Finding The Expression With An Estimated Product Of 45

Hey everyone! Today, we're diving into a fun math problem where we need to figure out which expression gives us an estimated product of 45. This means we're not looking for the exact answer, but rather which one is closest to 45 when we round the numbers and multiply. We've got three options to explore: A. $4.67 imes 21$, B. $8.4^2 + 5.20$, and C. $3.81 imes 7.8$. Let's break down each one step-by-step to see which one fits the bill.

Understanding Estimated Products

Before we jump into the calculations, let's quickly recap what an estimated product is. When we estimate, we're essentially rounding numbers to make the math easier. This is super useful in real life when you need a quick answer without pulling out a calculator. For example, if you're at the grocery store and want to know if you have enough money, you might round the prices of the items to the nearest dollar and add them up in your head. That's estimation in action! In our case, we're estimating the product, which means the result of multiplication. So, we'll be rounding the numbers in our expressions and then multiplying to see which gives us a result close to 45. This is not about finding the exact answer but making a reasonable guess based on simplified numbers. Think of it as a mental math workout! You're training your brain to quickly approximate values, which is a fantastic skill for everyday situations. Plus, it helps you develop a better number sense, understanding how numbers relate to each other and how operations affect them. Estimating products also helps in checking the reasonableness of your actual calculations later on. If your exact answer is wildly different from your estimated product, it might be a sign that you've made a mistake somewhere. So, let's get started and see which expression lands closest to our target of 45.

A. $4.67 imes 21$

Let's tackle option A: $4.67 imes 21$. The first step in estimating is to round the numbers. We can round $4.67$ to the nearest whole number, which would be $5$. And $21$ is already a pretty clean number, so we can leave it as is. Now, we have a much simpler problem: $5 imes 21$. This is something we can easily do in our heads. $5$ times $20$ is $100$, and $5$ times $1$ is $5$, so $5 imes 21$ is $100 + 5 = 105$. Okay, so our estimated product for option A is $105$. Now, we compare this estimate to our target of $45$. $105$ is significantly larger than $45$, so it seems like option A is not the expression we're looking for. But hey, that's perfectly alright! We're just exploring the options and eliminating the ones that don't fit. This process of estimation and comparison is key to solving problems like this. We're not just blindly calculating; we're using our number sense to make informed decisions. Also, notice how rounding the numbers made the multiplication much easier. Instead of dealing with decimals, we had nice, whole numbers to work with. This is the power of estimation! It simplifies complex calculations and gives us a good ballpark figure. So, with option A out of the way, let's move on to the next one and see if we get closer to our target of 45.

B. $8.4^2 + 5.20$

Now, let's dive into option B: $8.4^2 + 5.20$. This one looks a little different because we have an exponent and addition involved. But don't worry, we'll break it down step by step. First, let's focus on the exponent, $8.4^2$. This means $8.4$ multiplied by itself, or $8.4 imes 8.4$. To estimate this, we can round $8.4$ to the nearest whole number, which is $8$. So, we're estimating $8 imes 8$, which is $64$. Now, let's move on to the second part of the expression, $+ 5.20$. We can round $5.20$ to the nearest whole number, which is $5$. So, now our estimated expression looks like this: $64 + 5$. This is a simple addition problem. $64$ plus $5$ is $69$. So, our estimated value for option B is $69$. Comparing this to our target of $45$, we can see that $69$ is quite a bit larger. This indicates that option B is also likely not the expression we're looking for. It's important to note here that the order of operations matters. We had to deal with the exponent first before doing the addition. This is a fundamental concept in math, and it's crucial to follow the correct order to get the right answer (or in this case, the right estimate). So, with options A and B both seeming too large, let's move on to the final option, C, and see if it brings us closer to our target of 45.

C. $3.81 imes 7.8$

Alright, let's examine option C: $3.81 imes 7.8$. This looks like a straightforward multiplication problem, just like option A. To estimate, we'll round each number to the nearest whole number. $3.81$ is close to $4$, so we'll round it up. And $7.8$ is close to $8$, so we'll round that up as well. Now, we have the simplified problem: $4 imes 8$. This is a multiplication fact we probably know by heart: $4$ times $8$ is $32$. So, our estimated product for option C is $32$. Now, let's compare this to our target of $45$. $32$ is closer to $45$ than $105$ (from option A) or $69$ (from option B). This makes option C the most likely candidate for the expression with an estimated product of $45$. It's important to remember that we're estimating, so the answer won't be exact. But $32$ is the closest we've gotten so far. This highlights the importance of considering all options and comparing their estimates to the target value. We didn't just stop at the first one that seemed close; we evaluated all three to make the most informed decision. And that's a great strategy for problem-solving in general! So, based on our estimations, option C seems to be the winner. But let's do a quick recap to make sure we're confident in our answer.

Final Answer: The Expression with an Estimated Product of 45

Okay, let's recap what we've done. We were looking for the expression with an estimated product of 45. We had three options:

• A. $4.67 imes 21$ (estimated to $105$)
• B. $8.4^2 + 5.20$ (estimated to $69$)
• C. $3.81 imes 7.8$ (estimated to $32$)

By rounding the numbers and performing the operations, we found that option C, $3.81 imes 7.8$, had the estimated product closest to 45. While the exact product isn't 45, our estimation process helped us identify the most likely candidate. Estimation is a powerful tool in mathematics. It allows us to quickly approximate answers and check the reasonableness of our calculations. In this case, it helped us narrow down the options and confidently choose the expression with the estimated product closest to 45. So, the final answer is C. $3.81 imes 7.8$.

I hope this detailed explanation helped you understand how to estimate products and solve this type of problem. Remember, practice makes perfect, so keep honing your estimation skills! You got this!