How Do I Divide A Fraction By Another Fraction

Dividing fractions might seem daunting at first, but it’s actually a straightforward process once you understand the underlying concept. The key is to remember the phrase "keep, change, flip," which summarizes the three steps involved in dividing fractions. This article will break down the process step-by-step, provide examples, and explain why this method works. Whether you’re a student learning fractions for the first time or someone looking to refresh your math skills, this guide will help you confidently divide any fraction by another fraction.

Understanding Fractions

Before diving into the division process, let's quickly review what fractions are and their basic components. A fraction represents a part of a whole. It consists of two numbers:

• Numerator: The top number, which indicates how many parts of the whole you have.
• Denominator: The bottom number, which indicates the total number of equal parts the whole is divided into.

For example, in the fraction 3/4, 3 is the numerator and 4 is the denominator. This means you have 3 parts out of a total of 4.

Fractions can be proper (numerator is less than the denominator, like 1/2), improper (numerator is greater than or equal to the denominator, like 5/3), or mixed numbers (a whole number and a fraction, like 2 1/4). When dividing fractions, it's often easier to work with improper fractions rather than mixed numbers.

The "Keep, Change, Flip" Method: A Step-by-Step Guide

Dividing fractions boils down to a simple three-step process known as "keep, change, flip," also sometimes referred to as "keep, change, invert." Let’s break down each step:

Step 1: Keep the First Fraction

The first fraction in the division problem remains unchanged. You simply rewrite it exactly as it is. This is the "keep" part of the mnemonic.

Step 2: Change the Division Sign to Multiplication

The division operation is transformed into multiplication. This is the "change" step. This transformation is crucial because dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

Step 3: Flip the Second Fraction (Invert)

The second fraction, the one you are dividing by, is inverted. This means you swap the numerator and the denominator. The numerator becomes the denominator, and the denominator becomes the numerator. This is the "flip" or "invert" step. The inverted fraction is called the reciprocal of the original fraction.

Example:

Let's say you want to divide 1/2 by 3/4.

1. Keep: Keep the first fraction (1/2) as it is.
2. Change: Change the division sign (÷) to a multiplication sign (×).
3. Flip: Flip the second fraction (3/4) to its reciprocal (4/3).

Now the problem becomes: 1/2 × 4/3

Performing the Multiplication

Once you've applied the "keep, change, flip" method, you're left with a multiplication problem. Multiplying fractions is straightforward:

1. Multiply the Numerators: Multiply the numerators of the two fractions together.
2. Multiply the Denominators: Multiply the denominators of the two fractions together.
3. Simplify (if possible): Reduce the resulting fraction to its simplest form by dividing both the numerator and the denominator by their greatest common factor (GCF).

Continuing the Example:

1. Multiply Numerators: 1 × 4 = 4
2. Multiply Denominators: 2 × 3 = 6
3. Result: 4/6

Simplifying the Result:

The fraction 4/6 can be simplified because both 4 and 6 are divisible by 2.

4 ÷ 2 = 2

6 ÷ 2 = 3

Therefore, the simplified answer is 2/3.

So, 1/2 ÷ 3/4 = 2/3

Examples with Different Types of Fractions

Let's explore more examples to solidify your understanding of dividing fractions.

Example 1: Dividing a Proper Fraction by a Proper Fraction

Problem: 2/5 ÷ 1/3

1. Keep: 2/5
2. Change: ÷ to ×
3. Flip: 1/3 to 3/1

New problem: 2/5 × 3/1

1. Multiply Numerators: 2 × 3 = 6
2. Multiply Denominators: 5 × 1 = 5
3. Result: 6/5

The answer is 6/5, which is an improper fraction. You can leave it as an improper fraction or convert it to a mixed number. To convert it to a mixed number, divide 6 by 5. The quotient is 1, and the remainder is 1. So, the mixed number is 1 1/5.

Example 2: Dividing an Improper Fraction by a Proper Fraction

Problem: 7/4 ÷ 2/3

1. Keep: 7/4
2. Change: ÷ to ×
3. Flip: 2/3 to 3/2

New problem: 7/4 × 3/2

1. Multiply Numerators: 7 × 3 = 21
2. Multiply Denominators: 4 × 2 = 8
3. Result: 21/8

The answer is 21/8, an improper fraction. Convert it to a mixed number by dividing 21 by 8. The quotient is 2, and the remainder is 5. So, the mixed number is 2 5/8.

Example 3: Dividing a Proper Fraction by an Improper Fraction

Problem: 3/8 ÷ 5/2

1. Keep: 3/8
2. Change: ÷ to ×
3. Flip: 5/2 to 2/5

New problem: 3/8 × 2/5

1. Multiply Numerators: 3 × 2 = 6
2. Multiply Denominators: 8 × 5 = 40
3. Result: 6/40

Simplify the fraction by dividing both numerator and denominator by their GCF, which is 2.

6 ÷ 2 = 3

40 ÷ 2 = 20

The simplified answer is 3/20.

Example 4: Dividing Mixed Numbers

Problem: 2 1/2 ÷ 1 3/4

Before applying "keep, change, flip," convert the mixed numbers to improper fractions.

