Division of Fractions

Division is the process of splitting quantities into equal parts. In division, we divide a number by the divisor to get the quotient. Here, we will learn the step-by-step process for dividing two fractions. For instance, \({5\over8} \div {1\over6}\). Now, let’s learn the division of fractions step by step.

Examples:

• \(\frac{3}{5} \div \frac{2}{3} = \frac{9}{10}\)
   
• \(6 \div \frac{3}{4} = 8\)
   
• \(\frac{4}{9} \div 2 = \frac{2}{9}\)
   
• \(\frac{7}{8} \div \frac{1}{4} = \frac{7}{2}\)
   
• \(1 \frac{1}{2} \div \frac{3}{5} = \frac{5}{2}\)

 The main rule for dividing fractions is actually to turn the problem into multiplication. Since dividing by a number is the same as multiplying by its opposite (reciprocal), you follow a three-step process often called Keep, Change, Flip.

Here is how it works:

1. Keep the first fraction exactly as it is.
2. Change the division symbol (\(\div\)) to a multiplication symbol (\times).
3. Flip the second fraction upside down (swap the numerator and denominator).

Once you have done that, you just multiply straight across—top times top, bottom times bottom—and simplify the answer if you can.

Examples:

• \({2 \over 5} \div {3 \over 2}\)
  
  \({2 \over 5} \times {2 \over 3} = {4 \over 15}\)
   
• \({7 \over 3} \div {6 \over 13}\)
  
  \({2 \over 3} \times {13 \over 6} = {13 \over 9}\)

Note: If you are working with mixed numbers (like \(\frac{1}{2}\)), you have to change them into improper fractions first. The Keep, Change, Flip rule only works on standard fractions. 

There are certain steps that need to be followed while dividing fractions. Let us learn about these steps here: 

Step 1: Taking the inverse of the second fraction.

Step 2: Simplify the fraction 

Step 3: Multiply the first fraction with the inverse of the second fraction. 

Example: \({5\over15} \div {6\over12} \)

Step 1: Taking the inverse of the second fraction
The inverse of the second fraction \(6\over12\) is \(12\over6\)

Step 2: Simplifying the fractions

\({5\over15} = {1\over3}\)

\({12\over6} = 2\)
 

Step 3: Multiplying the fraction with the inverse of the second fraction.

\({{1\over3} \times 2} = {2\over3}\)

Division of fractions isn't always straightforward because the numbers can look different every time. You might be working with whole numbers, decimals, or mixed numbers. Spotting exactly which type you have is the best way to start, so you know if you need to make a quick change—like turning a decimal into a fraction—before you solve it.

Fraction by a Fraction

• Dividing a proper fraction by another proper fraction.
• Example: \(\frac{1}{2} \div \frac{1}{4}\)
   

Fraction by a Whole Number

• Dividing a proper fraction by a whole integer.
• Example: \(\frac{3}{4} \div 2\)
   

Fraction by a Decimal

• Dividing a proper fraction by a decimal.
• Example: \(\frac{1}{2} \div 3.2\)

Fraction by a Mixed Numbers

• Dividing a proper fraction by a mixed number.
• Example: \(\frac{1}{2} \div 1 \frac{1}{3}\)

To divide a fraction with whole numbers, we multiply the whole number by the denominator of the fraction.

Step 1: Taking the inverse of the whole number, that is, 1 by the whole number.
 

Step 2: Multiply the fraction with the new divisor.

Example: \({3\over5} ÷ 4\)
 

Step 1: Replace the whole number (4) with its reciprocal
The reciprocal of 4 is \(1\over4\). 

Step 2: Multiply the fraction with the new divisor

\({3\over5} × {1\over4} = {3\over20}\)
 

To divide a fraction by a decimal, we should first convert the decimal into fraction. Given below are the steps to divide a fraction by a decimal:
 

Step 1: Converting the decimal to fraction.
 

Step 2: Taking the inverse of the divisor.
 

Step 3: Simplify the fraction and multiply the fractions. 

Example: \({3\over4} \div {0.5}\)
 

Step 1: Converting the 0.5 to fraction

0.5 in fraction is \({5\over10} = {1\over2}\) 

Step 2: Taking the inverse of the divisor

Dividing \({3\over4} \div {1\over2 }\)

The inverse of \(1\over 2 \)is \(2\over1\) 
 

Step 3: Simplify the fraction and multiply the fractions.

\({3\over4} \times {2\over1} = {6\over4}\) 

\(= {3\over2} \)
 

To divide a fraction by a mixed fraction, we should first convert the mixed fraction into an improper fraction. Then, continue with the division process as usual.  

For example: \({​2\over3} \div {1{1\over2}} \)

\({1{1\over2}}\) can be converted into an improper fraction as; 

\({1{1\over2}} ={1 \times 2 + {1\over 2}}\)

\(= {3\over 2}\)
 

So, \({2\over3} \div {3\over2} = {2\over 3} × {2\over3}  \)

\(= {4\over 9}\)

• Visualize the Problem: Start with physical objects like paper strips, chocolate bars, or LEGO bricks before moving to abstract numbers. Asking "how many halves are in 3 chocolate bars?" makes much more sense to a learner than seeing \(3 \div \frac{1}{2}\) on a page.
   
• Link the Operations: Help students see the direct relationship between multiplication and division of fractions. Show them that dividing by a number is mathematically the same as multiplying by its reciprocal (like how dividing by 2 is the same as multiplying by half), which demystifies why the method changes.
   
• Explain the "Why" Behind the Rules: Don't just force them to memorize the division of fractions rules like "Keep, Change, Flip." Explain that we flip the second fraction to turn a difficult division problem into an easier multiplication problem, giving them a logical hook to remember the steps.
   
• Use Story Problems: Frame problems in real-world contexts, like measuring ingredients for a recipe or cutting lengths of ribbon. This prevents the concept from feeling abstract and helps them understand when to apply division versus other operations.
   
• Provide Varied Practice: Move beyond standard equations by using division of fractions worksheets that include visual puzzles and word problems. Mixed practice ensures they aren't just mechanically applying a formula but are actually reading and understanding the math.
   
• Encourage Self-Checking: Let students use a division of fraction calculator, but only to verify their manual work. This provides instant feedback, helping them catch mistakes and learn from them without using the device as a crutch.
   
• Teach Estimation First: Before calculating, ask students to guess if the result will be bigger or smaller. This builds number sense and acts as a safety net, helping them spot illogical answers immediately.

The division of fractions happens every day, and thus it has varied applications. We use it for cooking, time management, construction, and so on. Here are a few real-life applications of dividing fractions. 

 

• In cooking, to find the amount of ingredients required for the recipe, we use the division of fractions. That is, if we require \(3\over 4\) cups of sugar for 1 cup of flour, then for \(1\over 2\) cup of flour the required amount of sugar is half of \(3\over 4\), which is \({3\over4} \times {1\over2} = {3\over8}\)

• To share any object among a group, we use division of fraction.
   

• In construction, to cut the wooden plank of \(3\over4\) meter long into pieces of \(1\over8\) meters, we need to divide. That is, \({3\over4} \div {1\over8} = {3\over4} \times {8\over1} = {6}\).
   

• To split the time into smaller intervals, we use the division of fractions. For example, if a task takes \(3\over4 \)hours and if we want to divide it into sessions of \(1\over8\) hours, we divide it, \({3\over4} \div {1\over8} = {3\over4} × {8\over1} = 6 \) sessions. 
   

• In classrooms, teachers use division of fractions to divide the materials or questions among students.