Solving (3x - 12) Divided By 3/8 Equals 12.5 Divided By (1 - 1/16)
Introduction
In this article, we will delve into the process of solving the equation (3x - 12) : 3/8 = 12.5 : (1 - 1/16). This equation involves fractions, ratios, and algebraic expressions, requiring a step-by-step approach to arrive at the solution for x. Understanding how to solve such equations is crucial in various fields of mathematics and its applications. This guide aims to provide a clear, concise, and comprehensive explanation of each step involved, making it accessible to students and enthusiasts alike. We will break down the equation, simplify it, and use algebraic manipulations to isolate the variable x. By the end of this article, you will have a solid understanding of how to tackle similar problems with confidence.
Understanding the Equation
Before we jump into solving the equation, let's understand its components. The equation (3x - 12) : 3/8 = 12.5 : (1 - 1/16) is a proportion, which means it equates two ratios. On the left side, we have the ratio of the expression (3x - 12) to the fraction 3/8. On the right side, we have the ratio of the decimal 12.5 to the result of the subtraction (1 - 1/16). To solve for x, we need to simplify both sides and then use the properties of proportions to isolate x. Understanding the order of operations (PEMDAS/BODMAS) is crucial here: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. This order will guide us in simplifying the expressions correctly. Additionally, familiarity with fraction manipulation, decimal-to-fraction conversion, and basic algebraic principles is essential. This foundational knowledge will ensure that we can navigate through the equation-solving process smoothly and accurately. Remember, each part of the equation plays a role in finding the final solution, and understanding these roles is the first step towards solving complex mathematical problems.
Step 1: Simplify the Right Side
The first step in solving the equation (3x - 12) : 3/8 = 12.5 : (1 - 1/16) is to simplify the right side. This involves two main tasks: converting the decimal 12.5 into a fraction and simplifying the expression (1 - 1/16). Converting 12.5 to a fraction, we see that 12.5 is the same as 12 1/2, which can be written as an improper fraction: (12 * 2 + 1) / 2 = 25/2. Next, we simplify (1 - 1/16). To subtract fractions, we need a common denominator, which in this case is 16. So, we rewrite 1 as 16/16, making the subtraction 16/16 - 1/16 = 15/16. Now, the right side of the equation becomes 25/2 : 15/16. Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the division as multiplication: (25/2) * (16/15). To simplify this multiplication, we can cancel out common factors. Both 25 and 15 are divisible by 5, and 2 and 16 share a factor of 2. After canceling out the common factors, we get (5/1) * (8/3) = 40/3. So, the simplified right side of the equation is 40/3. This simplification is crucial because it transforms a complex ratio into a manageable fraction, making the subsequent steps in solving the equation more straightforward. By breaking down the right side into its simplest form, we reduce the chances of errors and pave the way for efficiently solving for x.
Step 2: Rewrite the Equation
Having simplified the right side of the equation (3x - 12) : 3/8 = 12.5 : (1 - 1/16), we can now rewrite the equation using our simplified value. We found that the right side simplifies to 40/3. Thus, our equation now looks like this: (3x - 12) : 3/8 = 40/3. This step is important because it brings us closer to isolating x. The original equation, with its decimals and complex fractions, can be intimidating. By simplifying one side, we've made the entire equation more approachable. Now, we need to address the left side. The expression (3x - 12) : 3/8 represents the ratio of (3x - 12) to 3/8. To further simplify, we can rewrite this division as multiplication by the reciprocal. The reciprocal of 3/8 is 8/3. So, the left side becomes (3x - 12) * (8/3). Rewriting the equation with this simplification gives us (3x - 12) * (8/3) = 40/3. This form is much easier to work with. We've eliminated the division and now have a straightforward algebraic equation. This rewriting step is a key technique in solving equations; it allows us to manipulate the equation into a form that is easier to solve. By carefully rewriting the equation, we are setting ourselves up for the next steps in finding the value of x.