• To convert 2 1/2 to an improper fraction: (2 × 2) + 1 = 5. So, 2 1/2 = 5/2
• To convert 1 3/4 to an improper fraction: (1 × 4) + 3 = 7. So, 1 3/4 = 7/4

Now the problem is: 5/2 ÷ 7/4

1. Keep: 5/2
2. Change: ÷ to ×
3. Flip: 7/4 to 4/7

New problem: 5/2 × 4/7

1. Multiply Numerators: 5 × 4 = 20
2. Multiply Denominators: 2 × 7 = 14
3. Result: 20/14

Simplify the fraction by dividing both numerator and denominator by their GCF, which is 2.

20 ÷ 2 = 10

14 ÷ 2 = 7

The simplified answer is 10/7, an improper fraction. Convert it to a mixed number: 1 3/7.

Why Does "Keep, Change, Flip" Work? The Mathematical Explanation

The "keep, change, flip" method might seem like a trick, but it's based on sound mathematical principles. Dividing by a fraction is the same as multiplying by its reciprocal. To understand why, let’s consider the definition of division.

Division is the inverse operation of multiplication. When you divide a number by another number, you are essentially asking, "How many times does the second number fit into the first number?"

For example, 6 ÷ 2 = 3 because 2 fits into 6 three times. This can also be written as 6 × (1/2) = 3. Notice that dividing by 2 is the same as multiplying by its reciprocal, 1/2.

Now, let's generalize this to fractions. Suppose you want to divide a/b by c/d. This can be written as:

(a/b) ÷ (c/d)

To eliminate the fraction in the denominator (c/d), you can multiply both the numerator and the denominator by the reciprocal of c/d, which is d/c:

[(a/b) ÷ (c/d)] × [(d/c) / (d/c)]

Since (d/c) / (d/c) = 1, multiplying by it doesn't change the value of the expression. Now, the expression becomes:

[(a/b) × (d/c)] / [(c/d) × (d/c)]

In the denominator, (c/d) × (d/c) = 1, because c cancels with c, and d cancels with d. So, the expression simplifies to:

(a/b) × (d/c)

This shows that dividing a/b by c/d is the same as multiplying a/b by the reciprocal of c/d, which is d/c. This is exactly what the "keep, change, flip" method achieves.

Common Mistakes to Avoid

When dividing fractions, it’s easy to make small errors that can lead to incorrect answers. Here are some common mistakes to watch out for:

1. Forgetting to Flip: The most common mistake is forgetting to flip the second fraction (the divisor). Remember, you only flip the second fraction, not the first.
2. Flipping the Wrong Fraction: Make sure you are flipping the second fraction and not the first. The first fraction stays the same.
3. Not Converting Mixed Numbers to Improper Fractions: Before you can apply "keep, change, flip," you must convert any mixed numbers to improper fractions. Failing to do so will result in an incorrect answer.
4. Forgetting to Simplify: Always simplify your final answer if possible. Leaving a fraction in an unsimplified form is not considered a complete answer.
5. Incorrectly Applying the Multiplication: After flipping the second fraction and changing the division to multiplication, double-check that you are multiplying the numerators together and the denominators together.
6. Confusing Division with Multiplication: Sometimes students mix up the rules for dividing and multiplying fractions. Remember, you don't need to find a common denominator when multiplying fractions, but you do need to "keep, change, flip" when dividing.

Real-World Applications of Dividing Fractions

Dividing fractions isn't just a theoretical math skill; it has practical applications in everyday life. Here are a few examples:

1. Cooking and Baking: Recipes often need to be scaled up or down. If a recipe calls for 2/3 cup of flour and you only want to make half the recipe, you would divide 2/3 by 2 (or multiply by 1/2) to find the new amount of flour needed.
2. Construction and Carpentry: When measuring materials for a project, you might need to divide a length into equal parts. For example, if you have a 5 1/2-foot board and need to cut it into pieces that are 3/4 foot long, you would divide 5 1/2 by 3/4 to determine how many pieces you can cut.
3. Sharing and Portioning: If you have a pizza that is 3/4 of a whole and you want to share it equally among 4 people, you would divide 3/4 by 4 (or multiply by 1/4) to determine how much each person gets.
4. Calculating Travel Time: If you are traveling a distance of 150 miles and you've covered 1/3 of the distance in 1 hour, you might want to know how long it will take to cover the remaining 2/3 of the distance, assuming you maintain the same speed.
5. Financial Planning: If you are allocating a portion of your budget to different categories, you might need to divide fractions. For instance, if you allocate 1/4 of your income to rent and you want to split that rent amount equally between two roommates, you would divide 1/4 by 2 (or multiply by 1/2) to determine each roommate’s share.

Practice Problems

To further enhance your understanding, try solving these practice problems. Answers are provided below, but try to work through them on your own first.

1. 3/5 ÷ 2/7
2. 1/4 ÷ 5/8
3. 9/10 ÷ 3/4
4. 2 2/3 ÷ 1/2
5. 4 1/5 ÷ 2 1/3

Answers:

1. 21/10 or 2 1/10
2. 2/5
3. 6/5 or 1 1/5
4. 16/3 or 5 1/3
5. 63/35 or 9/5 or 1 4/5

Conclusion

Dividing fractions doesn't have to be intimidating. By understanding the "keep, change, flip" method and practicing regularly, you can master this essential math skill. Remember to convert mixed numbers to improper fractions before dividing, and always simplify your final answer. With a solid understanding of the underlying principles and a bit of practice, you'll be able to confidently tackle any fraction division problem that comes your way. Whether it's for school, work, or everyday life, knowing how to divide fractions is a valuable skill that will serve you well.