Find Odd Number 'ab' Where Ab + Bb + 2a = 204

by Dimemap Team 46 views

Hey guys! Today, we're diving into a cool math problem where we need to find an odd number in a specific form. The problem states: Determine an odd number of the form 'ab' if ab + bb + 2a = 204. Sounds intriguing, right? Let's break it down step by step and figure out how to solve it. We will explore the problem, break down the equation, and use some logical steps to find our solution. Let's get started and unravel this mathematical puzzle together!

Understanding the Problem

So, let's start by really understanding what the problem is asking. We've got this equation: ab + bb + 2a = 204. Now, 'ab' isn't just a times b; it's a two-digit number where 'a' is the tens digit and 'b' is the units digit. Similarly, 'bb' is another two-digit number, but this time both digits are the same – 'b'. And we need to find an odd number 'ab' that makes this whole equation work.

To really dig into this, we need to remember what place value means. The number 'ab' can actually be written as 10a + b because 'a' is in the tens place. Likewise, 'bb' is really 10b + b. And remember, the final answer, the number 'ab', needs to be odd. What does that tell us about 'b'? Well, 'b' has to be an odd digit (1, 3, 5, 7, or 9) because that's what makes the entire number 'ab' odd. Keeping these key ideas in mind – place value and the odd nature of 'b' – will help us simplify the equation and move closer to finding the values of 'a' and 'b'. We're setting the stage to make the math a whole lot easier!

Breaking Down the Equation

Okay, now that we understand the problem, let's dive into the equation itself and break it down into something more manageable. Remember, we have ab + bb + 2a = 204. We figured out that 'ab' is really 10a + b, and 'bb' is 10b + b. So, let's substitute those into our equation. This gives us:

10a + b + 10b + b + 2a = 204

Now we can simplify things by combining like terms. We've got 10a and 2a, which add up to 12a. And we have b, 10b, and another b, which combine to give us 12b. So our equation now looks like this:

12a + 12b = 204

See how much cleaner that looks? We're not done yet, though. Notice that both terms on the left side of the equation have a common factor of 12. This means we can factor out the 12, which is a neat trick to simplify things further. Factoring out the 12 gives us:

12(a + b) = 204

Now, to isolate (a + b), we can divide both sides of the equation by 12. This will help us get closer to finding the individual values of 'a' and 'b'. Let's do that division and see what we get. This step-by-step simplification is key to solving the problem without getting lost in the numbers. Next, we’ll find out what a + b actually equals.

Solving for a + b

Alright, we've simplified our equation to a point where it's much easier to handle. We've got 12(a + b) = 204. To find out what (a + b) is, we need to get rid of that 12 that's multiplying it. How do we do that? Simple – we divide both sides of the equation by 12. This is a fundamental rule in algebra: whatever you do to one side, you have to do to the other to keep things balanced.

So, let's divide both sides by 12:

12(a + b) / 12 = 204 / 12

The 12s on the left side cancel each other out, leaving us with just (a + b). Now we need to figure out what 204 divided by 12 is. If you do the math, you'll find that 204 / 12 = 17. So, our equation now looks like this:

a + b = 17

This is a significant step! We've taken a more complex equation and boiled it down to something very straightforward. We now know that the sum of our digits 'a' and 'b' is 17. This gives us a clear relationship between 'a' and 'b' that we can use to narrow down the possibilities. Remember, we're looking for an odd number 'ab', and we already know that 'b' must be an odd digit. With this new information, we're well on our way to cracking the problem. Next up, we'll use the fact that 'b' is odd to find the specific values of 'a' and 'b'.

Using the Odd Digit Clue

Okay, remember our key clue? We're looking for an odd number 'ab', which means 'b' has to be an odd digit. This is super helpful because it drastically narrows down our options. We know that 'b' can only be 1, 3, 5, 7, or 9. These are the only odd digits we have to consider.

Now, we also know that a + b = 17. So, we can use this equation along with our possible values for 'b' to figure out what 'a' would be in each case. Let's go through each odd digit for 'b' and see what we get:

  • If b = 1, then a + 1 = 17, which means a = 16. But 'a' has to be a single digit (0-9), so this doesn't work.
  • If b = 3, then a + 3 = 17, which means a = 14. Again, 'a' can't be 14, so this doesn't work either.
  • If b = 5, then a + 5 = 17, which means a = 12. Nope, 'a' still isn't a single digit.
  • If b = 7, then a + 7 = 17, which means a = 10. Still too big for 'a'.
  • If b = 9, then a + 9 = 17, which means a = 8. Bingo! This works!

We've found a solution that fits all our conditions. When b = 9, a = 8. This means our number 'ab' is 89. We used the clue about 'b' being odd to systematically test possibilities and find the right one. Now, to be absolutely sure, we should plug these values back into our original equation and check if they work. Let’s do that next to confirm our answer.

Verifying the Solution

Fantastic! We think we've found our odd number 'ab', which we believe is 89. But in math, it's always a great idea to double-check your work. To verify our solution, we need to plug the values we found, a = 8 and b = 9, back into the original equation: ab + bb + 2a = 204. Remember, 'ab' is 89, and 'bb' is 99.

So, let's substitute these values into the equation:

89 + 99 + 2(8) = 204

Now, let's do the math. First, we calculate 2 times 8, which is 16. So our equation becomes:

89 + 99 + 16 = 204

Next, we add the numbers on the left side: 89 + 99 = 188, and then 188 + 16 = 204. So, our equation now looks like this:

204 = 204

It checks out! The left side equals the right side, which means our solution is correct. We've successfully verified that the odd number 'ab' is indeed 89. This step is super important because it gives us confidence that we haven't made any mistakes along the way. Plus, it’s super satisfying to see everything work out perfectly! So, we've found our answer, verified it, and now we can confidently say we've solved the problem.

Final Answer

Alright guys, we did it! After carefully breaking down the problem, simplifying the equation, and using some logical deduction, we've arrived at our final answer. The odd number 'ab' that satisfies the equation ab + bb + 2a = 204 is 89.

We started by understanding what the problem was asking, and then we transformed the equation into something easier to work with. We figured out that a + b = 17 and used the clue that 'b' had to be an odd digit to narrow down our possibilities. By testing each odd digit, we found that when b = 9, a = 8, giving us the number 89. Finally, we verified our solution by plugging the values back into the original equation and confirming that everything balanced out.

This problem was a fun mix of algebra and logical thinking, and it shows how breaking a complex problem into smaller steps can make it much easier to solve. Great job sticking with it, and I hope you enjoyed this math adventure as much as I did! Keep practicing, and you'll become a math whiz in no time! This whole process really highlights the power of systematic problem-solving. Remember, when you're faced with a tough math problem, take a deep breath, break it down, and tackle it one step at a time. You've got this!