Solving Equations: Step-by-Step Guide With Fractions

Hey guys! Today, we're diving into the world of equations, specifically tackling one that involves both decimal and mixed fractions. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you can confidently solve similar problems in the future. Our mission is to solve the equation: (𝑥 − 5.2) + 3 3/5 = 7 1/20. The key here is to convert everything into a common format, making the equation easier to handle. We’ll start by understanding the basics and then move to more complex steps. Remember, math is like building blocks – each step relies on the previous one. This means a strong foundation is crucial for success. So, let's roll up our sleeves and get started!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what this equation is all about. We have a variable, x, which represents the unknown value we're trying to find. The equation states that if we subtract 5.2 from x, then add 3 3/5, the result should be 7 1/20. To solve for x, we need to isolate it on one side of the equation. This involves performing operations on both sides to maintain the balance. The equation includes a decimal fraction (5.2) and mixed fractions (3 3/5 and 7 1/20). To make our lives easier, we'll convert the decimal fraction into a common fraction and the mixed fractions into improper fractions. This will give us a uniform playing field, allowing us to perform operations more smoothly. Think of it like this: imagine trying to add apples and oranges – it’s much simpler if you convert them both into pieces of fruit. Similarly, converting all fractions into a common format simplifies the equation. So, let’s break down the components and prepare for the conversion process. Understanding the structure of the equation is the first step towards conquering it. Now, let’s see how we can transform these numbers into more manageable forms.

Converting Decimal to Fraction

First things first, let's convert the decimal fraction 5.2 into a common fraction. Remember, decimals are just another way of representing fractions with denominators that are powers of 10. In this case, 5.2 can be read as “five and two-tenths.” To convert it to a fraction, we write the decimal part (0.2) as a fraction: 2/10. So, 5.2 becomes 5 2/10. But we're not done yet! We can simplify 2/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 1/5. Therefore, 5.2 is equivalent to the mixed fraction 5 1/5. Now, to make it an improper fraction (where the numerator is greater than the denominator), we multiply the whole number (5) by the denominator (5) and add the numerator (1). This gives us (5 * 5) + 1 = 26. So, 5 1/5 becomes 26/5. Voila! We've successfully converted 5.2 into the improper fraction 26/5. This conversion is crucial because it allows us to work with fractions consistently throughout the equation. Now that we’ve handled the decimal, let’s move on to the next step: converting the mixed fractions.

Converting Mixed Fractions to Improper Fractions

Next up, we need to convert the mixed fractions 3 3/5 and 7 1/20 into improper fractions. This process is similar to what we did with 5.2. For 3 3/5, we multiply the whole number (3) by the denominator (5) and add the numerator (3). So, (3 * 5) + 3 = 18. This gives us the improper fraction 18/5. Easy peasy! Now let’s tackle 7 1/20. We multiply the whole number (7) by the denominator (20) and add the numerator (1). Thus, (7 * 20) + 1 = 141. This gives us the improper fraction 141/20. Converting mixed fractions to improper fractions makes it easier to perform addition and subtraction because we're dealing with consistent fractional forms. Now that we've converted both the decimal and mixed fractions, our equation is starting to look a lot more manageable. We’re one step closer to solving for x. With these conversions complete, we can rewrite the original equation using improper fractions, setting the stage for the next phase of our solving adventure.

Rewriting the Equation

Now that we've converted 5.2 to 26/5, 3 3/5 to 18/5, and 7 1/20 to 141/20, we can rewrite the original equation using these improper fractions. Our equation now looks like this: (𝑥 − 26/5) + 18/5 = 141/20. See? Much cleaner and easier to work with! Rewriting the equation in this way allows us to perform operations on fractions directly, without the added complexity of mixed numbers or decimals. This is a crucial step in solving the equation efficiently. By having all terms in fractional form, we can easily find a common denominator and combine like terms. It’s like having all the ingredients for a recipe properly measured and prepped – the cooking process becomes much smoother. This new form of the equation sets the stage for the next step: isolating x by performing algebraic manipulations. With the equation in this format, we can focus on the operations needed to get x by itself and find its value. So, let's move forward and see how we can isolate x.

Isolating the Variable

Our next goal is to isolate x on one side of the equation. To do this, we need to get rid of the other terms that are cluttering the left side. We start by focusing on the (𝑥 − 26/5) + 18/5 = 141/20 equation. Notice that we have +18/5 on the left side. To eliminate this, we need to subtract 18/5 from both sides of the equation. This keeps the equation balanced, which is a golden rule in algebra. So, let's subtract 18/5 from both sides: (𝑥 − 26/5) + 18/5 − 18/5 = 141/20 − 18/5. This simplifies to 𝑥 − 26/5 = 141/20 − 18/5. Now we have x almost isolated, but there's still that pesky -26/5 term. To get rid of it, we'll add 26/5 to both sides of the equation: 𝑥 − 26/5 + 26/5 = 141/20 − 18/5 + 26/5. This simplifies to 𝑥 = 141/20 − 18/5 + 26/5. Now, x is finally isolated! But we're not quite done yet. We still need to simplify the right side of the equation to find the value of x. The next step involves combining the fractions on the right side, which requires us to find a common denominator. So, let's move on to that.

