Solve Math Mystery: Finding Two Numbers From Division Clues

Hey guys! Let's dive into a fun math problem that might seem a bit challenging at first, but trust me, we'll break it down step by step and make it super easy to understand. We're dealing with two mystery numbers, and we have some clues about how they relate to each other when we divide them. The goal? To find out what these numbers actually are. So, grab your thinking caps, and let's get started!

Understanding the Problem

Our math problem presents a classic division puzzle. We need to decipher the relationships between the numbers based on the quotients and remainders we get from division.

First off, let's break down the information we have. We're told that when the larger of the two numbers is divided by the smaller one, the result (or quotient) is 3, and there's a remainder of 3. Think of it like sharing: if you're dividing a pile of candies, the quotient is how many each person gets, and the remainder is how many are left over. In math terms, this means we can express the larger number as 3 times the smaller number, plus the remainder of 3. This is a crucial first step because it gives us a way to link the two numbers together in an equation.

But that's not all! The problem throws another curveball our way. It says that if we triple the larger number and then divide it by the smaller number, we get a new quotient of 10, with a remainder of 4. Tripling the larger number changes things up, giving us a new perspective on the relationship between the two numbers. This second piece of information is like a second clue in a detective story – it gives us another equation that involves both numbers. Having two equations is super helpful because it means we can use methods like substitution or elimination (don't worry if these sound scary, we'll go through them!) to solve for our two unknowns. So, we've got a system of clues here, and now we need to figure out how to use them together to crack the code and find our mystery numbers.

Setting Up the Equations

Okay, let's translate those words into math! This is where the problem becomes less abstract and more concrete. To do this effectively, we will use variables to represent the unknowns:

• Let's call the larger number 'x'.
• And the smaller number? We'll name it 'y'.

Now, we can rewrite the information from the problem using these variables. This is like translating from one language to another – we're taking the problem's language (words) and turning it into the language of math (equations). It makes things much clearer and easier to work with. The beauty of algebra is that it gives us a way to express these relationships precisely.

From the first part of the problem, we know that when x is divided by y, the quotient is 3, and the remainder is also 3. Remember how division works? The dividend (the number being divided, which is x in our case) is equal to the divisor (the number we're dividing by, which is y) times the quotient, plus the remainder. This is a fundamental concept, and it's the key to turning the word problem into an equation. So, we can write this as:

x = 3y + 3

This equation is like a mathematical sentence that describes the relationship between x and y based on the first clue. It tells us exactly how the larger number x is related to the smaller number y.

Now for the second part of the problem! It tells us that if we triple the larger number (that's 3x) and divide it by the smaller number (y), the quotient is 10, and the remainder is 4. We can use the same division principle here to create our second equation. Just like before, we multiply the divisor by the quotient and add the remainder. So, this translates to:

3x = 10y + 4

This is our second mathematical sentence, and it gives us another piece of the puzzle. It shows us a different relationship between x and y, this time involving tripling the larger number. Now, we have two equations, two pieces of information about the same two numbers. This is what we call a system of equations, and it's a powerful tool for solving problems like this. Next up, we'll see how to solve this system and actually find the values of x and y.

Solving the System of Equations

Alright, guys, we've got our two equations, and now it's time to put on our detective hats and solve them! There are a few different ways we could go about this, but one common and effective method is using substitution. Substitution is like replacing one thing with an equivalent thing – in this case, we'll replace one of the variables in one equation with an expression from the other equation. This will leave us with a single equation with just one variable, which is much easier to solve.

Let's take a look at our equations again:

x = 3y + 3
3x = 10y + 4

The first equation, x = 3y + 3, looks like a good candidate for substitution because it already has x isolated on one side. This means we know exactly what x is in terms of y. We can take this expression (3y + 3) and substitute it in place of x in the second equation. It's like we're saying, "Hey, we know x is the same as 3y + 3, so let's use that information in the other equation!"

So, let's do it! We'll replace x in the second equation (3x = 10y + 4) with (3y + 3). This gives us:

3(3y + 3) = 10y + 4

Notice how we've put (3y + 3) in parentheses? That's super important because we need to make sure we're multiplying the entire expression by 3, not just the first term. Now we've got an equation that only involves y, which means we're one step closer to finding our numbers!

Next, we need to simplify this equation. This means getting rid of the parentheses by distributing the 3 and then combining like terms. It's like tidying up the equation to make it easier to read and solve. Once we've simplified, we'll have a straightforward equation that we can solve for y. Then, once we know y, we can plug it back into one of our original equations to find x. It's like a chain reaction – solving for one variable leads us to the other. So, let's keep going, and we'll have our mystery numbers in no time!

