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# Dividing Fractions

## Brief Overview of Fraction

Fraction is part of a whole. It could be anything such as half of a cake or quarter of pizza. In mathematical language, a fraction is a decimal number that can be written in a/b form, where a is called ‘numerator’ and b is called ‘denominator’. Hence, we can write half of cake as ‘1/2nd cake’ where 1 is the numerator and 2 is the denominator. Similarly, we will write a quarter of pizza as ‘1/4th pizza’ where again 1 is the numerator and 4 is the denominator.

Fractions can be of two types mainly. Proper fraction and Improper Fractions. The former is a kind of fraction when the numerator is less than the denominator (a < b), whereas in the latter, the numerator is greater than denominator (a > b).

There is another type of fraction called mixed fraction which is a combination of whole number and proper fraction. For example, $3\frac{1}{4},4\frac{1}{5},9\frac{3}{8}$ etc. In these examples 3,4, and 9 are whole numbers whereas 1/4, 1/5, 3/8, are proper fractions. Mixed fraction can be also solved into improper fractions and similarly improper fractions can be solved into mixed fractions.

Suppose we have the mixed fraction $2\frac{1}{3}$. We can convert it into improper fraction in three steps. Firstly, multiply the denominator 3 with the whole part 2 i.e., 3 × 2 = 6. Next, we add the numerator 1 with the answer of first step i.e., 1 + 6 = 7. Finally, we write it in a/b form where a is the sum obtained in second step and b is the denominator ‘3’. The result is 7/3 an improper fraction.

We can write the improper fraction into mixed fraction by dividing the numerator with the denominator. The remainder is obtained, and the mixed fraction is written as quotient followed by proper fraction of remainder/devisor.

## Dividing Fractions

### Definition

Dividing fractions means to divide one fraction with other. We have been familiar with dividing one number with other number such as a ÷ b or a/b. However, in this section we will divide one fraction with other fraction, for example a/b ÷ c/d where a/b and c/d are two fractions.

### Method / Steps

To divide one fraction with the other fraction we will be using few steps that are mentioned below.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (a / b) ÷ (c / d) where a / b is first fraction (dividend), and c / d is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., c/d becomes d/c.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (a / b) × (d / c).

Step 4: Multiply the numerators (a × d) and denominators (b × c) such that the result is (ad) / (bc).

Step 5: Simplify or solve the fraction if required.

Figure 1 Dividing fractions [1]

### Why? Significance of Dividing Fractions

We are now able to divide one fraction with the other but let’s take a minute to think what does it mean to divide one fraction with the other?

We can explain it using a simple example of a cake. For example, one person eats 1/2 of cake whereas other had 1/4 of a cake. Now, if we want to know how many 1/4 (quarters) are there in 1/2 (half). We divide 1/2 with 1/4 such that 1/4 × ___ = 1/2. The answer is 2. Hence, we can say that there are two quarters in a half. This is the significance of dividing fractions.

We can say that when we divide fractions, we are finding number of groups of divisors (second fraction) that are there in the dividend (first fraction), or simply how many times of the divisors gives us dividend.

### Techniques / Tricks

To divide fraction with other fraction, we can use the five steps mentioned above. However, there is also another technique by which we can simplify or divide fractions. We write the two fractions as numerator and denominator (2 / 3) / (4 / 12). Simplify the first and third term that are 2 and 4 by using the 2-times table 2 × __ = 4. The answer is 2. Similarly, simplify the second and fourth term that are 3 and 12 using the 3-times table 3 × ___ = 12. The answer is 4. Now, we can write the fraction as (1 / 1) / (2 / 4). Next, we can write (1 / 1) as 1 and we will also simply the new denominator fraction which is (2 / 4) using 2-times table 2 × __ = 4. The answer is 2 and we can write the expression as (1) / (1 / 2). Finally, we can move 2 to the numerator place by taking its inverse twice i.e., (2-1)-1. So, the expression simplifies to just 2.

## Dividing Proper Fractions with Improper Fractions

### Examples No. 1

Divide the fraction 5/9 with 12/11.

#### Solution

To divide 5/9 with the 12/11 we will be using following steps.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (5 / 9) ÷ (12 / 11) where 5 / 9 is first fraction (dividend), and 12 / 11 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 12/11 becomes 11/12.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (5 / 9) × (11 / 12).

Step 4: Multiply the numerators (5 × 11) and denominators (9 × 12) such that the result is 55/108.

Step 5: Since it is a proper fraction in reduced form and further division can lead to decimals. Hence, it is already simplified.

(5 / 9) / (12 / 11) = 55 / 108

### Example No. 2

Divide the fraction 12/24 with 48/6.

#### Solution

We use the following steps to divide the proper fraction 12/24 with the improper fraction 48/6.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (12 / 24) ÷ (48 / 6) where 12 / 24 is first fraction (dividend), and 48 / 6 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 48/6 becomes 6/48.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (12 / 24) × (6 / 48).

Step 4: Multiply the numerators (12 × 6) and denominators (24 × 48) such that the result is 72/1152.

