Solve Fraction Problem: Numerator + Denominator = 18
Hey guys! Let's dive into a classic math problem involving fractions. It might seem tricky at first, but we'll break it down step by step so it's super easy to understand. Our main goal here is to figure out how to find an original fraction when we know some key things about its numerator and denominator. So, stick with me, and let's get started!
Understanding the Problem: The Sum of Numerator and Denominator
Okay, let’s kick things off by really understanding the core problem. In this type of question, you're often given a relationship between the numerator (the top number) and the denominator (the bottom number) of a fraction. For instance, in our case, we know the sum of the numerator and the denominator is 18. This is crucial information because it's the first piece of the puzzle we need to solve. To handle these kinds of problems effectively, it’s super important to translate these word problems into mathematical expressions. This is where algebra becomes our best friend. We can represent the numerator with one variable (let's say 'x') and the denominator with another (maybe 'y').
Now, how does this translate mathematically? Well, if the sum of the numerator and the denominator is 18, we can write this down as a simple equation: x + y = 18. See? We’ve turned a sentence into a neat little equation. This equation now gives us a solid foundation to work from. When you encounter problems like these, always look for these initial relationships – they're your starting points. Next up, we're told something else happens to the fraction: we add 10 to both the numerator and the denominator. This is another key piece of information we need to factor in. So, in the following sections, we will see how to handle these changes and what they tell us about the original fraction. Remember, math problems are like puzzles; each clue gets us closer to the solution. So, let's keep piecing it together!
Modified Fractions: Adding to Numerator and Denominator
Now, let's tackle the next part of the problem: what happens when we modify the fraction? In our specific case, we're adding 10 to both the numerator and the denominator. This is a pretty common twist in these kinds of fraction problems, so it's good to know how to handle it. When we add the same number to both the top and bottom of a fraction, we're essentially creating a new fraction. This new fraction might look different, but it's still related to the original one. This is where things get interesting, because this change gives us more information to work with.
So, let's break it down. If our original fraction has a numerator of 'x' and a denominator of 'y', then after adding 10 to both, our new numerator becomes x + 10, and our new denominator is y + 10. This means our modified fraction looks like (x + 10) / (y + 10). Now, here’s the crucial part: the problem tells us that this new fraction has a special property – it can be reduced, or simplified. Specifically, the modified fraction is said to reduce to 2/3. This is a super important piece of information because it gives us another equation to work with. We can now say that (x + 10) / (y + 10) = 2/3. This is our second equation, and it’s just as important as the first one (x + y = 18). With these two equations, we're setting the stage for solving for our unknowns, 'x' and 'y'. Next, we'll look at how to use these equations together to actually find the values of the numerator and the denominator. So, stick around – we’re getting closer to cracking this problem!
Setting Up the Equations: A Mathematical Representation
Alright, guys, let's formalize our understanding by writing down the equations we've established so far. This is a critical step in solving any word problem, especially when we're dealing with fractions and unknowns. Remember, we've identified two key pieces of information that can be turned into mathematical equations. The first one comes from the fact that the sum of the numerator and the denominator of the original fraction is 18. We represented the numerator as 'x' and the denominator as 'y', so this translates directly into our first equation: x + y = 18. This equation is like a map giving us the relationship between the top and bottom numbers of our fraction. It tells us that if we know one, we can figure out the other, at least to some extent.
The second equation comes from what happens when we modify the fraction. We added 10 to both the numerator and the denominator, resulting in a new fraction (x + 10) / (y + 10). The problem tells us that this new fraction simplifies to 2/3. This gives us our second equation: (x + 10) / (y + 10) = 2/3. This equation is super valuable because it introduces a new constraint on the relationship between 'x' and 'y'. It tells us not only their sum, but also how they relate to each other after a certain operation. Now, with these two equations in hand, we have what's called a system of equations. Solving a system of equations means finding the values of the variables (in this case, 'x' and 'y') that make both equations true at the same time. This is a fundamental concept in algebra, and it's the key to unlocking our fraction problem. In the next section, we’re going to explore how to solve this system of equations and find the actual values of 'x' and 'y'. Let's get to it!
Solving the Equations: Finding Numerator and Denominator
Okay, the real fun begins now: solving the system of equations! This is where we put our algebra skills to the test to actually find the values of our numerator ('x') and denominator ('y'). Remember, we've got two equations: x + y = 18 and (x + 10) / (y + 10) = 2/3. There are a couple of different ways we can approach solving this system, but a common and effective method is called substitution. The idea behind substitution is to solve one equation for one variable, and then substitute that expression into the other equation. This way, we turn our two-variable problem into a one-variable problem, which is much easier to handle.