Solving for y

Alright, let's continue simplifying the equation we got after substitution. We had:

3(3y + 3) = 10y + 4

Our first step here is to distribute the 3 on the left side of the equation. Remember, distributing means multiplying the 3 by each term inside the parentheses. So, 3 times 3y is 9y, and 3 times 3 is 9. This gives us:

9y + 9 = 10y + 4

Now, we need to isolate y on one side of the equation. This means getting all the terms with y on one side and all the constant terms (the numbers without variables) on the other side. A common strategy is to move the smaller y term to the side with the larger y term to avoid dealing with negative numbers. In this case, we have 9y on the left and 10y on the right, so let's move the 9y to the right side. To do this, we subtract 9y from both sides of the equation:

9y + 9 - 9y = 10y + 4 - 9y

This simplifies to:

9 = y + 4

See how the 9y terms canceled out on the left side, leaving us with just 9? We're getting closer!

Now, we need to get y completely by itself on the right side. We have y + 4, so to isolate y, we need to get rid of the + 4. We do this by subtracting 4 from both sides of the equation:

9 - 4 = y + 4 - 4

This simplifies to:

5 = y

Woohoo! We've solved for y! We know that the smaller number, y, is 5. That's a big step forward. But we're not done yet – we still need to find x, the larger number. But now that we know y, finding x is going to be a piece of cake.

Solving for x

Awesome, guys! We've figured out that y, the smaller number, is 5. Now, let's use this information to find x, the larger number. Remember our first equation? It was:

x = 3y + 3

This equation tells us exactly how x is related to y. Since we now know that y is 5, we can simply substitute 5 for y in this equation. It's like plugging in a known value to find an unknown one. This is one of the coolest things about algebra – we can use what we've already learned to discover even more!

So, let's plug it in! Replacing y with 5 in the equation gives us:

x = 3(5) + 3

Now, it's just a matter of simplifying. First, we multiply 3 by 5, which gives us 15:

x = 15 + 3

Then, we add 3 to 15:

x = 18

And there we have it! We've solved for x. The larger number, x, is 18. We now know both of our mystery numbers: x is 18, and y is 5. It's like we've cracked the code and unlocked the solution to the puzzle.

The Solution and Verification

Alright, math detectives, let's recap what we've found! After carefully working through the problem, setting up equations, and solving them, we've discovered our two mystery numbers:

• The larger number, x, is 18.
• The smaller number, y, is 5.

But before we celebrate too much, it's always a good idea to verify our solution. This is like double-checking our work to make sure we haven't made any mistakes along the way. Verification is a crucial step in problem-solving because it gives us confidence that our answer is correct. It also helps us catch any errors we might have made, which is super important in math and in life!

To verify our solution, we'll go back to the original problem and make sure that our numbers fit the conditions given. The problem told us two things:

1. When the larger number is divided by the smaller number, the quotient is 3, and the remainder is 3.
2. When three times the larger number is divided by the smaller number, the quotient is 10, and the remainder is 4.

Let's check the first condition. If we divide 18 (our larger number) by 5 (our smaller number), we get a quotient of 3 (because 5 goes into 18 three times) and a remainder of 3 (because 3 is left over). So, our numbers satisfy the first condition. That's a good sign!

Now, let's check the second condition. Three times our larger number (18) is 54. If we divide 54 by 5, we get a quotient of 10 (because 5 goes into 54 ten times) and a remainder of 4 (because 4 is left over). So, our numbers also satisfy the second condition. Woohoo!

Since our numbers satisfy both conditions of the problem, we can be confident that our solution is correct. We've successfully found the two mystery numbers! Give yourselves a pat on the back – you've done some awesome math sleuthing today!

Conclusion

So, guys, we successfully tackled a pretty cool math problem today! We started with some clues about two mystery numbers and used our problem-solving skills to figure out what those numbers were. We learned how to translate word problems into algebraic equations, how to solve a system of equations using substitution, and how to verify our solution to make sure it's correct. That's a lot of math power packed into one problem!

Remember, the key to solving these kinds of problems is to break them down into smaller, more manageable steps. Don't be intimidated by the whole problem at once – just focus on understanding each piece of information and how it relates to the others. And always remember to check your work! Verification is your friend.

Math is like a puzzle, and every problem is a new challenge. The more puzzles you solve, the better you get at it. So, keep practicing, keep exploring, and keep having fun with math! You've got this!