Step 5: Since it is a proper fraction and can be reduced or simplified therefore, we will divide 72 with 1152 such that 1152 is divisor and 72 is dividend. Since, we cannot directly divide 72 with 1152 because 72 does not belongs to the 1152 – times table. Therefore, we will divide both 72 and 1152 by another number such as 72. We chose 72 because both 72 and 1152 belong to 72 – times table. We will now use the 72 – times table because 72 × __ = 72 and 72 × __ = 1152. The answers are 1 (obviously) and 16. So, the expression simplifies into 1/16.

(12 / 24) / (48 / 6) = 1 / 16

### Example No. 3

Divide the fraction 3/9 with 9/3.

#### Solution

We must follow the following steps to divide 3/9 with the improper fraction 9/3.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (3 / 9) ÷ (9 / 3) where 3 / 9 is first fraction (dividend), and 9 / 3 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 9/3 becomes 3/9.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (3 / 9) × (3 / 9).

Step 4: Multiply the numerators (3 × 3) and denominators (9 × 9) such that the result is 9/81.

Step 5: Since it is a proper fraction and can be reduced or simplified therefore, we will divide 9 with 81 such that 81 is divisor and 9 is dividend. Since, we cannot directly divide 9 with 81 because 9 does not belongs to the 81 – times table. Therefore, we will divide both 9 and 81 by another number such as 9. We chose 9 because both 9 and 81 belong to 9 – times table. We will now use the 9 – times table because 9 × __ = 9 and 9 × __ = 81. The answers are 1 (obviously) and 9. So, the expression simplifies into 1/9.

(3 / 9) / (9 / 3) = 1 / 9

## Dividing Proper Fractions with Proper Fractions

### Example

Divide the fraction 4/2 with 5/15.

#### Solution:

To divide 4/2 with the 5/15 we will be using following steps.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (4 / 2) ÷ (5 / 15) where 4 / 2 is first fraction (dividend), and 5 / 15 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 5/15 becomes 15/5.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (4 / 2) × (15 / 5).

Step 4: Multiply the numerators (4 × 15) and denominators (2 × 5) such that the result is 60/10.

Step 5: Since it is an improper fraction and can be reduced or simplified therefore, we will divide 60 with 10 such that 10 is divisor and 60 is dividend. We will now start revising our 10-times table because 10 × __ = 60. It is 6, and 10 × 6 = 60. So, the expression simplifies into 6.

(4 / 2) / (5 / 15) = 6

## Dividing Improper Fractions with Improper Fractions

### Example No. 1

Divide the fraction 10/3 with 16/9.

#### Solution

To divide 10/3 with the 16/9 we will be using following steps.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (10 / 3) ÷ (16 / 9) where 10 / 3 is first fraction (dividend), and 16 / 9 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 16/9 becomes 9/16.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (10 / 3) × (9 / 16).

Step 4: Multiply the numerators (10 × 9) and denominators (3 × 16) such that the result is 90/48.

Step 5: Since it is an improper fraction and can be reduced or simplified therefore, we will divide 90 with 48 such that 48 is divisor and 90 is dividend. Since, we cannot directly divide 90 with 48 because 90 does not belongs to the 48 – times table. Therefore, we will divide both 90 and 48 by another number such as 6. We chose 6 because both 90 and 48 belong to 6 – times table. We will now start revising our 6-times table because 6 × __ = 90 and 6 × __ = 48. The answers are 15 and 8. So, the expression simplifies into 15/8.

(10 / 3) / (16 / 9) = 15 / 8

### Example No. 2

Divide the fraction 41/8 with 68/2.

#### Solution

To divide 41/8 with the 68/2 we will be using following steps.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (41 / 8) ÷ (68 / 2) where 41 / 8 is first fraction (dividend), and 68 / 2 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 68/2 becomes 2/68.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (41 / 8) × (2 / 68).

Step 4: Multiply the numerators (41 × 2) and denominators (8 × 68) such that the result is 82/544.

Step 5: Since it is an improper fraction and can be reduced or simplified therefore, we will divide 82 with 544 such that 544 is divisor and 82 is dividend. Since, we cannot directly divide 82 with 544 because 82 does not belongs to the 544 – times table. Therefore, we will divide both 82 and 544 by another number such as 2. We chose 2 because both 82 and 544 belong to 2 – times table. We will now start revising our 2 – times table because 2 × __ = 82 and 2 × __ = 544. The answers are 41 and 272. So, the expression simplifies into 41/272.

(41 / 8) / (68 / 2) = 41 / 272

### Example No. 3

Divide the fractions 18/2 with 16/2.

#### Solution

To divide 18/2 with the 16/2 we will be using following steps.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (18 / 2) ÷ (16 / 2) where 18 / 2 is first fraction (dividend), and 16 / 2 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 16/2 becomes 2/16.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (18 / 2) × (2 / 16).

Step 4: Multiply the numerators (18 × 2) and denominators (2 × 16) such that the result is 36/32.

