Find Expressions With A Quotient Of 9

Hey math whizzes! Today, we're diving into the world of division and looking for some specific results. Our mission, should we choose to accept it, is to identify which of the given expressions result in a quotient of exactly 9. This means we need to perform the division for each option and see if it hits the bullseye of 9. It's like a mathematical treasure hunt, and only the correct divisions will lead us to the treasure!

We've got four expressions lined up: 1.35 ÷ 1.5, 1.35 ÷ 0.15, 13.5 ÷ 1.5, and 13.5 ÷ 0.15. Each of these involves decimals, which can sometimes throw people off, but don't you worry! We'll break them down step-by-step. Remember, a quotient is simply the answer you get when you divide one number by another. So, we're essentially asking: Which of these division problems equals 9?

Let's get our calculators ready (or our trusty mental math skills!) and start crunching those numbers. This isn't just about finding the answer; it's about understanding how decimal division works and reinforcing our arithmetic abilities. So, grab a snack, get comfy, and let's tackle this challenge together. We'll explore each option, show the work, and finally reveal which ones have that magical quotient of 9. Ready to jump in? Let's go!

Diving into the Divisions: A Step-by-Step Approach

Alright guys, let's get down to business and actually do the math. For each expression, we'll calculate the quotient and see if it matches our target of 9. It's super important to be careful with the decimal points when dividing, as a misplaced decimal can lead you down the wrong path entirely. We'll tackle them one by one, so no need to feel overwhelmed. Think of it as a guided tour through these calculations.

Expression 1: 1.35 ÷ 1.5

Our first contestant is 1.35 divided by 1.5. When dealing with decimal division, a common trick is to make the divisor (the second number) a whole number. We can do this by multiplying both numbers by the same power of 10. In this case, since 1.5 has one decimal place, we'll multiply both 1.35 and 1.5 by 10. So, the expression becomes (1.35 * 10) ÷ (1.5 * 10), which simplifies to 13.5 ÷ 1.5.

Now, how do we divide 13.5 by 1.5? We can again make the divisor a whole number by multiplying by 10: (13.5 * 10) ÷ (1.5 * 10) = 135 ÷ 15.

Let's think about our multiplication facts. What number multiplied by 15 gives us 135? We know that 15 * 10 = 150, which is a bit too high. Let's try a number slightly less than 10. How about 15 * 9?

15 * 9 = (10 + 5) * 9 = (10 * 9) + (5 * 9) = 90 + 45 = 135.

Bingo! So, 135 ÷ 15 = 9. Therefore, 1.35 ÷ 1.5 = 0.9.

Wait a minute! Our target quotient is 9, and we got 0.9. That means this first expression is not one of our answers. It's close, but not quite there. Keep that in mind, guys, because sometimes a decimal place difference is all it takes!

Expression 2: 1.35 ÷ 0.15

Next up, we have 1.35 divided by 0.15. Again, let's make our divisor, 0.15, a whole number. It has two decimal places, so we'll multiply both numbers by 100.

(1.35 * 100) ÷ (0.15 * 100) = 135 ÷ 15.

Look familiar? We just calculated this in the previous step! We found that 135 divided by 15 is exactly 9.

So, 1.35 ÷ 0.15 = 9.

Hooray! We've found our first expression that has a quotient of 9. Make a note of this one, because it's definitely a winner in our hunt.

Expression 3: 13.5 ÷ 1.5

Moving on to our third expression: 13.5 divided by 1.5. Let's make the divisor, 1.5, a whole number by multiplying both numbers by 10.

(13.5 * 10) ÷ (1.5 * 10) = 135 ÷ 15.

And again, we've landed on the same division problem: 135 ÷ 15. We already know the answer to this one, don't we? It's 9!

So, 13.5 ÷ 1.5 = 9.

Awesome! Another one for the books. This expression also gives us a quotient of 9. We're on a roll now!

Expression 4: 13.5 ÷ 0.15

Finally, let's check out our last expression: 13.5 divided by 0.15. To make the divisor, 0.15, a whole number, we need to multiply by 100 because it has two decimal places.

