Math Problem: Finding Two Numbers With Given Conditions

Oct 12, 2025 by ADMIN 56 views

Hey guys! Let's dive into a fascinating math problem today. This one involves finding two numbers based on some interesting relationships between them. We're going to break it down step by step, so it's super easy to follow. So, grab your thinking caps, and let’s get started!

Understanding the Problem

To really nail this math problem, we first need to understand the information we've been given. One number is 72 greater than the other, and when we divide their sum by their difference, we get a quotient of 5 with a remainder of 2. Sounds a bit complex, right? Don't worry! We'll simplify it.

First, let's identify our main keywords. We're looking at "two numbers," a "difference" of 72, a "sum," a "quotient" of 5, and a "remainder" of 2. These are our clues. The key here is translating these words into mathematical expressions. We need to represent the unknowns (the two numbers) with variables, form equations based on the given conditions, and then solve those equations.

Why is this kind of problem important? Well, it's not just about finding two random numbers. It's about developing critical thinking skills, enhancing our ability to translate real-world scenarios into mathematical models, and honing our problem-solving abilities. These are skills that are valuable not just in math class, but in everyday life! Figuring out budgets, planning projects, or even just splitting a bill with friends – all these situations can benefit from a good understanding of algebra and problem-solving strategies.

So, as we go through this, remember we're not just aiming for the answer. We're building a toolbox of skills that will help us tackle all sorts of challenges. Let's keep that in mind as we move forward.

Setting Up the Equations

Now that we've dissected the problem, let's translate those words into some solid mathematical equations. This is where the magic happens, guys! We'll use algebra to represent the unknowns and relationships, which will give us a clear path to the solution.

First, let's assign variables. Since we have two unknown numbers, let’s call them x and y. It’s a classic choice, but hey, it works! We'll say x is the larger number and y is the smaller number. This distinction is important because the problem tells us one number is greater than the other, and we need to keep track of that in our equations.

The first piece of information we have is that one number is 72 greater than the other. Since we said x is the larger number, we can write this as an equation: x = y + 72. See how we've turned a sentence into a mathematical statement? This is the power of algebra!

Now, for the second part. We know that when we divide the sum of the numbers by their difference, we get a quotient of 5 and a remainder of 2. The sum of the numbers is simply x + y. The difference, since x is larger, is x - y. The division part is a bit trickier, but we can express it using the division algorithm: Dividend = Divisor * Quotient + Remainder. In our case, the dividend is the sum (x + y), the divisor is the difference (x - y), the quotient is 5, and the remainder is 2. So, we can write the equation: x + y = 5(x - y) + 2.

Boom! We've just created a system of two equations with two variables. This is a huge step because now we have a well-defined problem that we can solve using algebraic techniques. These two equations hold the key to unlocking the values of x and y, and by extension, the solution to our problem. In this section, we've carefully crafted our equations, laying the groundwork for the next step: solving for x and y. So, let's move on to the exciting part of solving this system and finally reveal those mysterious numbers!

Solving the Equations

Alright, guys, here comes the exciting part – actually solving the equations we set up earlier! This is where we put our algebra skills to the test and finally figure out what those two numbers are. We've got a system of two equations:

x = y + 72
x + y = 5(x - y) + 2

There are a couple of ways we can tackle this, but the substitution method is a great choice here since the first equation is already solved for x. This means we can easily substitute the expression for x from the first equation into the second equation, leaving us with an equation with just one variable (y).

Let's do that: Substitute (y + 72) for x in the second equation:

(y + 72) + y = 5((y + 72) - y) + 2

Now, we simplify and solve for y. First, combine like terms on the left side:

2y + 72 = 5((y + 72) - y) + 2

Next, simplify inside the parentheses on the right side:

2y + 72 = 5(72) + 2

2y + 72 = 360 + 2

2y + 72 = 362

Subtract 72 from both sides:

2y = 290

Divide both sides by 2:

y = 145

Great! We've found y, which is one of our numbers. Now we just need to find x. Luckily, we have the first equation, x = y + 72, which makes this super easy. Substitute y = 145 into this equation:

x = 145 + 72

x = 217

So, there you have it! We've found both numbers. x is 217, and y is 145. We’ve successfully navigated the algebraic terrain and pinpointed the values that satisfy our initial conditions. Pat yourselves on the back, guys – this is a big win!

Checking the Solution

Before we declare victory, it's super important to check our solution. You know what they say: double-checking is always a good idea, especially in math! We need to make sure that the numbers we found actually fit the conditions given in the problem. This is our chance to catch any silly mistakes and ensure we've got the right answer.

Remember, the problem stated two things: first, that one number is 72 greater than the other, and second, that dividing their sum by their difference gives a quotient of 5 with a remainder of 2. Let's see if our numbers, 217 and 145, meet these criteria.

First condition: Is 217 seventy-two more than 145? Let's subtract: 217 - 145 = 72. Yep, that checks out! Our numbers satisfy the first condition.

Second condition: Does dividing their sum by their difference give a quotient of 5 with a remainder of 2? Let's calculate the sum and difference:

Sum: 217 + 145 = 362
Difference: 217 - 145 = 72

Now, let's divide the sum by the difference: 362 ÷ 72. If you do the division, you'll find that 362 = 5 * 72 + 2. This means the quotient is 5, and the remainder is 2. Perfect! Our numbers satisfy the second condition as well.

Since our numbers satisfy both conditions, we can confidently say that our solution is correct. We’ve not only solved the problem but also verified our answer, a testament to our thoroughness and mathematical prowess!

Conclusion

So, guys, we've successfully tackled a challenging math problem today! We've journeyed from understanding the problem statement, setting up equations, solving for the unknowns, and finally, verifying our solution. The two numbers we found are 217 and 145. But more than just finding the answer, we've honed our problem-solving skills, enhanced our understanding of algebraic concepts, and reinforced the importance of checking our work.

This type of problem is a fantastic exercise in translating words into mathematical expressions and using algebra to unravel the mysteries hidden within. Remember, math isn't just about numbers and formulas; it's about critical thinking, logical reasoning, and approaching challenges with a structured mindset.

Whether you're a student grappling with algebra, a math enthusiast seeking a mental workout, or just someone who enjoys a good puzzle, I hope you found this exploration helpful and engaging. Keep practicing, keep exploring, and remember that every problem is an opportunity to learn and grow. Until next time, happy solving!