Step 5: Since it is an improper fraction and can be reduced or simplified therefore, we will divide 36 with 32 such that 32 is divisor and 36 is dividend. Since, we cannot directly divide 36 with 32 because 36 does not belongs to the 32 – times table. Therefore, we will divide both 36 and 32 by another number such as 4. We chose 4 because both 36 and 32 belong to 4 – times table. We will now start revising our 4 – times table because 4 × __ = 36 and 4 × __ = 32. The answers are 9 and 8. So, the expression simplifies into 9/8.

(18 / 2) / (16 / 2) = 9 / 8

## Dividing Mixed Fractions

### Example No.1

Divide the mixed fraction 3(1/2) with 2/3.

#### Solution

To divide 3(1/2) with the 2/3 we will be using the same steps with addition of one step which is to be called as step 0. Since, in this section we will be dealing with mixed fractions. So, in step 0, we convert all mixed fractions to improper fractions.

Step 0: There is one mixed fraction which is 3(1/2). We will convert it into improper fraction by multiplying the denominator 2 with whole 3 and adding 1 to it i.e., (2 × 3) + 1. The result is 7 which becomes the new numerator. Hence, mixed fraction can be written as 7/2

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (7 / 2) ÷ (2 / 3) where 7 / 2 is first fraction (dividend), and 2 / 3 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 2/3 becomes 3/2.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (7 / 2) × (3 / 2).

Step 4: Multiply the numerators (7 × 3) and denominators (2 × 2) such that the result is 21/4.

Step 5: Since it is an improper fraction and cannot be reduced or simplified further. Therefore, we will leave it here.

(3 (1 / 2)) / (2 / 3) = 21 / 4

### Example No. 2

Divide the mixed fraction 4(9/12) with 12/2.

#### Solution:

To divide 4(9/12) with the 12/2 we will be using the same steps with addition of one step which is to be called as step 0. Since, in this section we will be dealing with mixed fractions. So, in step 0, we convert all mixed fractions to improper fractions.

Step 0: There is one mixed fraction which is 4(9/12). We will convert it into improper fraction by multiplying the denominator 12 with whole 4 and adding 9 to it i.e., (12 × 4) + 9. The result is 57 which becomes the new numerator. Hence, mixed fraction can be written as 57/12

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (57 / 12) ÷ (12 / 2) where 57 / 12 is first fraction (dividend), and 12 / 2 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 12/2 becomes 2/12.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (57 / 12) × (2 / 12).

Step 4: Multiply the numerators (57 × 2) and denominators (12 × 12) such that the result is 114/144.

Step 5: Since it is an improper fraction and can be reduced or simplified therefore, we will divide 114 with 144 such that 144 is divisor and 114 is dividend. Since, we cannot directly divide 114 with 144 because 114 does not belongs to the 144 – times table. Therefore, we will divide both 114 and 144 by another number such as 6. We chose 6 because both 114 and 144 belong to 6 – times table. We will now start revising our 6 – times table because 6 × __ = 114 and 6 × __ = 144. The answers are 19 and 24. So, the expression simplifies into 19/24.

(4(9 / 12)) / (12 / 2) = 19 / 24

### Example No. 3

Divide the mixed fraction 6(7/2) with 7(6/2)

#### Solution:

To divide 6(7/2) with the 7(6/2) we will be using the same steps with addition of one step which is to be called as step 0. Since, in this section we will be dealing with mixed fractions. So, in step 0, we convert all mixed fractions to improper fractions.

Step 0: There are two mixed fraction 6(7/2) and 7(6/2). We will convert them into improper fraction one by one. In the first mixed fraction, we multiply the denominator 2 with whole 6 and adding 7 to it i.e., (2 × 6) + 7. The result is 19 which becomes the new numerator. Hence, mixed fraction can be written as 19/2. Similarly, the other mixed fraction can be written as 20/2.

Step 1: Write the two fractions as numerator and denominator and put the divide ( ÷ ) symbol in between the two fractions. For example, (19 / 2) ÷ (20 / 2) where 19 / 2 is first fraction (dividend), and 20 / 2 is second fraction (devisor).

Step 2: Take reciprocal of the second fraction (devisor) which is the second fraction. Reciprocal is to change numerator and denominator of a fraction i.e., 20/2 becomes 2/20.

Step 3: Replace division between the two fractions by multiplications i.e., simply replace ÷ sign by × sign. The term then becomes (19 / 2) × (2 / 20).

Step 4: Multiply the numerators (19 × 2) and denominators (2 × 20) such that the result is 38/40.

Step 5: Since it is a proper fraction and can be reduced or simplified therefore, we will divide 38 with 40 such that 40 is divisor and 38 is dividend. Since, we cannot directly divide 38 with 40 because 38 does not belongs to the 40 – times table. Therefore, we will divide both 38 and 40 by another number such as 2. We chose 2 because both 38 and 40 belong to 2 – times table. We will now start revising our 2 – times table because 2 × __ = 38 and 2 × __ = 40. The answers are 19 and 20. So, the expression simplifies into 19/20.

(6 (7 / 2)) / (7 (6 / 2)) = 19 / 20