Cost Of 15 Apples: Direct Proportionality Explained!
Hey guys! Let's dive into a super common math problem that you've probably encountered before: direct proportionality. It sounds fancy, but it's actually quite simple. Today, we're tackling a classic example: the cost of apples. We'll break down the problem step-by-step, making sure you understand not just the answer, but the why behind it. So, grab your thinking caps and let's get started!
Understanding Direct Proportionality
First things first, let's make sure we're all on the same page about direct proportionality. In simple terms, two quantities are directly proportional if they increase or decrease together at a constant rate. Think of it like this: if you buy more of something, you'll pay more. The number of items and the total cost increase together. Conversely, if you buy fewer items, the cost goes down. This consistent relationship is the heart of direct proportionality.
Mathematically, we can express this relationship as y = kx, where:
- y is one quantity (in our case, the total cost)
- x is the other quantity (the number of apples)
- k is the constant of proportionality (the cost per apple)
The constant of proportionality, k, is the magic number that links the two quantities. It tells us how much one quantity changes for every unit change in the other. In our apple example, k represents the price of a single apple. Once we know k, we can easily calculate the cost for any number of apples.
Identifying direct proportionality in word problems is key. Look for phrases like "directly proportional," "varies directly," or situations where an increase in one quantity clearly leads to a proportional increase in the other. Recognizing this relationship allows us to set up the problem correctly and solve it efficiently. Direct proportionality is a fundamental concept in math and science, showing up in various real-world scenarios, from calculating recipe ingredients to understanding the relationship between distance, speed, and time.
Setting Up the Problem: 5 Apples for $10
Okay, let's get back to our apple dilemma. The problem states: "If the cost of 5 apples is $10, what is the cost of 15 apples if the cost varies directly with the number of apples purchased?" The first crucial step is identifying the given information and what we need to find. We know the cost of 5 apples, and we want to determine the cost of 15 apples. The key phrase here is "cost varies directly," which tells us we're dealing with direct proportionality. This is our signal to use the concept we just discussed.
Now, let's define our variables. Let's use:
- 'C' to represent the total cost
- 'A' to represent the number of apples
Based on the information given, we can write our first equation. We know that when A = 5 (5 apples), C = $10. So, we have:
$10 = k * 5
This equation is super important because it allows us to find the constant of proportionality, k. Remember, k is the cost per apple, and once we know this, we can solve for any number of apples. Think of it like finding the fundamental unit price. It’s like knowing the price of one Lego brick so you can calculate the price of a whole set. This foundational step sets the stage for solving the entire problem. Setting up the problem correctly is often half the battle, and with a clear understanding of our variables and the relationship between them, we're well on our way to finding the solution. This meticulous approach ensures accuracy and helps avoid common pitfalls in problem-solving.
Finding the Constant of Proportionality (k)
Alright, we've set up our equation: $10 = k * 5. Now, it's time to find the constant of proportionality, k. Remember, k represents the cost per apple, which is the key to solving our problem. To isolate k, we need to get it by itself on one side of the equation. This involves using a little bit of algebraic manipulation – don't worry, it's easier than it sounds!
Currently, k is being multiplied by 5. To undo this multiplication, we'll perform the inverse operation, which is division. We'll divide both sides of the equation by 5. This is a crucial step because it maintains the balance of the equation. What we do to one side, we must do to the other to keep things equal. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.
So, we have:
$10 / 5 = (k * 5) / 5
On the left side, $10 divided by 5 is simply 2. On the right side, the 5s cancel each other out, leaving us with just k. This is exactly what we wanted!
Therefore, we find that:
k = 2
This means the constant of proportionality, k, is 2. In the context of our problem, this tells us that each apple costs $2. We've now successfully unlocked a critical piece of information. Knowing the value of k allows us to calculate the cost for any number of apples. It's like having the key to a treasure chest – we can now unlock the answer to our original question and any similar questions that might come our way. This step demonstrates the power of algebraic manipulation in solving real-world problems.
Calculating the Cost of 15 Apples
Great job, guys! We've figured out that the cost per apple (k) is $2. Now, we're ready to tackle the main question: What is the cost of 15 apples? We'll use our understanding of direct proportionality and the value of k to find the answer. Remember our direct proportionality equation: C = k * A, where C is the total cost, A is the number of apples, and k is the constant of proportionality.
We know:
- k = $2 (cost per apple)
- A = 15 (number of apples)
We want to find C, the total cost of 15 apples. To do this, we simply substitute the values of k and A into our equation:
C = $2 * 15
Now, it's just a matter of simple multiplication:
C = $30
Therefore, the cost of 15 apples is $30. We've successfully used the concept of direct proportionality and our calculated value of k to solve the problem. This step highlights the practical application of mathematical concepts. By understanding the relationship between quantities and using the right tools (like our equation and the constant of proportionality), we can solve real-world problems efficiently. This clear and straightforward calculation demonstrates the power of a systematic approach to problem-solving, making even seemingly complex problems manageable.
Conclusion: Direct Proportionality in Action
Awesome work, everyone! We've successfully solved the apple problem and, more importantly, gained a deeper understanding of direct proportionality. To recap, we started with the information that 5 apples cost $10 and the knowledge that the cost varies directly with the number of apples. We identified the relationship as direct proportionality, which allowed us to use the equation C = k * A.
We then found the constant of proportionality, k, which represents the cost per apple. By dividing the total cost ($10) by the number of apples (5), we determined that k = $2. This means each apple costs $2.
Finally, we used the value of k and the desired number of apples (15) to calculate the total cost. By substituting the values into our equation, C = $2 * 15, we found that the cost of 15 apples is $30.
This problem beautifully illustrates how direct proportionality works in the real world. We saw how the cost increases proportionally with the number of apples purchased. This concept isn't just limited to apples; it applies to countless situations, from calculating fuel costs based on distance traveled to determining the amount of ingredients needed for a recipe based on the number of servings. Understanding direct proportionality gives you a powerful tool for solving a wide range of problems. So, the next time you encounter a situation where quantities change together proportionally, remember the principles we've discussed, and you'll be well-equipped to find the solution! Keep practicing, and you'll become a direct proportionality pro in no time!
