Given That The Entrance Fee For 7 People In A Movie Theater Is $392, What Is The Constant Of Proportionality Representing The Price Per Entry?
In this article, we will explore how to determine the constant of proportionality when given the total cost for a certain number of movie tickets. This is a fundamental concept in mathematics, particularly in the study of ratios and proportions. Understanding how to calculate this constant can help us solve various real-world problems, such as determining the cost of additional tickets or comparing prices at different theaters.
Understanding Proportionality
Before diving into the specific problem, let's first understand what proportionality means. Two quantities are said to be proportional if they vary in such a way that one of the quantities is a constant multiple of the other. This constant multiple is known as the constant of proportionality. In simpler terms, if we have two variables, say x and y, and y is directly proportional to x, then we can express this relationship as:
y = kx
where y represents the total cost, x represents the number of tickets, and k is the constant of proportionality. This constant, k, represents the price per ticket in our scenario. To find the constant of proportionality, we need to rearrange the formula:
k = y / x
This formula tells us that the constant of proportionality is equal to the total cost divided by the number of tickets. By understanding this basic concept, we can easily apply it to solve our problem.
Problem Statement: Finding the Constant of Proportionality
Our problem states that the cost of 7 movie tickets is $392. We need to find the constant of proportionality, which represents the price per ticket. To solve this, we can use the formula we discussed earlier:
k = y / x
In this case:
- y (total cost) = $392
- x (number of tickets) = 7
Now, let's substitute these values into the formula:
k = 392 / 7
By performing this division, we can find the constant of proportionality.
Step-by-Step Calculation
To calculate the constant of proportionality, we need to divide the total cost ($392) by the number of tickets (7). Let's perform the division:
392 ÷ 7
-
First, divide 39 by 7. The quotient is 5, and the remainder is 4.
5
7|392 -35
4
```
-
Bring down the next digit (2) to form 42.
5
7|392 -35
42
```
-
Divide 42 by 7. The quotient is 6, and the remainder is 0.
56
7|392 -35
42
-42
0
```
So, 392 ÷ 7 = 56. Therefore, the constant of proportionality, k, is 56.
Interpretation of the Result
The constant of proportionality, k, is 56. This means that the price per movie ticket is $56. In other words, for every one ticket purchased, the cost is $56. This constant allows us to easily calculate the cost for any number of tickets. For instance, if we wanted to buy 10 tickets, we could simply multiply the number of tickets by the constant of proportionality:
Cost of 10 tickets = 10 * 56 = $560
Similarly, if we wanted to buy 2 tickets:
Cost of 2 tickets = 2 * 56 = $112
Understanding the constant of proportionality makes it straightforward to determine the cost for any quantity of tickets.
Practical Applications and Further Exploration
The concept of the constant of proportionality is not limited to movie ticket prices. It can be applied in various real-world scenarios, such as:
- Calculating the cost of items in bulk: If you know the price of one item, you can find the cost of multiple items using this constant.
- Converting units: Converting between different units, such as kilometers to miles or Celsius to Fahrenheit, often involves a constant of proportionality.
- Scaling recipes: When increasing or decreasing a recipe, the constant of proportionality helps maintain the correct ratios of ingredients.
- Understanding currency exchange rates: The exchange rate between two currencies is a constant of proportionality that allows you to convert amounts from one currency to another.
To further explore this concept, you can consider more complex problems involving multiple variables or different types of proportionality, such as inverse proportionality. Understanding these concepts will enhance your mathematical skills and your ability to solve real-world problems.
Conclusion
In conclusion, we have successfully calculated the constant of proportionality for the given scenario, where the cost of 7 movie tickets is $392. By dividing the total cost by the number of tickets, we found that the constant of proportionality is $56, which represents the price per ticket. This constant allows us to easily determine the cost for any number of tickets. The concept of proportionality is a fundamental mathematical principle with numerous practical applications, making it an essential skill to understand and apply. Remember, the constant of proportionality is a powerful tool for solving a wide range of problems involving ratios and proportions. By mastering this concept, you can confidently tackle various real-world scenarios and enhance your mathematical proficiency. Practice and application are key to solidifying your understanding, so be sure to explore different problems and contexts where proportionality comes into play. This will not only improve your problem-solving skills but also deepen your appreciation for the practical relevance of mathematics in everyday life. Keep exploring and keep learning!