Combining Fractions

Now that we have x isolated, our equation looks like this: 𝑥 = 141/20 − 18/5 + 26/5. To simplify the right side, we need to combine the fractions. But before we can add or subtract fractions, they need to have a common denominator. In this case, the denominators are 20 and 5. The least common multiple (LCM) of 20 and 5 is 20. So, we'll use 20 as our common denominator. The fraction 141/20 already has the denominator 20, so we don't need to change it. But we need to convert -18/5 and 26/5 to equivalent fractions with a denominator of 20. To do this, we multiply both the numerator and denominator of -18/5 by 4: (-18 * 4) / (5 * 4) = -72/20. Similarly, we multiply both the numerator and denominator of 26/5 by 4: (26 * 4) / (5 * 4) = 104/20. Now we can rewrite the equation as: 𝑥 = 141/20 − 72/20 + 104/20. With all the fractions having the same denominator, we can now combine them by adding and subtracting the numerators. So, let's do that in the next step!

Adding and Subtracting Fractions

With a common denominator of 20, our equation is now: 𝑥 = 141/20 − 72/20 + 104/20. To find the value of x, we need to add and subtract the numerators: 141 − 72 + 104. Let's break it down: 141 − 72 = 69, and then 69 + 104 = 173. So, the numerator of our result is 173. The denominator remains 20. Therefore, 𝑥 = 173/20. We've found the value of x as an improper fraction! But it's often helpful to convert the improper fraction back to a mixed fraction to get a better sense of its magnitude. So, let's do that now. To convert 173/20 to a mixed fraction, we divide 173 by 20. The quotient is the whole number part, and the remainder is the numerator of the fractional part. When we divide 173 by 20, we get a quotient of 8 and a remainder of 13. So, 173/20 is equivalent to the mixed fraction 8 13/20. We've successfully found the value of x and expressed it in both improper and mixed fraction forms. Great job! But before we celebrate, let's take one more step to ensure our answer is correct.

Checking the Solution

Alright, we've found that 𝑥 = 8 13/20. But before we declare victory, it's crucial to check our solution. This ensures that our answer is correct and that we haven't made any mistakes along the way. To check, we'll substitute 8 13/20 back into the original equation: (𝑥 − 5.2) + 3 3/5 = 7 1/20. Remember, we converted 5.2 to 26/5, 3 3/5 to 18/5, and 7 1/20 to 141/20. So, we'll use these improper fractions in our check. Our equation with the substitution looks like this: (173/20 − 26/5) + 18/5 = 141/20. First, let's simplify the expression inside the parentheses. We need a common denominator to subtract 26/5 from 173/20. As we found earlier, the common denominator is 20. So, we convert 26/5 to an equivalent fraction with a denominator of 20: (26 * 4) / (5 * 4) = 104/20. Now we can subtract: 173/20 − 104/20 = 69/20. Next, we add 18/5 to 69/20. Again, we need a common denominator, which is 20. We convert 18/5 to an equivalent fraction with a denominator of 20: (18 * 4) / (5 * 4) = 72/20. Now we can add: 69/20 + 72/20 = 141/20. So, the left side of the equation simplifies to 141/20, which is exactly the same as the right side of the equation. This confirms that our solution, 𝑥 = 8 13/20, is indeed correct! Checking our work is a vital step in problem-solving, and it gives us confidence in our answer.

Conclusion

Phew! We made it! We successfully solved the equation (𝑥 − 5.2) + 3 3/5 = 7 1/20. We started by converting the decimal and mixed fractions to improper fractions. Then, we rewrote the equation, isolated the variable x, combined the fractions, and found the value of x. Finally, we checked our solution to make sure it was correct. Remember, the key to solving equations with fractions is to convert everything to a common format, find a common denominator, and perform operations step-by-step. With practice, you'll become a fraction-solving pro! I hope this step-by-step guide has been helpful. Keep practicing, and you'll conquer any equation that comes your way. You've got this! This journey through the equation underscores the importance of methodical problem-solving. Each step, from converting decimals to checking the solution, plays a crucial role in achieving accuracy. By breaking down complex problems into manageable steps, we make them less intimidating and more approachable. And remember, math isn't just about finding the right answer; it's about the process of critical thinking and problem-solving that you develop along the way. So, embrace the challenge, enjoy the process, and keep honing your skills. You're well on your way to mastering equations and so much more!