Dividing Mixed Numbers: A Step-by-Step Guide

Hey there, math enthusiasts! Ever found yourself scratching your head when faced with dividing mixed numbers? Don't worry; you're not alone! Mixed numbers can seem a bit intimidating at first, but with the right approach, dividing them becomes a piece of cake. In this guide, we'll break down the process step by step, using the example of $8 \frac{2}{3} \div 1 \frac{7}{9}$. So, let's dive in and conquer those fractions!

Understanding Mixed Numbers and Division

Before we jump into the division process, let's quickly recap what mixed numbers are and how division works with fractions. This foundational knowledge will make the steps much clearer and easier to remember.

What are Mixed Numbers?

A mixed number is simply a combination of a whole number and a proper fraction. Think of it as having some whole units and a part of another unit. For instance, $8 \frac{2}{3}$ means we have 8 whole units and an additional two-thirds of another unit. The whole number part is 8, and the fractional part is $\frac{2}{3}$. Mixed numbers are common in everyday life, from measuring ingredients in a recipe to figuring out how much time you've spent on a task.

Dividing Fractions: The Basics

Dividing by a fraction might seem a bit strange at first. Instead of thinking about how many times one number fits into another, we're essentially asking, "How many of this fraction are there in this other number?" The key to dividing fractions is to remember the phrase "keep, change, flip." This means we keep the first fraction, change the division sign to multiplication, and flip (find the reciprocal of) the second fraction. For example, $\frac{1}{2} \div \frac{1}{4}$ becomes $\frac{1}{2} \times \frac{4}{1}$.

Step 1: Convert Mixed Numbers to Improper Fractions

This is the crucial first step when dividing mixed numbers. Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Converting to improper fractions makes the division process much smoother.

How to Convert

To convert a mixed number to an improper fraction, follow these steps:

1. Multiply the whole number by the denominator of the fraction.
2. Add the numerator of the fraction to the result from step 1.
3. Write this sum as the new numerator, keeping the same denominator as the original fraction.

Let's apply this to our example, $8 \frac{2}{3}$:

1. Multiply the whole number (8) by the denominator (3): 8 * 3 = 24
2. Add the numerator (2) to the result: 24 + 2 = 26
3. Write this sum (26) as the new numerator, keeping the denominator (3): $\frac{26}{3}$

So, $8 \frac{2}{3}$ is equivalent to the improper fraction $\frac{26}{3}$.

Now, let's convert $1 \frac{7}{9}$ to an improper fraction:

1. Multiply the whole number (1) by the denominator (9): 1 * 9 = 9
2. Add the numerator (7) to the result: 9 + 7 = 16
3. Write this sum (16) as the new numerator, keeping the denominator (9): $\frac{16}{9}$

Thus, $1 \frac{7}{9}$ is equivalent to the improper fraction $\frac{16}{9}$.

Step 2: Apply "Keep, Change, Flip"

Now that we have our mixed numbers converted to improper fractions, we can apply the "keep, change, flip" rule. This will transform our division problem into a multiplication problem, which is much easier to handle.

Keep, Change, Flip in Action

Remember, "keep, change, flip" means:

• Keep the first fraction as it is.
• Change the division sign ($\div$) to a multiplication sign ($\times$).
• Flip the second fraction (find its reciprocal).

Our problem is now $\frac{26}{3} \div \frac{16}{9}$. Applying the rule, we get:

• Keep $\frac{26}{3}$
• Change $\div$ to $\times$
• Flip $\frac{16}{9}$ to $\frac{9}{16}$

So, our new problem is $\frac{26}{3} \times \frac{9}{16}$.

Step 3: Multiply the Fractions

Multiplying fractions is straightforward: simply multiply the numerators together and the denominators together.

Multiplying Numerators and Denominators

In our case, we have $\frac{26}{3} \times \frac{9}{16}$.

• Multiply the numerators: 26 * 9 = 234
• Multiply the denominators: 3 * 16 = 48

This gives us the fraction $\frac{234}{48}$.

Step 4: Simplify the Resulting Fraction

The fraction $\frac{234}{48}$ is a bit unwieldy. We need to simplify it to its simplest form. This means finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

Finding the Greatest Common Factor (GCF)

The GCF is the largest number that divides evenly into both the numerator and the denominator. There are several ways to find the GCF, but one common method is to list the factors of each number and identify the largest one they have in common.

• Factors of 234: 1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234
• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The greatest common factor of 234 and 48 is 6.

Dividing by the GCF

Now, divide both the numerator and the denominator by the GCF (6):

• 234 \div 6 = 39
• 48 \div 6 = 8

This simplifies our fraction to $\frac{39}{8}$.

Step 5: Convert Back to a Mixed Number (If Necessary)

The problem asked for the answer as a fraction or a mixed number in simplest form. Since $\frac{39}{8}$ is an improper fraction, we should convert it back to a mixed number.

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction to a mixed number, follow these steps:

1. Divide the numerator by the denominator.
2. Write the quotient (the whole number result of the division) as the whole number part of the mixed number.
3. Write the remainder as the numerator of the fractional part, keeping the same denominator as the original fraction.

Let's apply this to $\frac{39}{8}$:

1. Divide 39 by 8: 39 \div 8 = 4 with a remainder of 7
2. Write the quotient (4) as the whole number part.
3. Write the remainder (7) as the numerator of the fractional part, keeping the denominator (8).

This gives us the mixed number $4 \frac{7}{8}$.

Final Answer

So, $8 \frac{2}{3} \div 1 \frac{7}{9} = 4 \frac{7}{8}$.

Practice Makes Perfect

Dividing mixed numbers might seem like a lot of steps at first, but with practice, it becomes second nature. The key is to follow each step carefully and understand why you're doing it. Remember to convert to improper fractions, apply "keep, change, flip," multiply, simplify, and convert back to a mixed number if needed. Keep practicing, and you'll master dividing mixed numbers in no time!

For further learning and practice, you might find helpful resources on websites like Khan Academy, which offers numerous lessons and exercises on fractions and mixed numbers.