Find The Factors: Product Is 620, Increased By 155

Hey guys! Let's dive into a cool math problem today that involves finding two numbers based on their product and how the product changes when we tweak one of the numbers. It's like a little puzzle, and who doesn't love puzzles, right? We’re going to break it down step-by-step so you can totally nail it. So, let's get started and make math a bit more fun!

Understanding the Problem

The heart of the problem is this: We're hunting for two numbers. When you multiply them together, you get 620. Now, here’s the twist: If you add 5 to just one of those numbers, the result of the multiplication goes up by 155. That's our clue, and we're going to use it to crack the case. To make sure we're all on the same page, let's put this into math terms. Think of our two mystery numbers as 'x' and 'y.' So, we know that:

• x * y = 620 (This is our starting point, the basic relationship between the two numbers)
• (x + 5) * y = 620 + 155 (This tells us what happens when we add 5 to one of the numbers)

This is where the fun begins because we've turned a word problem into something we can actually play with using math. The beauty of algebra is how it helps us take real-world situations and turn them into equations we can solve. We're not just guessing here; we're using logic and math rules to find our answer. This is super important in math – the ability to translate a scenario into an equation. It’s like learning a secret code, and once you know the code, you can unlock all sorts of problems!

Setting Up the Equations

Alright, let's put on our math hats and get these equations looking slick and ready to solve. We've already got our main players: x * y = 620 and (x + 5) * y = 620 + 155. The second equation might look a bit chunky, but we can tidy it up a bit. First things first, let’s simplify 620 + 155. That's an easy peasy 775, right? So, our second equation now reads (x + 5) * y = 775. Now, let's do a little algebra magic and expand that equation. Remember the distributive property? It’s going to be our best friend here. We're going to multiply y by both x and 5. This gives us x * y + 5 * y = 775. Okay, things are starting to look interesting. Notice anything familiar? Yep, we've got an x * y in this equation, and we know from our first equation that x * y = 620. This is like finding a piece of the puzzle that fits perfectly! We can substitute 620 for x * y in our expanded equation. So, we get 620 + 5 * y = 775. See how we're turning a complicated problem into simpler steps? That’s a key strategy in problem-solving. By breaking things down, we make the challenge way less daunting. We're not just blindly crunching numbers here; we're strategically manipulating our equations to reveal the secrets they hold. It’s like being a math detective, and the equations are our clues.

Solving for 'y'

Okay, let's get down to business and solve for 'y'. We've got our equation nice and set up: 620 + 5 * y = 775. Our goal here is to isolate 'y' on one side of the equation, which means we want to get it all by its lonesome. To do that, we need to start undoing the operations around it. The first thing we see is that 620 is being added to 5 * y. So, let's subtract 620 from both sides of the equation. This keeps the equation balanced, like a see-saw, and gets us closer to our goal. 620 + 5 * y - 620 = 775 - 620. Simplifying that gives us 5 * y = 155. We're on a roll now! We've managed to get the term with 'y' all by itself on one side. Now, we just need to deal with that 5 that's multiplying 'y'. The opposite of multiplication is division, so we're going to divide both sides of the equation by 5. (5 * y) / 5 = 155 / 5. And when we do that calculation, we find that y = 31. Woohoo! We've found one of our mystery numbers. But hold on, our quest isn't over yet. We've got 'y', but we still need to find 'x'. But now that we know 'y', finding 'x' is going to be a piece of cake. Remember, math isn’t just about getting the right answer; it’s about the journey and the steps you take to get there. Each step we take builds on the previous one, and that’s how we solve these puzzles.

Solving for 'x'

Alright, we've nailed down 'y' as 31, which is awesome! But remember, we're on a mission to find two numbers, so 'x' is still playing hide-and-seek with us. But don't worry, we've got the perfect tool to flush 'x' out: our trusty original equation, x * y = 620. We know what 'y' is now, so we can just plug it into the equation and solve for 'x'. It's like we're connecting the dots to reveal the final picture. So, let's swap 'y' for 31 in our equation: x * 31 = 620. Now, this looks much more manageable, doesn't it? We've got 'x' multiplied by 31, and we need to get 'x' all by itself. Just like before, we'll use the opposite operation to undo the multiplication. That means we're going to divide both sides of the equation by 31. (x * 31) / 31 = 620 / 31. When we crunch those numbers, we get x = 20. Yes! We've found our other number. It’s like completing a treasure hunt, and the treasure is the solution to our problem. Finding 'x' was super smooth once we knew 'y', and that shows how important it is to use what you've already figured out to solve the rest of the problem. In math, every step forward makes the next step a little easier.

Verifying the Solution

We’ve got our two numbers, x = 20 and y = 31, but before we throw a math party, let's make absolutely sure they fit the bill. It's like double-checking your map before you declare you've reached your destination – just to be sure you didn't take a wrong turn somewhere. Our original problem had two conditions, so we need to test both of them. First up, the product of our two numbers should be 620. So, let's multiply them: 20 * 31. What do you know? It equals 620. So far, so good! Our numbers are playing nice with the first rule. But we've got that second condition to tackle: If we add 5 to one of the factors, the product increases by 155. Let's add 5 to x (which is 20) and see what happens. So, we're looking at (20 + 5) * 31, which is 25 * 31. That gives us 775. Now, is 775 really 155 more than our original product of 620? Let's check: 775 - 620 = 155. Bingo! Our numbers pass the second test with flying colors. We've officially verified our solution! This step is so crucial because it confirms that our hard work has paid off and that we haven't made any sneaky mistakes along the way. It’s like putting the final piece in a jigsaw puzzle and seeing the whole picture come together.

Final Answer

Drumroll, please! After all our math sleuthing, we've successfully uncovered the two mystery numbers. We set up our equations, we crunched the numbers, and we double-checked our answers to make sure everything was spot on. So, what are the two factors that fit the bill? The answer is 20 and 31. These two numbers not only multiply to give 620, but also play along perfectly with the condition that adding 5 to one of them increases the product by 155. We did it, guys! We took a math problem, broke it down into manageable steps, and emerged victorious. Remember, math problems might seem daunting at first, but with a bit of strategy and a step-by-step approach, you can conquer them all. And the feeling of getting to the right answer? Totally worth it. So, keep practicing, keep exploring, and keep having fun with math!