Dividing Mixed Numbers: A Step-by-Step Guide

Hey guys! Let's dive into a math problem together. Today, we're tackling the division of a mixed number by a fraction: $-5 \frac{2}{5} \div(-\frac{3}{5})$. Don't worry; it's not as scary as it looks! We'll break it down step by step so you can nail it every time. Ready? Let's get started!

Converting Mixed Numbers to Improper Fractions

First things first, we need to convert the mixed number $-5 \frac{2}{5}$ into an improper fraction. This is a crucial initial step because it simplifies the division process immensely. So, how do we do it?

Think of a mixed number as a combination of a whole number and a fraction. In our case, we have -5 wholes and an additional $\frac{2}{5}$. To convert this, we multiply the whole number by the denominator of the fraction and then add the numerator. This result becomes the new numerator, while the denominator stays the same. Mathematically, it looks like this:

$-5 \frac{2}{5} = -\frac{(5 \times 5) + 2}{5}$

Let's break it down further:

1. Multiply the whole number by the denominator: $5 \times 5 = 25$.
2. Add the numerator: $25 + 2 = 27$.
3. Place the result over the original denominator: $\frac{27}{5}$.
4. Don't forget the negative sign! Since our original mixed number was negative, our improper fraction is also negative: $-\frac{27}{5}$.

So, $-5 \frac{2}{5}$ converts to $-\frac{27}{5}$. Now, why is this important? Working with improper fractions allows us to perform mathematical operations, like division, much more easily than if we were to stick with mixed numbers. It transforms the problem into a straightforward fraction division, which we will tackle in the next section. Trust me; mastering this conversion is a game-changer for all your fraction-related calculations. It's like unlocking a secret level in a video game – suddenly, everything becomes much clearer and more manageable. Plus, you'll look like a math whiz to your friends!

Dividing Fractions: Keep, Change, Flip

Alright, now that we've transformed our mixed number into an improper fraction, let's talk about dividing fractions. The golden rule here is: "Keep, Change, Flip." This simple mnemonic is your best friend when dividing fractions, and it's super easy to remember. So, what does it mean?

We start with our problem: $-\frac{27}{5} \div(-\frac{3}{5})$.

1. Keep: Keep the first fraction exactly as it is. In our case, $-\frac{27}{5}$ stays the same.
2. Change: Change the division sign to a multiplication sign. So, $\div$ becomes $\times$.
3. Flip: Flip the second fraction (the divisor) by swapping its numerator and denominator. This is also known as finding the reciprocal. So, $-\frac{3}{5}$ becomes $-\frac{5}{3}$.

Now, our problem looks like this: $-\frac{27}{5} \times(-\frac{5}{3})$.

This transformation is crucial because dividing by a fraction is the same as multiplying by its reciprocal. Think of it like this: instead of asking how many times $\frac{3}{5}$ fits into $-\frac{27}{5}$, we're asking what is $-\frac{27}{5}$ multiplied by the inverse of $\frac{3}{5}$. This makes the calculation much more straightforward.

Now that we've "Kept, Changed, and Flipped," we're ready to multiply our fractions. Remember, multiplying fractions is generally easier than dividing them, so this step simplifies the whole process. We're on the home stretch now – keep going!

Multiplying Fractions: Numerator Times Numerator, Denominator Times Denominator

Now comes the fun part: multiplying fractions! Remember, when multiplying fractions, we simply multiply the numerators together and the denominators together. It's a straightforward process that turns our division problem into a much simpler multiplication problem.

So, we have: $-\frac{27}{5} \times(-\frac{5}{3})$.

1. Multiply the numerators: $-27 \times -5 = 135$. Remember, a negative times a negative equals a positive, so we get a positive 135.
2. Multiply the denominators: $5 \times 3 = 15$.

Now, we have a new fraction: $\frac{135}{15}$. This fraction represents the result of our multiplication. However, it's not in its simplest form yet. We need to simplify it to get our final answer. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD.

In our case, the GCD of 135 and 15 is 15. So, we divide both the numerator and the denominator by 15:

• $\frac{135}{15} = \frac{135 \div 15}{15 \div 15} = \frac{9}{1} = 9$

Therefore, $-\frac{27}{5} \times(-\frac{5}{3}) = 9$. This is our final answer! Multiplying fractions is a fundamental skill in mathematics, and it's essential for solving various problems, from basic arithmetic to more complex algebra. So, make sure you're comfortable with this process. With a little practice, you'll be multiplying fractions like a pro!

Simplifying the Result

After performing the multiplication, we ended up with the fraction $\frac{135}{15}$. While this is technically the correct answer, it's not in its simplest form. Simplifying fractions is essential because it presents the answer in its most concise and understandable form. Plus, it's generally considered good mathematical practice.

To simplify $\frac{135}{15}$, we need to find the greatest common divisor (GCD) of 135 and 15. The GCD is the largest number that divides both 135 and 15 without leaving a remainder. In this case, the GCD is 15.

Now, we divide both the numerator and the denominator by the GCD:

$\frac{135}{15} = \frac{135 \div 15}{15 \div 15} = \frac{9}{1}$

Since any number divided by 1 is just the number itself, $\frac{9}{1}$ simplifies to 9. Therefore, the simplified result of our division problem is 9.

Simplifying fractions is a critical skill in mathematics. It not only makes the answer easier to understand but also helps in further calculations. Always remember to simplify your fractions to their lowest terms to ensure accuracy and clarity in your work. Whether you're dealing with basic arithmetic or advanced calculus, simplifying fractions will always be a valuable tool in your mathematical arsenal.

Final Answer

So, after all the steps, we found that: $-5 \frac{2}{5} \div(-\frac{3}{5}) = 9$.

Isn't that awesome? You've successfully navigated through converting mixed numbers, dividing fractions, and simplifying the result. Give yourself a pat on the back; you've earned it!