Dividing Fractions with Whole Numbers

In this section, we will learn about Dividing Fractions With Whole Numbers. Before jumping into the main part let us understand the basic concepts of fractions and whole numbers.

What are Fractions?

The portion/part of the whole thing is represented by a fraction. There are 2 parts to the fraction a denominator and a numerator. The top number is called the numerator, and the denominator is the number on the bottom.

Ex: 4/6 is a fraction. 4 is the numerator which is represented above the line and 6 is the denominator which is represented below the line. Here 4/6 can be written as 1/3 which is part of the number 3.

What are Whole Numbers?

The whole numbers are described as the positive integers including zero. No decimal or fractional element is found in the whole number. In other words, a number that is not a fraction is a whole number.

The mathematical notation for whole numbers is

W = {0, 1, 2, 3, 4, 5, 6, 7 , ………..}

Now we have understood the basic concepts of Fractions and Whole numbers. Let us look into how to the fractions by whole numbers and vice versa.

How to Divide Fractions with Whole Numbers

Here let us discuss the step to dividing fractions by whole numbers.

Step 1: The first step in dividing fractions by whole numbers is simply to write the fraction followed by the sign of the division and the whole number by which we need to divide it. 

Ex: If we want to divide a fraction 5/4 by a whole number 3. We can represent this step as follows.

\[\frac{5}{4}\] ÷ 3

Step 2: Convert the whole number into a fraction: To convert a whole number to a fraction, simply place the number over the number 1. The whole number becomes the numerator and 1 becomes the fraction denominator.

Ex: Let us look at the same example which is discussed in step 1. Here we have to convert whole number 3 into a fraction just by replacing 1 in the denominator which doesn’t change the value of 3.

\[\frac{5}{4}\] ÷ \[\frac{3}{1}\]

Step 3: Take a reciprocal of the whole number which we are dividing the fraction. To find the reciprocal reverse the numerator and denominator.

Ex: \[\frac{3}{1}\]can be written as \[\frac{1}{3}\]

Step 4: After we take the reciprocal the division process will become a multiplication.

Ex: \[\frac{5}{4}\] x \[\frac{1}{3}\]

Step 5: Now multiply the numerator and denominator of the fractions to obtain the new fraction.

Ex: \[\frac{5}{4}\] x \[\frac{1}{3}\] = \[\frac{5}{12}\]

Step 6: Simplify the fraction if necessary. To simply find the lowest common denominator, which means that both the numerator and denominator can be separated by any number that is equally divided into both numbers.

Ex: \[\frac{2}{16}\] can be simplified as \[\frac{1}{8}\].

Now let us some problems on dividing Fractions with Whole Numbers by using the above-mentioned steps.

Problems on Dividing Fractions by Whole Numbers.

1) Divide the Fraction 3/7 by the Whole Number 3.

Ans: 

Step 1: Write the fraction followed by the sign of the division

\[\frac{3}{7}\] ÷ 3

Step 2: Convert the whole number into a fraction

3 can be written as 3/1.

Step 3: Take a reciprocal of the whole number.

3/1 can be written as ⅓.

Step 4: Division process becomes multiplication.

\[\frac{3}{7}\] x \[\frac{1}{3}\]

Step 5: Multiply numerator and denominator of the fractions.

\[\frac{3}{7}\] x \[\frac{1}{3}\] = \[\frac{3}{21}\] 

Step 6: Simplify the fraction.

\[\frac{3}{21}\] = \[\frac{1}{7}\]

The final fraction obtained after dividing 3/7 by the whole number 3 is 1/7.

2) Divide the Fraction 5/2 by the Whole Number 10.

Ans: 

Step 1: Write the fraction followed by the sign of the division

\[\frac{5}{2}\] ÷ 10

Step 2: Convert the whole number into a fraction

10 can be written as 10/1.

Step 3: Take a reciprocal of the whole number.

10/1 can be written as 1/10.

Step 4: Division process becomes multiplication.

\[\frac{5}{2}\] x \[\frac{1}{10}\]

Step 5: Multiply numerator and denominator of the fractions.

\[\frac{5}{2}\] x \[\frac{1}{10}\] = \[\frac{5}{20}\] 

Step 6: Simplify the fraction.

\[\frac{5}{20}\] = \[\frac{1}{4}\] 

The final fraction obtained after dividing 5/2 by the whole number 10 is 1/4.

How to Divide Numbers with Fractions

Here we find the steps to divide the whole number by a fraction.

Step 1: Make a fraction out of the whole number. Make the whole number the numerator of a fraction denominator as 1.

Ex: Whole number 5 can be written in fraction form as 5/1.

Step 2: Find the reciprocal of the fraction. To find the reciprocal of the fraction reverse the numerator and denominator.

Ex: The reciprocal of the fraction 5/7 is 7/5 which is obtained by reversing the numerator and the denominator.

Step 3: Since we have found the reciprocal of the fraction, the division process will now be a multiplication process.

Ex: \[\frac{5}{1}\] x \[\frac{7}{5}\]  

Step 4: Multiply the numerator and denominator to find the fraction.

Ex: \[\frac{5}{1}\] x \[\frac{7}{5}\] = \[\frac{35}{5}\]  

Step 5: Simplify the fraction if necessary. To simply find the lowest common denominator and divide both the numerator and denominator by that number.

Ex: \[\frac{35}{5}\] is having the lowest common denominator as 5. So dividing both numerator and denominator by 5 we get the simplified answer as 7/1 or 7.

Let us solve some problems on dividing whole numbers by fractions.

Problems on How to Divide Numbers with Fractions.

1) Divide the Whole Number 7 by the Fraction 3/4.

Ans: 

Step 1: Make a fraction out of the whole number.

Here the whole number 7 can be written as 7/1 in fraction form.

Step 2: Find the reciprocal of the fraction.

¾ reciprocal is 4/3. 

Step 3: Division process becomes multiplication.

\[\frac{7}{1}\] x \[\frac{4}{3}\]    

Step 4: Multiply the numerator and denominator.

\[\frac{7}{1}\] x \[\frac{4}{3}\] = \[\frac{28}{3}\] 

Further simplification cannot be done. So the final answer obtained after dividing the whole number 7 by the fraction ¾ is 28/3.

2) Divide the Whole Number 12 by the Fraction 8/3.

Ans: 

Step 1: Make a fraction out of the whole number.

Here the whole number 12 can be written as 12/1 in fraction form.

Step 2: Find the reciprocal of the fraction.

8/3 reciprocal is 3/8. 

Step 3: Division process becomes multiplication.

\[\frac{12}{1}\] x \[\frac{3}{8}\]

Step 4: Multiply the numerator and denominator.

\[\frac{12}{1}\] x \[\frac{3}{8}\] = \[\frac{36}{8}\]

Step 5: Simplify the fraction.

Here the lowest common denominator which divides both numerator and denominator is 4. So 36/8 can be simplified to 9/2. 

So the final answer obtained after dividing the whole number 12 by the fraction 8/3 is 9/2.

Conclusion

• When any fraction is divided by a whole number the final answer will always be a fraction.

• When a whole number is divided by a fraction the final answer will be either a fraction or a whole number.