Let's start with our simpler equation: x + y = 18. We can easily solve this for one variable in terms of the other. For example, let's solve for 'x'. Subtracting 'y' from both sides gives us x = 18 - y. Great! Now we have an expression for 'x' in terms of 'y'. Next, we'll take this expression and substitute it into our second equation. Replacing 'x' with '(18 - y)' in the equation (x + 10) / (y + 10) = 2/3 gives us ((18 - y) + 10) / (y + 10) = 2/3. See what we did there? We've eliminated 'x' and now we have an equation with just 'y'. This equation might look a bit messy, but don't worry, we can simplify it. The next step is to solve this equation for 'y'. This will involve a bit of algebraic manipulation, but we'll take it step by step. Once we find 'y', we can easily plug it back into our equation x = 18 - y to find 'x'. So, let's roll up our sleeves and get into the nitty-gritty of solving for 'y'.
Calculation Steps: Finding the Solution
Alright, let's crunch some numbers and get to the heart of the calculation! We left off with the equation ((18 - y) + 10) / (y + 10) = 2/3. The first thing we want to do is simplify the numerator on the left side. So, (18 - y) + 10 becomes 28 - y. Our equation now looks like (28 - y) / (y + 10) = 2/3. To get rid of the fractions, we can cross-multiply. This means multiplying the numerator of the left side by the denominator of the right side, and vice versa. So, we multiply (28 - y) by 3 and (y + 10) by 2. This gives us 3 * (28 - y) = 2 * (y + 10). Now, we need to distribute the multiplication on both sides. Multiplying out, we get 84 - 3y = 2y + 20.
We're getting closer! Now, let's gather the 'y' terms on one side and the constants on the other. Add 3y to both sides, and we get 84 = 5y + 20. Next, subtract 20 from both sides, and we have 64 = 5y. Finally, to solve for 'y', divide both sides by 5: y = 64 / 5. Oops! It seems we've made a mistake somewhere because 'y' should be an integer (since it's a denominator). Let's backtrack and check our steps. (After reviewing the steps, the mistake was not made in calculation but in the setup. It will be corrected in the next step.) Okay, after a careful review, it seems we’ve made a classic mistake – a small error in copying the equation or a sign! These things happen, guys, and it’s why checking your work is super important. Let's go back and correct it. Instead of dwelling on the error, we’ll use it as a learning opportunity. Math is all about precision, and even small mistakes can lead to big differences in the final answer. So, let’s dust ourselves off, retrace our steps, and nail the correct solution. In the next section, we'll revisit our calculations with a fresh perspective, making sure we don't miss anything this time. Let’s get it right!
Finding the Original Fraction: The Final Answer
Let's correct our course and find the accurate solution. After reviewing, let's go back to 3 * (28 - y) = 2 * (y + 10). This expands to 84 - 3y = 2y + 20. Now, gathering like terms, we add 3y to both sides and subtract 20 from both sides, which gives us 64 = 5y. Here's where the earlier error likely occurred in the subsequent steps (or was anticipated to occur). Dividing both sides by 5, we get y = 64/5 which is incorrect since y should be an integer.
Let's pinpoint the exact mistake by re-examining the equation (x + 10) / (y + 10) = 2/3 and the substitution x = 18 - y. We correctly substituted to get ((18 - y) + 10) / (y + 10) = 2/3, which simplifies to (28 - y) / (y + 10) = 2/3. Cross-multiplying gives us 3(28 - y) = 2(y + 10). Expanding this, we have 84 - 3y = 2y + 20. Adding 3y to both sides and subtracting 20 from both sides correctly leads to 64 = 5y.
The actual mistake is realizing this outcome indicates there might be an issue with the problem statement itself, as it leads to a non-integer solution for 'y', which is unusual for this type of problem. However, let's proceed assuming there's a valid integer solution and that we need to find the closest possible answer given the equations. The correct algebraic manipulation led us to 64 = 5y. Dividing both sides by 5 indeed gives y = 64/5 = 12.8. Since we need an integer value and given the context, let's reassess the entire approach. The system of equations is:
- x + y = 18
- (x + 10) / (y + 10) = 2/3
From equation 1, x = 18 - y. Substituting into equation 2:
***((18 - y) + 10) / (y + 10) = 2/3***
***(28 - y) / (y + 10) = 2/3***
***3(28 - y) = 2(y + 10)***
***84 - 3y = 2y + 20***
***5y = 64***
The equation 5y = 64 is correct based on the problem statement. The issue is that it doesn't yield an integer solution for y. This suggests either a mistake in the problem statement or that we should look for the closest integer solution if the problem implies integer numerators and denominators.
However, assuming the problem is correctly stated and we made no algebraic errors, the non-integer result is mathematically consistent. If y = 64/5, then substituting back into x = 18 - y, we get x = 18 - 64/5 = (90 - 64) / 5 = 26/5. Thus, the original fraction would be (26/5) / (64/5) which simplifies to 26/64 or 13/32. Let's check if adding 10 to both numerator and denominator gives us 2/3:
*(13 + 10) / (32 + 10) = 23/42* which does *not* simplify to 2/3.
Conclusion:
Given the inconsistency and the non-integer results, there's likely an issue with the problem statement. The original problem might have a typo or require a different approach that isn't immediately obvious with standard algebraic methods for solving systems of equations. Our correct algebraic steps led to a mathematically sound but contextually inconsistent result, suggesting a problem in the initial conditions provided.