Adding To Fractions: Find The Number To Add To 15/135 To Get 2/7

Hey guys! Ever found yourself scratching your head over a fraction problem that just seems impossible? Don't worry, we've all been there. Today, we're going to break down a tricky one: finding the number you need to add to 15/135 to get 2/7. It might sound intimidating, but trust me, with a step-by-step approach, you'll be solving these problems like a pro in no time. So, let’s dive into this math adventure together!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. We have a fraction, 15/135, and we want to add another number (let's call it 'x') to it. The goal is to end up with a new fraction that equals 2/7. Think of it like this: you've got a slice of pizza that's 15/135 of the whole pie, and you want to add another slice (our 'x') so you have 2/7 of the whole pizza. The core concept we're dealing with here is fraction addition, and to effectively tackle it, we'll need to be comfortable with finding common denominators and simplifying fractions. This problem isn’t just about finding a single answer; it’s about understanding the relationship between fractions and how they interact with each other. Imagine you are baking a cake, and you need to adjust the ingredient quantities. This problem is conceptually similar – you're adjusting a fraction to reach a desired proportion. Remember, math isn't just about numbers; it's about problem-solving and critical thinking. Each step we take here builds a foundation for more complex mathematical challenges in the future. So, let’s put on our math hats and get started! We'll break down each step, making sure it’s crystal clear and easy to follow. Remember, the key is to take it one step at a time, and soon, you’ll be a fraction-solving whiz.

Step 1: Simplify the Fraction 15/135

Our first step is to simplify the fraction 15/135. Simplifying fractions makes them easier to work with, kind of like decluttering your workspace before starting a project. You want everything neat and organized, right? Same goes for fractions! To simplify, we need to find the greatest common divisor (GCD) of both the numerator (15) and the denominator (135). The GCD is the largest number that divides evenly into both numbers. Think of it as finding the biggest piece you can cut both numbers into without any leftovers. In this case, the GCD of 15 and 135 is 15. We can find this by listing the factors of each number: Factors of 15: 1, 3, 5, 15; Factors of 135: 1, 3, 5, 9, 15, 27, 45, 135. The largest number they have in common is 15. Now, we divide both the numerator and the denominator by 15: (15 ÷ 15) / (135 ÷ 15) = 1/9. So, 15/135 simplifies to 1/9. This simplified fraction is much easier to handle in our equation. Simplifying fractions isn’t just a mathematical trick; it's a practical skill. Imagine you're sharing a pizza with friends. Saying you're going to eat 15/135 of the pizza sounds way more confusing than saying you'll eat 1/9, even though they represent the same amount. Simplifying makes numbers more digestible, both literally and figuratively! This step is crucial because it sets us up for smoother calculations in the following steps. By working with smaller numbers, we reduce the chances of making errors and make the overall process less cumbersome. So, give yourself a pat on the back for simplifying – you've just made the rest of the problem a whole lot easier.

Step 2: Set Up the Equation

Now that we've simplified 15/135 to 1/9, we can set up our equation. Remember, we want to find a number, which we'll call 'x', that we can add to 1/9 to get 2/7. We can write this as an equation: 1/9 + x = 2/7. This equation is the heart of our problem. It translates the word problem into a mathematical statement that we can solve. Think of it as creating a roadmap for our journey to the solution. Each part of the equation represents a piece of the puzzle: 1/9 is our starting point, 'x' is the unknown number we're trying to find, and 2/7 is our destination. Setting up the equation correctly is super important. If we mess up the equation, we'll end up solving the wrong problem! It’s like entering the wrong address into your GPS – you might end up somewhere completely different. So, double-check that your equation accurately reflects the original problem. In this case, we're saying that adding 'x' to 1/9 will give us 2/7, which is exactly what the problem asks. Equations are the language of mathematics. They allow us to express relationships between numbers and variables in a clear and concise way. Mastering the art of setting up equations is a fundamental skill in algebra and beyond. It’s like learning the grammar of math – it enables you to communicate mathematical ideas effectively. Now that we have our equation, we're ready to roll up our sleeves and solve for 'x'. The next step involves a bit more fraction magic, but don't worry, we'll tackle it together.

Step 3: Isolate 'x'

Our next task is to isolate 'x' on one side of the equation. This means we want to get 'x' all by itself, so we can see what it equals. To do this, we need to get rid of the 1/9 that's being added to 'x'. The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other. It's like a balancing scale – you need to keep both sides equal. So, to get rid of the 1/9, we subtract 1/9 from both sides of the equation: 1/9 + x - 1/9 = 2/7 - 1/9. On the left side, 1/9 and -1/9 cancel each other out, leaving us with just 'x'. That's exactly what we wanted! Now our equation looks like this: x = 2/7 - 1/9. Isolating the variable is a fundamental technique in algebra. It allows us to solve for unknowns in a wide variety of problems. Think of it like detective work – you're trying to uncover a secret, and isolating the variable is like cornering your suspect! This step might seem simple, but it's incredibly powerful. It transforms our equation from one where 'x' is mixed up with other terms to one where 'x' is all alone, ready to be revealed. By subtracting 1/9 from both sides, we've maintained the balance of the equation while moving closer to our solution. Now, we have a new problem: subtracting two fractions. But don't worry, we've got the skills to handle it! The next step will involve finding a common denominator, and once we do that, the subtraction will be a breeze. So, let's keep up the momentum and move on to the next step.