(13.5 * 100) ÷ (0.15 * 100) = 1350 ÷ 15.

This one is a bit different. We have 1350 divided by 15. We know that 135 ÷ 15 = 9. Since 1350 is 10 times larger than 135, the result of dividing 1350 by 15 should also be 10 times larger than 9.

So, 1350 ÷ 15 = 9 * 10 = 90.

Therefore, 13.5 ÷ 0.15 = 90.

This result, 90, is not our target quotient of 9. So, this last expression is not part of our solution set.

The Grand Reveal: Which Expressions Have a Quotient of 9?

After carefully working through each division, we can now confidently state which expressions meet our criteria. It's been a journey, exploring the nuances of decimal division, and hopefully, you've gained a clearer understanding of how it all works. Remember, the key is often to manipulate the numbers so you're dividing whole numbers, making the calculation much more straightforward.

Let's recap our findings:

• 1.35 ÷ 1.5 = 0.9 (Not 9)
• 1.35 ÷ 0.15 = 9 (Success!)
• 13.5 ÷ 1.5 = 9 (Success!)
• 13.5 ÷ 0.15 = 90 (Not 9)

So, the expressions that have a quotient of 9 are:

• 1.35 ÷ 0.15
• 13.5 ÷ 1.5

These are the two expressions that, when you perform the division, give you the answer 9. Pretty neat how changing the decimal places can significantly alter the quotient, right? It highlights the importance of precision in mathematics.

Why Understanding Decimal Division Matters

Guys, this exercise isn't just about picking out answers from a list. It's about building a solid foundation in arithmetic, especially with decimals. Understanding how to correctly divide decimals is a crucial skill that pops up in tons of real-world scenarios. Think about splitting a bill at a restaurant where the total has cents, calculating recipes where you need to divide ingredients, or even managing your budget. Accurate division is key!

The process of making the divisor a whole number by multiplying both the dividend and the divisor by the same power of 10 is a fundamental technique. It doesn't change the value of the division; it just makes it easier for us to compute. For example, dividing 13.5 by 1.5 is the same as dividing 135 by 15. This simplification allows us to rely on our knowledge of whole number division, which we've likely practiced much more.

It's also a fantastic way to build number sense. When you look at 1.35 ÷ 0.15, you can estimate. You know that 0.15 is a small number, so dividing by it should result in a larger number than the dividend (1.35). Comparing it to 1.35 ÷ 1.5, where the divisor is larger, you'd expect a smaller quotient. This kind of estimation and comparison helps you develop an intuitive feel for numbers and their relationships.

Furthermore, mastering these types of problems reinforces the concept of place value. The position of a digit in a number determines its value. When you multiply or divide by powers of 10, you're essentially shifting those digits and their values. Recognizing this helps you understand why the decimal division rules work the way they do.

So, next time you encounter a division problem with decimals, don't shy away from it! Break it down, use the techniques we discussed, and remember the underlying mathematical principles. It’s all part of becoming a math superstar! Keep practicing, and you'll be a decimal division pro in no time.

Conclusion: Your Math Skills Are Awesome!

We’ve successfully navigated through the division expressions and pinpointed the ones that yield a quotient of 9. It’s really satisfying when you can solve a problem like this, isn't it? The expressions 1.35 ÷ 0.15 and 13.5 ÷ 1.5 are your winners. They both result in that clean, crisp answer of 9.

Remember the strategy we used: converting the division problem into one involving whole numbers by multiplying both the dividend and the divisor by the appropriate power of 10. This simple yet powerful technique makes decimal division manageable and less intimidating. It’s a fundamental skill that underpins many mathematical concepts and real-world applications.

Keep flexing those mathematical muscles, guys! The more you practice problems like these, the more confident and capable you’ll become. Math is a journey, and every problem solved is a step forward. Don't be afraid to experiment with different numbers and scenarios. The world of mathematics is vast and full of fascinating discoveries waiting for you. So, keep exploring, keep questioning, and most importantly, keep learning!