Step 3: Simplify the Left Side
Now that we have the equation in the form (3x - 12) * (8/3) = 40/3, the next step is to simplify the left side. This involves distributing the fraction 8/3 across the terms inside the parentheses (3x - 12). When we distribute 8/3 to 3x, we get (8/3) * 3x. The 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with 8x. Next, we distribute 8/3 to -12, resulting in (8/3) * -12. To perform this multiplication, we can think of -12 as -12/1. Multiplying the fractions, we get (8 * -12) / (3 * 1) = -96/3. Simplifying this fraction, we find that -96/3 = -32. So, the simplified left side of the equation is 8x - 32. Our equation now looks like this: 8x - 32 = 40/3. This simplification is a crucial step because it eliminates the parentheses and combines like terms, making the equation easier to solve. By simplifying the left side, we've transformed a more complex expression into a linear equation, which is a standard form that we can easily manipulate to isolate x. The ability to simplify expressions is fundamental in algebra, and this step demonstrates how it can make solving equations more manageable. With the left side simplified, we are now in a good position to isolate x and find its value.
Step 4: Isolate the Term with x
With the equation simplified to 8x - 32 = 40/3, our next goal is to isolate the term containing x, which is 8x. To do this, we need to eliminate the constant term on the left side, which is -32. We can eliminate -32 by adding 32 to both sides of the equation. This maintains the balance of the equation, ensuring that the equality remains true. Adding 32 to the left side, 8x - 32 + 32, simplifies to 8x. On the right side, we have 40/3 + 32. To add these terms, we need a common denominator. We can rewrite 32 as a fraction with a denominator of 3: 32 = 32/1 = (32 * 3) / 3 = 96/3. Now we can add the fractions: 40/3 + 96/3 = (40 + 96) / 3 = 136/3. So, our equation now becomes 8x = 136/3. This step is crucial because it brings us closer to isolating x. By adding 32 to both sides, we've successfully moved the constant term to the right side, leaving only the term with x on the left. This is a fundamental technique in solving algebraic equations. Isolating the term with the variable is a key step towards finding the value of the variable. With 8x isolated, we are now just one step away from solving for x itself.
Step 5: Solve for x
Having isolated the term with x, our equation is now 8x = 136/3. To solve for x, we need to get x by itself. Since x is being multiplied by 8, we can isolate x by dividing both sides of the equation by 8. This maintains the balance of the equation and allows us to find the value of x. Dividing the left side, 8x / 8, simplifies to x. On the right side, we have (136/3) / 8. Dividing by 8 is the same as multiplying by its reciprocal, which is 1/8. So, we can rewrite the division as multiplication: (136/3) * (1/8). Now, we multiply the fractions: (136 * 1) / (3 * 8) = 136 / 24. This fraction can be simplified. Both 136 and 24 are divisible by 8. Dividing both the numerator and the denominator by 8, we get (136 / 8) / (24 / 8) = 17 / 3. Therefore, the solution for x is x = 17/3. This final step is the culmination of all the previous steps. By dividing both sides of the equation by 8, we successfully isolated x and found its value. The solution x = 17/3 is the answer to the original equation. It represents the value of x that makes the equation true. This process demonstrates the power of algebraic manipulation in solving equations. By carefully following each step, we were able to break down a complex equation and find the solution for x. The result x = 17/3 can also be expressed as a mixed number, which is 5 2/3, or as a decimal approximation, which is approximately 5.67. Depending on the context, one form may be more suitable than another.
Conclusion
In conclusion, solving the equation (3x - 12) : 3/8 = 12.5 : (1 - 1/16) involves a series of steps that require careful attention to detail and a solid understanding of algebraic principles. We began by simplifying the right side of the equation, converting the decimal to a fraction and performing the subtraction within the parentheses. This gave us a simplified ratio that was easier to work with. Next, we rewrote the entire equation using this simplified value, which set the stage for further manipulation. We then simplified the left side of the equation by distributing and combining like terms, transforming the equation into a more manageable form. After simplifying both sides, we isolated the term containing x by adding the appropriate constant to both sides of the equation. This brought us one step closer to finding the value of x. Finally, we solved for x by dividing both sides of the equation by the coefficient of x. This gave us the solution x = 17/3, which can also be expressed as the mixed number 5 2/3 or the decimal approximation 5.67. The process of solving this equation highlights the importance of breaking down complex problems into smaller, more manageable steps. By following a systematic approach and applying the rules of algebra, we were able to find the solution. This skill is valuable not only in mathematics but also in many other areas of life where problem-solving is essential. The ability to manipulate equations and solve for unknowns is a fundamental tool in various fields, including science, engineering, and economics. The solution x = 17/3 is the value that satisfies the original equation, and the steps we took to find it provide a clear illustration of the equation-solving process. Understanding these steps will help you tackle similar problems with confidence and accuracy.