Step 4: Find a Common Denominator

Now we need to subtract 1/9 from 2/7. But, we can't directly subtract fractions unless they have the same denominator. It's like trying to add apples and oranges – they're different units! We need to find a common denominator, which is a number that both 7 and 9 divide into evenly. The easiest way to find a common denominator is to find the least common multiple (LCM) of the two denominators. The LCM is the smallest number that is a multiple of both numbers. In this case, the LCM of 7 and 9 is 63 (since 7 x 9 = 63 and neither number shares any factors). Think of the common denominator as a common language for our fractions. It allows us to express them in the same units, making it possible to add or subtract them. Without a common denominator, we're stuck with fractions that are speaking different languages! Finding a common denominator is a crucial step in fraction arithmetic. It's a skill that you'll use again and again in math, so it's worth mastering. It’s like learning the rules of grammar in writing – it allows you to express your ideas clearly and effectively. Once we have our common denominator, we need to convert both fractions to have this denominator. This involves multiplying both the numerator and the denominator of each fraction by a suitable number. In the next step, we'll see exactly how to do this conversion. So, we've identified our common denominator as 63. Now, let's get ready to transform our fractions so they both have this denominator. We're on the home stretch now!

Step 5: Convert the Fractions

Okay, we've found our common denominator: 63. Now we need to convert both 2/7 and 1/9 into equivalent fractions with a denominator of 63. Let's start with 2/7. To get the denominator from 7 to 63, we need to multiply by 9 (since 7 x 9 = 63). Remember, whatever we do to the denominator, we must also do to the numerator to keep the fraction equivalent. So, we multiply both the numerator and denominator of 2/7 by 9: (2 x 9) / (7 x 9) = 18/63. Now let's convert 1/9. To get the denominator from 9 to 63, we need to multiply by 7 (since 9 x 7 = 63). Again, we multiply both the numerator and the denominator by 7: (1 x 7) / (9 x 7) = 7/63. So, we've successfully converted 2/7 to 18/63 and 1/9 to 7/63. These new fractions are equivalent to the original ones, but they have a common denominator, which means we can finally subtract them! Think of converting fractions as changing currencies. You might have the same amount of money, but it looks different depending on whether it's in dollars or euros. Similarly, 2/7 and 18/63 represent the same quantity, just expressed with different denominators. This step is like preparing our ingredients before we start cooking. We need to have everything in the right form before we can combine them. Converting fractions to a common denominator is a fundamental skill in fraction arithmetic, and it’s essential for solving problems like this one. Now that we have our fractions in a common language, we're ready to perform the subtraction. The finish line is in sight!

Step 6: Subtract the Fractions

Now that we have our fractions with a common denominator, we can finally subtract! We have x = 18/63 - 7/63. To subtract fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. It's like subtracting slices from the same pizza – you're only concerned with how many slices are being taken away. So, 18/63 - 7/63 = (18 - 7) / 63 = 11/63. Therefore, x = 11/63. This means that the number we need to add to 1/9 (which is the simplified form of 15/135) to get 2/7 is 11/63. We've done it! We've successfully solved for 'x' and found the answer to our problem. Subtracting fractions with a common denominator is a straightforward process, but it's built upon all the previous steps we've taken. We had to simplify the original fraction, set up the equation, isolate the variable, find a common denominator, and convert the fractions. Each step was crucial in getting us to this final answer. Think of this subtraction as the final piece of the puzzle clicking into place. We've gathered all the information we need, and now we're putting it together to reveal the solution. It’s like the climax of a good story – everything has been building up to this moment! Now that we've found our answer, it's always a good idea to check it. In the next step, we'll do just that to make sure our solution is correct.

Step 7: Check Your Answer

It's always a good idea to check your answer to make sure it's correct. Think of it as proofreading your work before you submit it. We want to be confident that our solution is accurate. To check our answer, we'll substitute x = 11/63 back into our original equation: 1/9 + x = 2/7. Substituting x = 11/63, we get: 1/9 + 11/63 = 2/7. Now we need to see if this equation holds true. To add 1/9 and 11/63, we need a common denominator, which we already know is 63. We converted 1/9 to 7/63 in a previous step, so we have: 7/63 + 11/63 = 2/7. Adding the numerators, we get: (7 + 11) / 63 = 18/63. So, our equation now looks like this: 18/63 = 2/7. To see if these fractions are equal, we can simplify 18/63. Both 18 and 63 are divisible by 9: (18 ÷ 9) / (63 ÷ 9) = 2/7. So, 18/63 simplifies to 2/7. This means that our equation 2/7 = 2/7 is true! Our answer checks out. We've successfully verified that adding 11/63 to 1/9 gives us 2/7. Checking your answer is a crucial step in problem-solving. It gives you peace of mind and helps you catch any mistakes you might have made along the way. Think of it as the final inspection before you launch a rocket – you want to make sure everything is working perfectly! This step reinforces the importance of accuracy in math. It’s not just about getting an answer; it’s about getting the right answer. By checking our work, we’re demonstrating a commitment to precision and thoroughness. So, give yourself a big pat on the back – you've not only solved the problem, but you've also verified your solution. That's what we call a job well done!

Conclusion

Guys, we did it! We successfully found the number to add to 15/135 to get 2/7. We took a potentially daunting problem and broke it down into manageable steps. We simplified fractions, set up an equation, isolated the variable, found a common denominator, converted fractions, subtracted fractions, and checked our answer. Each step was a building block in our journey to the solution. This problem wasn't just about numbers; it was about problem-solving, critical thinking, and perseverance. We learned valuable skills that we can apply to other math problems and even to real-life situations. Think of this journey as climbing a mountain. The summit might seem far away at first, but with each step, you get closer and closer. And when you finally reach the top, the view is amazing! Learning math can be challenging, but it's also incredibly rewarding. It's like learning a new language – it opens up a whole new world of understanding and possibilities. So, keep practicing, keep asking questions, and keep exploring the fascinating world of math. You've got this! And remember, every problem you solve makes you a stronger and more confident mathematician. Now go out there and conquer those math challenges!