Solving For The Larger Number When The Difference Is 150 And The Quotient Is 4

Hey guys! Today, let's dive into a classic math problem that involves finding two numbers when we know their difference and their quotient. This kind of problem is super common in algebra, and once you get the hang of it, you'll be solving them like a pro. We're going to break it down step by step, so it's crystal clear. We will approach this task with a conversational style, ensuring that the solution is not only mathematically sound but also easily understandable. We will emphasize the importance of translating word problems into mathematical equations, a fundamental skill in algebra. Additionally, we will explore the real-world applications of such problems, highlighting how they help in developing critical thinking and problem-solving skills.

Understanding the Problem

So, the problem states: "The difference of two numbers is 150 and their quotient is 4. Find the larger of the two numbers." To kick things off, let's dissect what this means. We've got two mystery numbers. When you subtract the smaller one from the bigger one, you get 150. Also, if you divide the bigger number by the smaller number, you end up with 4. Our mission? To figure out what that larger number is. This is a typical algebraic problem where we'll need to use variables to represent the unknowns and set up equations based on the given information. Let’s begin by identifying the key components: the difference, the quotient, and the two unknown numbers. We'll then translate these components into algebraic expressions, which will form the foundation for our equations. Remember, the goal here isn't just to find the answer but to understand the process, which is a valuable skill in mathematics and beyond.

Setting Up the Equations

Alright, let's get those math muscles flexing! First, we're going to assign variables. Let's call the larger number x and the smaller number y. This makes it easier to write out our equations. Remember, choosing the right variables is crucial in simplifying the problem. Now, let's translate the given information into equations. The problem tells us that "the difference of two numbers is 150." This translates directly to our first equation: x - y = 150. This equation represents the subtraction aspect of the problem, showing the relationship between the larger and smaller numbers in terms of their difference. Next, we know that "their quotient is 4." In math speak, that means x / y = 4. This is our second equation, representing the division aspect of the problem. It shows the ratio between the two numbers. Now we have two equations with two unknowns, which is a classic setup for solving simultaneous equations. This is where the fun begins, as we get to use algebra to unravel the mystery and find the values of x and y.

Solving the Equations

Okay, guys, now for the juicy part – cracking these equations! We've got two equations: x - y = 150 and x / y = 4. There are a couple of ways we could tackle this, but let's use the substitution method. It's pretty straightforward and easy to follow. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the problem to a single equation with one variable, which is much easier to solve. Let’s start by rearranging the second equation to solve for x. Multiply both sides of x / y = 4 by y, and you get x = 4y. Now we have an expression for x in terms of y. The next step is to substitute this expression into our first equation. This is where we replace x in the first equation with 4y, creating a new equation that only involves y. This new equation will allow us to solve for y, which is half the battle. Once we find y, we can easily find x using the relationship we've already established.

Substitute x = 4y into the first equation: 4y - y = 150. See what we did there? Now we've got an equation with just y! Let's simplify this. 4y minus y is 3y, so we have 3y = 150. To find y, we divide both sides of the equation by 3: y = 150 / 3, which means y = 50. Awesome! We've found the smaller number. Now, let's use this to find the larger number, x. Remember, we already figured out that x = 4y. So, just plug in y = 50: x = 4 * 50, which gives us x = 200. Boom! We've got both numbers. But remember, the question asks for the larger number, so we're almost there.

Identifying the Larger Number

We've done the math, and we've got our two numbers: x = 200 and y = 50. Remember, x is the larger number and y is the smaller number. So, which one is the larger number? It's pretty clear, right? 200 is definitely bigger than 50. So, the larger number is 200. We've successfully navigated through the equations and found the solution. But before we wrap up, it's always a good idea to check our answer. This is a crucial step in problem-solving, as it helps to ensure that our solution is correct and that we haven't made any mistakes along the way. Checking our answer involves plugging the values we found back into the original equations and verifying that they hold true. This not only confirms our solution but also reinforces our understanding of the problem and the process we used to solve it.

Checking Our Solution

To make sure we nailed it, let's plug our numbers back into the original equations. This is like the final boss level of the problem – we want to make sure everything checks out. First, let's check the difference: x - y = 150. We found x to be 200 and y to be 50. So, 200 - 50 = 150. Bingo! That equation holds true. We've confirmed that the difference between the two numbers is indeed 150. Now, let's move on to the next equation, which involves the quotient. This step is crucial to ensure that our solution is consistent with all the information provided in the problem. If our solution doesn't satisfy both equations, it means we need to revisit our calculations and identify any errors. However, if it checks out for both equations, we can confidently say that we've found the correct answer. Let's continue with the second equation.

Now, let's check the quotient: x / y = 4. Plugging in our values, we get 200 / 50 = 4. Double bingo! That equation also holds true. We've confirmed that the quotient of the larger number divided by the smaller number is 4. So, our solution perfectly fits both conditions of the problem. We've not only found the answer but also verified its correctness, which is a testament to our understanding of the problem and our ability to solve it accurately. This is a great feeling, knowing that we've tackled the problem effectively and arrived at the correct solution. It's a moment to pat ourselves on the back and celebrate our success. Now, let's confidently state our final answer.

Final Answer

So, after all that brainpower, we've arrived at the final answer. The larger of the two numbers is 200. We broke down the problem, set up our equations, solved them like pros, and even checked our work to make sure we were spot on. Math problems like these are all about taking it step by step and staying organized. You guys got this! Remember, the key to mastering algebra and problem-solving is practice. The more you practice, the more comfortable you'll become with the process, and the easier these problems will seem. So, keep challenging yourself with new problems, and don't be afraid to make mistakes along the way. Mistakes are just opportunities to learn and grow. With consistent effort and a positive attitude, you'll be able to tackle even the most complex math problems with confidence.

Great job, guys! You've successfully navigated this problem. Remember, the key to these types of problems is to break them down into smaller, manageable steps. Keep practicing, and you'll become a math whiz in no time! This journey through the problem highlights the importance of persistence and attention to detail. Each step, from translating the word problem into algebraic equations to solving those equations and verifying the solution, requires careful consideration and precision. The satisfaction of arriving at the correct answer is a reward for the effort invested and a motivator to continue learning and improving our problem-solving skills. So, let's carry this positive experience forward and embrace new challenges with the same enthusiasm and determination.