Find Numbers: Difference 90, Quotient 3, Remainder 10

Introduction

Hey guys! Let's dive into a fun math problem today. We're going to explore how to find two mystery numbers based on some interesting clues. Specifically, we need to discover two numbers where their difference is 90, their quotient is 3, and when we divide them, we get a remainder of 10. Sounds like a puzzle, right? Well, that's because it is! But don't worry, we'll break it down step by step and make sure it's super clear. We'll use some basic algebraic principles and a bit of logical thinking to crack this code. Math can seem daunting sometimes, but when you approach it like a detective solving a case, it becomes much more engaging and, dare I say, fun. So, let's put on our thinking caps and get started on this numerical quest. Understanding how to solve these types of problems not only sharpens your math skills but also enhances your problem-solving abilities in general. Remember, math isn't just about numbers; it's about logic, patterns, and critical thinking. Stick with me, and we'll unravel this mystery together! We'll see how equations can be powerful tools in figuring out the unknown. Let’s make sure that by the end of this explanation, you're not only able to solve this particular problem but also confident in tackling similar challenges. Are you ready to embark on this mathematical adventure? Let's do it!

Setting Up the Equations

Okay, let's translate the problem's clues into math language. This is a crucial step in solving any word problem. First, let’s call our two mystery numbers x and y. To make things easier, we'll assume that x is the larger number and y is the smaller one. This assumption helps us set up our equations correctly. Now, let’s look at our first clue: the difference between the two numbers is 90. In math terms, this means x minus y equals 90. So, we can write our first equation as: x - y = 90. Easy peasy, right? Next up is the clue about the quotient and remainder. The problem states that the quotient is 3 and the remainder is 10 when we divide the larger number (x) by the smaller number (y). Remember how division works? We can express this relationship using the division algorithm. The division algorithm basically says that dividend = (divisor × quotient) + remainder. In our case, x is the dividend, y is the divisor, 3 is the quotient, and 10 is the remainder. Plugging these values into the division algorithm, we get our second equation: x = 3y + 10. So, we've successfully transformed the word problem into two algebraic equations. We have a system of equations now, and that's our roadmap to finding x and y. We're one step closer to cracking the code! Remember, the key to solving these kinds of problems is to break them down into smaller, manageable parts. We took the verbal clues and translated them into mathematical statements. Now, we have two equations and two unknowns, which is a classic setup for solving. Let's move on to the next step: actually solving these equations. Are you feeling confident? You should be! We've got this. Let's go!

Solving the System of Equations

Alright, now comes the fun part: actually solving for our mystery numbers! We have two equations: x - y = 90 and x = 3y + 10. There are a couple of ways we can tackle this, but the substitution method is a great fit here. Why? Because we already have x expressed in terms of y in the second equation. This makes our job a little easier. So, what we're going to do is take the expression for x from the second equation (which is 3y + 10) and substitute it into the first equation wherever we see x. This means our first equation, x - y = 90, becomes (3y + 10) - y = 90. See what we did there? We replaced x with its equivalent expression. Now we have a single equation with just one variable, y. This is something we can definitely solve! Let's simplify this equation. Combine the y terms: 3y - y gives us 2y. So our equation now looks like this: 2y + 10 = 90. Next, we want to isolate the term with y in it. To do that, we subtract 10 from both sides of the equation. This gives us 2y = 80. Almost there! To find y, we need to get y all by itself. Since y is being multiplied by 2, we do the opposite: we divide both sides of the equation by 2. This gives us y = 40. Hooray! We've found one of our numbers. But we're not done yet. We still need to find x. Remember our second equation, x = 3y + 10? Now that we know y, we can plug it into this equation to find x. So, x = 3(40) + 10. Let's do the math: 3 times 40 is 120, and 120 plus 10 is 130. So, x = 130. We've done it! We've solved the system of equations and found both of our mystery numbers: x = 130 and y = 40. But before we celebrate too much, let's make sure our answers make sense in the context of the original problem. This is a crucial step in problem-solving: checking your work.

Checking the Solution

Okay, we've found our two numbers, x = 130 and y = 40. But before we declare victory, let's make absolutely sure these numbers fit all the clues from the original problem. This is like the detective making sure all the pieces of the puzzle fit together – it’s a crucial step! First, let's check the difference: Is the difference between x and y equal to 90? Well, 130 - 40 does indeed equal 90. So far, so good! Our first clue checks out. Now, let's tackle the second clue: When we divide x by y, is the quotient 3 and the remainder 10? To check this, we'll perform the division: 130 ÷ 40. 40 goes into 130 three times (3 times 40 is 120), and the remainder is 130 - 120, which is 10. Awesome! The quotient is 3, and the remainder is 10, just like the problem stated. Both clues are satisfied! This gives us a lot of confidence that we've found the correct solution. Checking our work is not just about confirming our answer; it's also a way to deepen our understanding of the problem and the relationships between the numbers. By going back to the original clues and verifying our solution, we're reinforcing our understanding of the mathematical concepts involved. Plus, it feels pretty great to know we've nailed it! So, we've successfully navigated this mathematical puzzle, found the two numbers, and verified that they fit all the criteria. What's next? Let's wrap up our findings and highlight the key steps we took to get here.

Conclusion

Alright, guys, we did it! We successfully found the two numbers that fit all the clues: 130 and 40. We started with a word problem that seemed a bit tricky, but we broke it down step by step and conquered it. Let's recap the journey we took. First, we carefully read the problem and identified the key information. We translated the verbal clues into mathematical equations. This is a crucial skill in problem-solving: turning words into math. Then, we set up a system of equations: x - y = 90 and x = 3y + 10. We chose the substitution method to solve this system because it seemed like the most efficient way to go. We substituted the expression for x from the second equation into the first equation, which allowed us to solve for y. Once we found y, we plugged it back into one of the original equations to find x. And voila! We had our two numbers. But we didn't stop there. We checked our solution against the original problem to make sure everything made sense. This is a non-negotiable step in math – always verify your answers! By checking, we gained confidence that our solution was correct and reinforced our understanding of the problem. Solving problems like this is not just about finding the right answer; it's about developing critical thinking skills, logical reasoning, and problem-solving strategies. These are skills that will serve you well in all areas of life, not just in math class. So, the next time you encounter a challenging problem, remember our approach: break it down, translate it into math language, solve it step by step, and always check your work. You've got this! And remember, math can be fun when you approach it like a puzzle. Keep practicing, keep exploring, and keep challenging yourself. You'll be amazed at what you can achieve.

This problem was a great example of how algebra can be used to solve real-world puzzles. We took a situation described in words and turned it into a solvable mathematical problem. This is a powerful skill that opens doors to understanding and solving many different kinds of challenges. So, keep honing your math skills, and you'll be well-equipped to tackle whatever comes your way. Great job today, everyone!