Dividing Fractions with Whole Numbers

A fraction represents a part of a whole and is written using two numbers separated by a line. These two numbers are important in a fraction. The numerator, which is the topmost part, shows how many parts are taken, while the denominator, the bottom number, indicates the whole of the parts. For example, \(\frac{4}{9}\) means 4 parts out of 9 equal parts of the whole. 
 

Parts of a Fraction
 

A fraction has two main parts, each serving a specific purpose: 

• Numerator: It is the number on the top of the fraction. A numerator shows how many parts of the whole are being counted or considered. 

• Denominator: The number on the bottom of the fraction is the denominator. It shows the total number of equal parts, the whole is divided into. 

• For example, in \(\frac{5}{8}\), the numerator is 5, meaning 5 equal parts are taken. And the denominator is 8, meaning the whole is divided into 8 equal parts. 

Reciprocal of a Fraction
 

The reciprocal of a fraction is found by flipping the numerator and the denominator. In simple terms, you just interchange their positions. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). Make sure that the numerator or denominator is a non-zero number, because division by zero is not allowed. 

For example, the reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\).
Reciprocals are especially useful in division of fractions, where dividing by a fraction is the same as multiplying by its reciprocal. 
For example: \(6÷\frac{2}{3}=6×\frac{3}{2} \).

Whole numbers are the series of numbers starting from 0,1, 2, 3, and so on. These numbers are not fractions or decimals. They are part of the number system used for counting and basic arithmetic. Whole numbers do not include negative numbers or fractions.

To divide a fraction by a whole number, follow the steps given below:

Step 1:  Write the whole number as a fraction:
Write the whole number as a fraction with 1 as the denominator.

Step 2: Take the reciprocal of the given whole number:
The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Step 3: Multiply the fractions:
Multiply the numerators and the denominators together.

Step 4: Simplify the answer (if needed):
Reduce the fraction by dividing the numerator and denominator by their common factors.

Example: Divide \(\frac{8}{9}÷3\).
 

Solution: 

Step 1: Write the whole number as a fraction. Hence, 3 will become \(\frac{3}{1}\).
 

Step 2: Take the reciprocal of the whole number. 

Reciprocal of \(\frac{3}{1}\) is \(\frac{1}{3}\).

Step 3: Now multiply the fractions: 

\(\frac{8}{9}×\frac{1}{3} = \frac{8×1}{9×3} = \frac{8}{27} \)

Step 4: Simplify the answer, if needed. 

As \(\frac{8}{27}\) is the simplest form, the final answer for \(\frac{8}{9}÷3=\frac{8}{27}\).

To divide a whole number by a fraction, we must follow the following steps:

Step 1: Write the whole number as a fraction:
This can be done by making the denominator as 1 and putting the whole number above the 1.

Step 2: Find the reciprocal of the fraction given.
The reciprocal of a fraction is found by swapping its numerator and denominator. 

Step 3: Multiply the fractions
Multiply the numerators and the denominators.

Step 4: Simplify the answer (if needed):
Divide the numerator and denominator by its greatest common factor. 
 

Example: Divide \(8÷\frac{2}{3}\)
 

Solution: 

Step 1: Write the whole number as a fraction. 

\(8 = \frac{8}{1}\)
 

Step 2: Find the reciprocal of the fraction. 

Reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
 

Step 3: Multiply the fractions. 

\(\frac{8}{1}×\frac{3}{2}=\frac{24}{2}\).
 

Step 4: Simplify the result.

Therefore, \(8÷\frac{2}{3}=12\).

Learning to divide fractions by whole numbers helps simplify everyday math tasks and improves problem-solving speed and accuracy.

 

• Write the whole number as a fraction with denominator 1.
   
• Flip the second fraction and change the division sign to multiplication.
   
• Simplify the fractions before multiplying to make it easier.
   
• Multiply the numerators and denominators to get the answer.
   
• Convert improper fractions to mixed numbers if needed.
   

• Parents and teachers can use real life examples such as sharing food, poring liquids, or dividing objects into equal parts to make the concept easier to understand for students. 
   

• Encourage estimation first, so that students can predict whether the answer should be bigger or smaller. This will be useful for building number sense in students.
   

• Relate to multiplication facts to strengthen fluency in students, since dividing fractions is essentially multiplying by a reciprocal. 
   

• Incorporate learning with digital tools like online math games, and help students gain practice through fraction calculators and division of fractions worksheets available online. 
   

• Check the level of understanding of students through open discussions by asking them to explain why the reciprocal method works, not just how. 
   

   

   

  
   

   

   

   

   

  
   

  
   

   

  
   

Dividing fractions by whole numbers has many practical applications. Let's explore some examples:

• Cooking and baking: Dividing fractions by whole numbers is commonly used when adjusting recipes. If a recipe calls for 3/4 cup of sugar but needs to be divided into two smaller batches, divide 3/4 by 2. This results in 3/8 cup of sugar per batch.

• Sharing food: If half a pizza is shared among three people, each gets \( \frac{1}{6} \).

• Cutting and Measuring Fabric or Wood: In tailoring, carpentry, and construction, dividing fractions helps when cutting materials into equal parts. Similarly, a carpenter divides a wooden plank into equal sections using the same principle.
  
   
• Time management: Dividing fractions helps when splitting time equally. For example, if you have ¾ of an hour to complete 3 tasks, divide ¾ by 3 to find ¼ hour (15 minutes) per task.
  
   
• Budgeting and finance: When splitting a fractional amount of money, such as half of your monthly savings between two goals. Dividing fractions helps distribute the funds fairly.