Simplifying Fractions: Find The Equivalent Expression!

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Hey math enthusiasts! Let's dive into a fun problem today: finding an equivalent expression for the fraction −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}}. Don't worry, it might look a little intimidating at first, but we'll break it down step by step and make it super easy to understand. We'll explore the given options and figure out which one is mathematically the same as our original expression. Get ready to flex those fraction muscles! So, which expression is equivalent to $\frac{-2 \frac{1}{4}}{-\frac{2}{3}} $? Let's solve this together!

Understanding the Problem: Equivalent Expressions

Alright, guys, before we jump into the calculations, let's make sure we're all on the same page about what an equivalent expression means. Basically, it's an expression that has the same value as another expression, even if it looks different. Think of it like this: you can have different amounts of money in different types of bills, but the total value is still the same. So, our goal here is to find an expression that, when simplified, gives us the same answer as −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}}. Understanding equivalent expressions is key here, we're not just looking for something that looks similar; it has to be mathematically identical. We're on the hunt for a twin, a doppelganger, a mathematical mirror image! Any questions? Okay, let's keep going.

Converting the Mixed Number

First things first, we need to convert the mixed number in our original expression, −214-2 \frac{1}{4}, into an improper fraction. Remember how to do this? You multiply the whole number by the denominator and add the numerator. Then, you put that result over the original denominator. In our case, it's:

  • (−2×4)+1=−8+1=−9(-2 \times 4) + 1 = -8 + 1 = -9
  • So, −214=−94-2 \frac{1}{4} = -\frac{9}{4}

Now, our original expression looks like this: −94−23\frac{-\frac{9}{4}}{-\frac{2}{3}}. Much cleaner, right? This step is super important. Mess up the conversion, and the whole thing goes sideways. Always double-check your work – it’s easy to make a small error, and those small errors can lead to big problems down the line.

Dividing Fractions: The Key Operation

Next, we have a fraction divided by a fraction. How do we handle this? Well, the trick is to remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is just flipping it over – the numerator becomes the denominator, and the denominator becomes the numerator. So, the reciprocal of −23-\frac{2}{3} is −32-\frac{3}{2}. Now, let's rewrite our expression using multiplication:

  • −94−23=−94÷−23=−94×−32\frac{-\frac{9}{4}}{-\frac{2}{3}} = -\frac{9}{4} \div -\frac{2}{3} = -\frac{9}{4} \times -\frac{3}{2}

See how we changed the division to multiplication and flipped the second fraction? Magic! Keep this rule in mind; it's a fundamental concept when working with fractions. You can't skip this step. Trust me. You'll thank me later.

Simplifying the Multiplication

Now we have a multiplication problem: −94×−32-\frac{9}{4} \times -\frac{3}{2}. When multiplying fractions, we multiply the numerators together and the denominators together. Also, remember that a negative times a negative is a positive. So, let's do it:

  • (−9×−3)=27(-9 \times -3) = 27
  • (4×2)=8(4 \times 2) = 8
  • Therefore, −94×−32=278-\frac{9}{4} \times -\frac{3}{2} = \frac{27}{8}

We now have 278\frac{27}{8} as our answer. But the original question asks for equivalent expressions, not the final simplified answer. So now we've done the hardest part. Congratulations, we're almost there! Pat yourselves on the back, you all deserve it!

Evaluating the Answer Choices

Okay, now that we've simplified our original expression, let's look at the answer choices. Remember, we're looking for an expression that, when simplified, equals 278\frac{27}{8}. Let's go through the choices one by one.

Analyzing Option A: 94÷32\frac{9}{4} \div \frac{3}{2}

First, let's deal with Option A: 94÷32\frac{9}{4} \div \frac{3}{2}.

  • To solve this, we convert the division into multiplication by the reciprocal: 94÷32=94×23\frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3}
  • Now multiply: 9×24×3=1812\frac{9 \times 2}{4 \times 3} = \frac{18}{12}
  • Simplify the fraction by dividing both numerator and denominator by 6: 1812=32\frac{18}{12} = \frac{3}{2}

32\frac{3}{2} is not equal to 278\frac{27}{8}, so Option A is incorrect. I know that may seem a little difficult, but don't worry, you guys can do this! We are almost done.

Analyzing Option B: −94÷(−23)-\frac{9}{4} \div \left(-\frac{2}{3}\right)

Next up, Option B: −94÷(−23)-\frac{9}{4} \div \left(-\frac{2}{3}\right). Let's break it down.

  • Convert division to multiplication using the reciprocal: −94÷(−23)=−94×−32-\frac{9}{4} \div \left(-\frac{2}{3}\right) = -\frac{9}{4} \times -\frac{3}{2}
  • Multiply the fractions: −9×−3=27-9 \times -3 = 27 and 4×2=84 \times 2 = 8, so the result is 278\frac{27}{8}.

This matches our simplified original expression. Therefore, Option B is correct. Let's keep going to verify our answer, just to be sure!

Analyzing Option C: −94÷23-\frac{9}{4} \div \frac{2}{3}

Let's evaluate option C: −94÷23-\frac{9}{4} \div \frac{2}{3}.

  • Convert division to multiplication: −94÷23=−94×32-\frac{9}{4} \div \frac{2}{3} = -\frac{9}{4} \times \frac{3}{2}
  • Multiply the fractions: −9×34×2=−278\frac{-9 \times 3}{4 \times 2} = -\frac{27}{8}

Since −278-\frac{27}{8} is not equal to 278\frac{27}{8}, Option C is incorrect. We're getting closer to our conclusion. This is going to be easy.

Analyzing Option D: 94÷(−32)\frac{9}{4} \div \left(-\frac{3}{2}\right)

Finally, let's look at Option D: 94÷(−32)\frac{9}{4} \div \left(-\frac{3}{2}\right).

  • Change to multiplication using the reciprocal: 94÷(−32)=94×−23\frac{9}{4} \div \left(-\frac{3}{2}\right) = \frac{9}{4} \times -\frac{2}{3}
  • Multiply: 9×−24×3=−1812\frac{9 \times -2}{4 \times 3} = -\frac{18}{12}
  • Simplify: −1812=−32- \frac{18}{12} = -\frac{3}{2}

Since −32-\frac{3}{2} is not equal to 278\frac{27}{8}, Option D is also incorrect.

Conclusion: The Answer Revealed!

Alright, guys, after all that work, we've found our answer. The expression equivalent to −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}} is Option B: −94÷(−23)-\frac{9}{4} \div \left(-\frac{2}{3}\right). Remember, the key is to take it slow, convert mixed numbers to improper fractions, and remember how to divide fractions (multiply by the reciprocal!). Keep practicing, and you'll become fraction masters in no time! So, which expression is equivalent to $\frac{-2 \frac{1}{4}}{-\frac{2}{3}} $? Option B is correct. We did it!

We took the time to explain everything. We hope this was helpful! We'll see you in the next one!