Find X Third Proportional Explained With Example

Hey there, math enthusiasts! Let's dive into a fascinating problem involving proportions and natural numbers. This is a common type of question you might encounter in algebra or number theory, and it's a great way to sharpen your problem-solving skills. We are going to solve the problem, If $x$ is a natural number such that 18 is the third proportional to $(x+2)$ and $(x+6)$, what is the value of $x$? Before we get to the solution, let's make sure we understand all the key concepts involved. So, grab your thinking caps, and let's get started!

Understanding Proportions

Proportions are a fundamental concept in mathematics that describe the relationship between two ratios. Specifically, a proportion states that two ratios are equal. A ratio, in its simplest form, is a comparison of two quantities, often expressed as a fraction. When we say that two ratios are proportional, we mean that they represent the same relationship. Think of it like this: if you double one quantity in a ratio, you must also double the other quantity to maintain the same proportion.

For example, the ratio 1:2 is proportional to the ratio 2:4 because both represent the same relationship – one quantity is half the other. This concept is incredibly versatile and appears in various mathematical and real-world scenarios, from scaling recipes to calculating distances on maps. Now, let’s dig a little deeper into the concept of a third proportional, as it's the key to cracking our initial problem. The third proportional comes into play when we're dealing with continued proportions. A continued proportion is a sequence of three or more quantities where the ratio between any two consecutive quantities is the same. For instance, if we have three numbers, a, b, and c, they are in continued proportion if a:b = b:c. In this case, 'c' is called the third proportional to 'a' and 'b'. This means that 'c' maintains the proportional relationship established by 'a' and 'b'. Understanding this relationship is crucial for solving problems where we need to find an unknown third proportional. It's like finding the missing piece of a puzzle that fits perfectly within the existing proportional structure. The beauty of proportions lies in their ability to help us make predictions and solve problems where direct measurement or calculation might be difficult. They provide a framework for understanding how quantities relate to each other, allowing us to extrapolate and infer information based on known relationships. This makes proportions an indispensable tool in fields ranging from mathematics and physics to economics and everyday life.

Breaking Down the Problem Statement

The problem statement provides us with the core information we need to solve for the unknown. It tells us that $x$ is a natural number, which is a crucial piece of information because it restricts the possible values of $x$ to positive integers (1, 2, 3, and so on). This eliminates the possibility of $x$ being a fraction, decimal, or negative number. Then comes the key part: 18 is the third proportional to $(x+2)$ and $(x+6)$. This is where our understanding of proportions comes into play. It means that the ratio between $(x+2)$ and $(x+6)$ is the same as the ratio between $(x+6)$ and 18. Mathematically, we can express this relationship as:

$$\frac{x+2}{x+6} = \frac{x+6}{18}$$

This equation is the heart of the problem. It translates the verbal statement about proportions into a concrete mathematical expression that we can manipulate and solve. The equation encapsulates the proportional relationship described in the problem, and our goal is to find the value of $x$ that makes this equation true. To solve for $x$, we'll need to use algebraic techniques to isolate $x$ on one side of the equation. This will involve cross-multiplication and simplification. Before we jump into the algebraic manipulation, let's take a moment to appreciate how elegantly this problem combines the abstract concept of proportions with the concrete reality of numbers. It's a perfect example of how mathematics can be used to model and solve real-world problems. By carefully dissecting the problem statement and identifying the key relationships, we've set ourselves up for a successful solution.

Solving for x

Now, let's solve the equation we derived from the problem statement: $\frac{x+2}{x+6} = \frac{x+6}{18}$. The first step is to cross-multiply, which means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives us: $(x+2) \cdot 18 = (x+6) \cdot (x+6)$. Expanding both sides, we get: $18x + 36 = x^2 + 12x + 36$. Next, we want to rearrange the equation into a standard quadratic form, which is $ax^2 + bx + c = 0$. To do this, we subtract $18x$ and 36 from both sides of the equation: $0 = x^2 + 12x + 36 - 18x - 36$. Simplifying, we get: $0 = x^2 - 6x$. Now, we can factor out an $x$ from the right side of the equation: $0 = x(x - 6)$. This equation is satisfied if either $x = 0$ or $x - 6 = 0$. So, we have two possible solutions: $x = 0$ or $x = 6$. However, the problem states that $x$ is a natural number. Natural numbers are positive integers (1, 2, 3, ...), so $x = 0$ is not a valid solution in this context. Therefore, the only valid solution is $x = 6$. It's crucial to remember the constraints given in the problem statement, such as $x$ being a natural number, because they can significantly narrow down the possible solutions. In this case, the natural number constraint eliminated one of our solutions, leaving us with the correct answer. Solving for $x$ involved a series of algebraic manipulations, including cross-multiplication, expansion, rearrangement into quadratic form, and factoring. Each step was crucial in isolating $x$ and finding its value. This process demonstrates the power of algebra in solving problems involving proportions and other mathematical relationships.

Verifying the Solution

To ensure that our solution is correct, it's always a good practice to verify it. We found that $x = 6$, so let's substitute this value back into the original proportion: $\fracx+2}{x+6} = \frac{x+6}{18}$. Plugging in $x = 6$, we get $\frac{6+26+6} = \frac{6+6}{18}$. Simplifying, we have $\frac{812} = \frac{12}{18}$. Now, let's reduce both fractions to their simplest forms. Dividing both the numerator and denominator of the first fraction by 4, we get $\frac{23}$. Dividing both the numerator and denominator of the second fraction by 6, we also get $\frac{2{3}$. Since both fractions simplify to $\frac{2}{3}$, the proportion holds true. This confirms that $x = 6$ is indeed the correct solution. Verifying the solution is a critical step in problem-solving because it helps us catch any errors we might have made along the way. It's like a final check to make sure everything adds up correctly. By substituting our solution back into the original equation or proportion, we can confirm that it satisfies the given conditions. This not only gives us confidence in our answer but also reinforces our understanding of the problem and the solution process. In this case, verification involved substituting $x = 6$ into the original proportion, simplifying both sides, and confirming that they were equal. This simple check provided us with the assurance that we had solved the problem correctly.

Final Answer

So, the value of $x$ that satisfies the given conditions is 6. This corresponds to option (a) in the provided choices. We arrived at this answer by carefully dissecting the problem statement, translating it into a mathematical equation, solving the equation using algebraic techniques, and verifying our solution. This problem highlights the importance of understanding fundamental mathematical concepts like proportions and natural numbers, as well as the ability to apply algebraic skills to solve equations. The process of solving this problem also underscores the value of problem-solving strategies such as verifying solutions to ensure accuracy. Mathematics is not just about finding the right answer; it's about understanding the process and developing the skills to tackle a variety of problems. This particular problem, while seemingly straightforward, involves a combination of concepts that are essential for success in more advanced mathematical studies. By mastering these concepts and practicing problem-solving techniques, you can build a solid foundation for future mathematical endeavors. So, keep practicing, keep exploring, and keep challenging yourself with new mathematical problems!

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If $x$ is a natural number and 18 is the third proportional to $(x+2)$ and $(x+6)$, then what is the value of $x$?

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Find x Third Proportional Explained with Example