Divide Mixed Numbers: Simplify 5 1/2 ÷ 2 3/4

by Luna Greco 45 views

Hey guys! Today, we're diving into the world of mixed numbers and complex fractions. Specifically, we're going to tackle a problem where we need to divide mixed numbers and simplify the result to its lowest terms. Let's take the example of 5 1/2 divided by 2 3/4. This might seem a bit daunting at first, but don't worry, we'll break it down step by step so you'll be a pro at this in no time!

Understanding Mixed Numbers and Complex Fractions

Before we jump into solving the problem, let's make sure we're all on the same page about what mixed numbers and complex fractions actually are. Mixed numbers are numbers that combine a whole number and a fraction, like our 5 1/2 and 2 3/4. The whole number part tells us how many whole units we have, and the fraction part tells us what portion of another unit we have. So, 5 1/2 means we have five whole units and a half of another unit. On the other hand, Complex fractions are fractions where either the numerator, the denominator, or both contain a fraction. When we are dividing mixed numbers, we are essentially dealing with a complex fraction.

Step 1: Converting Mixed Numbers to Improper Fractions

The first thing we need to do when dividing mixed numbers is to convert them into improper fractions. This makes the division process much easier. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert a mixed number to an improper fraction, we follow these steps:

  1. Multiply the whole number by the denominator of the fraction.
  2. Add the numerator of the fraction to the result.
  3. Keep the same denominator as the original fraction.

Let's apply this to our example. For 5 1/2:

  1. Multiply 5 (the whole number) by 2 (the denominator): 5 * 2 = 10
  2. Add 1 (the numerator) to the result: 10 + 1 = 11
  3. Keep the same denominator (2): So, 5 1/2 becomes 11/2.

Now, let's do the same for 2 3/4:

  1. Multiply 2 (the whole number) by 4 (the denominator): 2 * 4 = 8
  2. Add 3 (the numerator) to the result: 8 + 3 = 11
  3. Keep the same denominator (4): So, 2 3/4 becomes 11/4.

Now we've successfully converted our mixed numbers into improper fractions: 5 1/2 = 11/2 and 2 3/4 = 11/4. This is a crucial first step because it transforms our problem into a more manageable form for division.

Step 2: Dividing Fractions: Keep, Change, Flip!

Okay, now that we have our improper fractions, we're ready to divide! Dividing fractions might seem tricky, but there's a simple rule that makes it super easy: Keep, Change, Flip!

What does that mean? Well:

  • Keep: Keep the first fraction the same.
  • Change: Change the division sign to a multiplication sign.
  • Flip: Flip the second fraction (this means swapping the numerator and the denominator – it's also called taking the reciprocal).

Let's apply this to our problem, which is now (11/2) ÷ (11/4):

  1. Keep: Keep the first fraction, 11/2, as it is.
  2. Change: Change the division sign (÷) to a multiplication sign (*).
  3. Flip: Flip the second fraction, 11/4, to become 4/11.

So, our problem now looks like this: (11/2) * (4/11). We've transformed a division problem into a multiplication problem, which is much easier to handle!

Step 3: Multiplying Fractions

Multiplying fractions is straightforward: we simply multiply the numerators together and the denominators together. So, for (11/2) * (4/11):

  • Multiply the numerators: 11 * 4 = 44
  • Multiply the denominators: 2 * 11 = 22

This gives us the fraction 44/22. We're almost there!

Step 4: Simplifying the Fraction to Lowest Terms

The final step is to simplify our fraction to its lowest terms. This means finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by it. The GCF is the largest number that divides evenly into both numbers.

In our case, we have the fraction 44/22. What's the greatest common factor of 44 and 22? Well, both numbers are divisible by 22! So, we divide both the numerator and the denominator by 22:

  • 44 ÷ 22 = 2
  • 22 ÷ 22 = 1

This gives us the simplified fraction 2/1, which is simply equal to 2. Therefore, the final answer to the problem 5 1/2 divided by 2 3/4, simplified to its lowest terms, is 2.

Conclusion

So, guys, we've successfully navigated through dividing mixed numbers and simplifying the result! Remember the key steps:

  1. Convert mixed numbers to improper fractions.
  2. Use "Keep, Change, Flip" to turn division into multiplication.
  3. Multiply the fractions.
  4. Simplify the result to its lowest terms.

With these steps in mind, you'll be able to tackle any complex fraction problem that comes your way. Keep practicing, and you'll become a math whiz in no time! Now you know that when reducing the complex fraction with mixed numbers (5 1/2 divided by 2 3/4) to the lowest terms, the solution will result in 2, which corresponds to option A. Keep up the great work, and I'll see you in the next math adventure!