ratio: The ratio between the two variables remains constant throughout the relationship. This constant ratio is represented by the constant of variation, k.
Proportional Change: As one variable changes, the other variable changes proportionally. If one variable doubles, the other variable also doubles; if one variable is halved, the other variable is also halved.

Direct variation finds applications in various real-world scenarios, such as calculating the distance traveled at a constant speed, determining the cost of items based on quantity, and understanding the relationship between force and acceleration. By grasping the principles of direct variation, we can effectively model and solve problems involving proportional relationships.

Problem Statement: Ham Sandwiches and Direct Variation

Now that we have a solid understanding of direct variation, let's tackle the specific problem at hand. The problem states:

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 $n = 3$ ?

This problem presents a classic example of direct variation. The cost of the sandwiches is directly proportional to the number of sandwiches purchased. This means that as the number of sandwiches increases, the cost also increases proportionally, and vice versa.

To solve this problem, we will utilize the principles of direct variation and the given information to determine the constant of variation. Once we have the constant of variation, we can easily calculate the cost for any number of sandwiches.

Solving the Problem: Step-by-Step

Let's break down the solution to this problem into a step-by-step process:

Step 1: Express the Direct Variation Relationship

As we know, the cost $c$ varies directly with the number of sandwiches $n$ . We can express this relationship mathematically using the direct variation equation:

c = kn

Where:

c is the cost of the sandwiches
n is the number of sandwiches
k is the constant of variation (the cost per sandwich)

Step 2: Determine the Constant of Variation (k)

To find the constant of variation, we can use the given information: when $n = 9$ , $c = $54$. Substitute these values into the equation:

$54 = k * 9

Now, solve for k:

k = $54 / 9

k = $6

Therefore, the constant of variation, k, is $6. This means that the cost per sandwich is $6.

Step 3: Calculate the Cost for n = 3

Now that we have the constant of variation, we can calculate the cost when $n = 3$ . Substitute $n = 3$ and $k = $6$ into the direct variation equation:

c = 6 * 3

c = $18

Therefore, the cost of the sandwiches when $n = 3$ is $18.

Answer and Conclusion

The correct answer to the problem is A. $18.

In conclusion, this problem demonstrates the application of direct variation in a real-world scenario. By understanding the principles of direct variation and utilizing the given information, we were able to determine the constant of variation and calculate the cost for a specific number of sandwiches. Direct variation is a powerful tool for modeling and solving problems involving proportional relationships, and its applications extend far beyond the realm of ham sandwiches.

Additional Examples and Applications of Direct Variation

To further illustrate the versatility of direct variation, let's explore some additional examples and applications:

Distance and Speed: The distance traveled by a vehicle at a constant speed varies directly with the time traveled. If a car travels 120 miles in 2 hours, it will travel 180 miles in 3 hours (assuming constant speed).
Work and Time: The amount of work done varies directly with the time spent working. If a person can complete 10 tasks in 4 hours, they can complete 25 tasks in 10 hours (assuming consistent work rate).
Hooke's Law: In physics, Hooke's Law states that the extension of a spring is directly proportional to the force applied to it. If a force of 10 Newtons stretches a spring by 5 centimeters, a force of 20 Newtons will stretch it by 10 centimeters.
Currency Exchange: The amount of foreign currency you can exchange for a certain amount of domestic currency varies directly with the exchange rate. If the exchange rate is 1.2 US dollars per Euro, then 100 Euros can be exchanged for 120 US dollars.

These examples highlight the wide range of applications of direct variation in various fields, including physics, economics, and everyday life. By recognizing and understanding direct variation relationships, we can make predictions, solve problems, and gain valuable insights into the world around us.

Practice Problems

To reinforce your understanding of direct variation, try solving these practice problems:

If y varies directly with x, and y = 24 when x = 8, find y when x = 5.
The weight of an object on the Moon varies directly with its weight on Earth. An object that weighs 180 pounds on Earth weighs 30 pounds on the Moon. How much would a person who weighs 150 pounds on Earth weigh on the Moon?
The number of pages a printer can print varies directly with the time it prints. If a printer can print 120 pages in 5 minutes, how many pages can it print in 12 minutes?

By working through these practice problems, you can solidify your understanding of direct variation and enhance your problem-solving skills.

Conclusion

Direct variation is a fundamental concept in mathematics that describes a proportional relationship between two variables. In this article, we explored the principles of direct variation, solved a problem involving the cost of ham sandwiches, and examined various real-world applications. By grasping the concept of direct variation, we can effectively model and solve problems involving proportional relationships in diverse scenarios. Remember, the key to understanding direct variation lies in recognizing the constant ratio between the variables and applying the direct variation equation:

y = kx

With practice and application, you can master direct variation and confidently tackle related problems in mathematics and beyond. This concept is not only crucial for academic success but also for navigating everyday situations where proportional relationships play a significant role. So, embrace the power of direct variation and unlock its potential in your mathematical journey!

Final Thoughts

As we conclude our exploration of direct variation, it's important to emphasize the value of this concept in mathematics and its relevance to real-world applications. Direct variation provides a framework for understanding and modeling proportional relationships, which are prevalent in various fields, from science and engineering to economics and finance. By mastering the principles of direct variation, you equip yourself with a powerful tool for problem-solving, decision-making, and critical thinking.

Remember, the beauty of mathematics lies in its ability to reveal the underlying patterns and relationships that govern our world. Direct variation is just one example of how mathematical concepts can provide insights into the way things work. So, continue to explore, learn, and apply your mathematical knowledge to make sense of the world around you. The journey of mathematical discovery is a rewarding one, and the more you delve into it, the more you will appreciate its power and elegance.