Get detailed information regarding the division of fractions here. Check all the important terms, formulae, and the usage of fractions division. Refer step by step procedure to solve the fraction division problems. Know the various methods to solve proper, improper, mixed fractions, etc. Follow the complete guide to know the day-to-day usage of fractions division problems. Go through the below sections to follow the importance and problems of division on fractions.

## Division of Fractions – Introduction

Dividing the fractions also involves various methods and also a lengthy process. First of all, you have to know the difference between proper, improper, and mixed fractions. Once you come to know the difference, then you can easily analyze the method to solve them. As we have seen, for the multiplication, we have to make the fractions equivalent.

In the same way, the division of fractions requires the equivalent fractions to solve them. First, make sure you make the fractions equivalent and then follow all the steps in dividing the fractions. Generally, fraction division in the direct method requires more effort. Hence, we are providing an alternative method here.

### How to Divide Fractions?

Division of fractions can be done by multiplying the fractions by writing the reciprocal of one of fraction numbers or by reversing one of two fraction integers. Reciprocal of the fraction number or reverse of the fraction number means, if the fraction is \(\frac { a }{ b } \), then \(\frac { b }{ a } \) is its reciprocal. Interchanging the position of numerator and denominator is nothing but reversing the fraction. Fractions division can be classified into 3 ways.

- Fractions division by a fraction
- Fractions division by a mixed fraction
- Fractions division by a whole number

#### Fractions Division by a Fraction

Fractions division can be done in three steps. For division, we convert into the multiplication of fractions and obtain the desired result. The three steps are as follows.

- Convert the second fraction into its reciprocal and multiply it with the first fraction.
- Multiply the denominator and numerator of both the fractions.
- Next, simplify the fraction numbers.

For example, suppose that \(\frac { a }{ b } \) is divided by the fraction \(\frac { c }{ d } \). We can solve the division as:

- \(\frac { a }{ b } \) ÷ \(\frac { c }{ d } \) = \(\frac { a }{ b } \) * \(\frac { d }{ c } \)
- \(\frac { a }{ b } \) ÷\(\frac { c }{ d } \) = \(\frac { a*d }{ b*c } \)
- \(\frac { a }{ b } \) ÷ \(\frac { c }{ d } \) = \(\frac { ad }{ bc } \)

As given above, we can divide the fractions in three ways. Further simplification has to be done to get the exact result.

### Fractions Division by a Whole Number

Fractions Division by a whole number is the easiest process. Follow the below steps to divide a whole number.

- The whole number is nothing but the real numbers which include zero and all positive integers. All the whole numbers can be written as the fraction values if we give the denominator value as 1.
- Find the reciprocal of the given number
- Now, multiply the value of the fraction by a given fraction.
- Then, simplify the equation to get its lowest terms.

**Example:**

Divide the fraction \(\frac { 6 }{ 5 } \) by 10?

**Solution:**

As given in the question,

The equation is \(\frac { 6 }{ 5 } \) by 10.

**Step 1: **Conversion into the whole number

As 10 is the whole number, we convert it into fractional value i.e., we write it as \(\frac { 10 }{ 1 } \)

**Step 2: **Find the reciprocal of the fraction

To find the reciprocal, we get \(\frac { 1 }{ 10 } \)

**Step 3:** Multiply both the fractions

Multiply the fractions, \(\frac { 6 }{ 5 } \) x \(\frac { 1 }{ 10 } \)

**Step 4: **Now, multiply the numertors and denominators of the fractions i.e., \(\frac { 6×1 }{ 15×10 } \)

**Step 5:** Simplify the equation

The resutant is \(\frac { 3 }{ 25 } \)

### Fractions Division by a Mixed Fraction

Fractions Division by a mixed fraction is almost similar to fractions division by fraction. The following are the steps to divide mixed fractions.

- Conversion of mixed fraction into an improper fraction
- Now, find the reciprocal of the improper fraction
- Multiply the resultant fraction by the given fraction value.
- Simplify the fractions.

**Example:**

Divide the fraction \(\frac { 2 }{ 5 } \) by 3\(\frac { 1 }{ 2 } \)?

**Solution:**

As given in the question,

The equation is \(\frac { 2 }{ 5 } \) by 3\(\frac { 1 }{ 2 } \)

**Step 1:** Conversion of mixed fraction into improper fraction

Convert, 3\(\frac { 1 }{ 2 } \), we get the result as 7/2

**Step 2:** Now, convert the reciprocal of the improper fraction, we get \(\frac { 2 }{ 7 } \)

**Step 3:** Multiply \(\frac { 2 }{ 5 } \) and \(\frac { 2 }{ 7 } \)

**Step 4:** Multiply the numertors and denominators of the fractions i.e., \(\frac { 2×2 }{ 5×7 } \)

**Step 5:** Simplify the fraction

the result value is \(\frac { 4 }{ 35 } \)

### Fractions Division of Decimal Values

In the above sections, we have seen how to divide the fractions into three steps. Now, we see how to divide decimals with examples.

To convert the decimal values into natural numbers, we multiply both the numerator and denominator by 10.

**Example:**

Divide \(\frac { 0.5 }{ 0.2 } \)

**Solution:**

As given in the question,

The equation is \(\frac { 0.5 }{ 0.2 } \)

To divide the decimal values, convert the values into natural numbers by multiplying both numerator and denominator with 10.

Therefore, \(\frac { 0.5×10 }{ 0.2×10 } \)

We get the solution as, \(\frac { 5 }{ 2 } \) = 2.5

Also, we use the method of dividing fractions to solve the problem.

We can write 0.5 as \(\frac { 5 }{ 10 } \) and 0.2 as \(\frac { 2 }{ 10 } \)

Therefore, to find the solution of \(\frac { 5÷10 }{ 2÷10 } \)

Take the reciprocal of the second fraction i.e., \(\frac { 10 }{ 2 } \)

= \(\frac { 5*10 }{ 10*2 } \)

= \(\frac { 50 }{ 20 } \)

= \(\frac { 5 }{ 2 } \)

= 2.5

### Dividing Fractions Examples

**Problem 1: **Lucy has \(\frac { 1 }{ 5 } \) of a bag of dog food left. She is splitting it between her 3 dogs evenly. What fraction of the original bag does each dog get?

**Solution:**

As given in the question,

Amount of dog food left = \(\frac { 1 }{ 5 } \)

No of dogs = 3

The fraction of the original bag each dog gets = \(\frac { 1/5 }{ 3 } \)

= \(\frac { 1 }{ 5 } \) x \(\frac { 1 }{ 3 } \)

= \(\frac { 1 }{ 15 } \) of the bag each dog gets.

Thus the final solution is \(\frac { 1 }{ 15 } \)

**Problem 2:** AJ has \(\frac { 1 }{ 4 } \) of a gallon of saltwater that he is using for an experiment. He needs to evenly separate the saltwater into separate beakers. How much salt water will be in each beaker?

**Solution:**

As given in the question,

Amount of salt water he is using for an experiment = \(\frac { 1 }{ 4 } \)

No of beakers = 3

Amount of salt water in each beaker = \(\frac { 1/4 }{ 3 } \)

= \(\frac { 1 }{ 4 } \) x \(\frac { 1 }{ 3 } \)

= \(\frac { 1 }{ 12 } \)

Therefore, \(\frac { 1 }{ 12 } \) gallons of salt water will be in each beaker.

**Problem 3:** Devin has a board that measures 4 ft in length. The board is going to be cut into \(\frac { 1 }{ 4 } \) ft pieces. How many pieces will Devin split the board into?

**Solution:**

As given in the question,

Length of the board = 4 ft

No of pieces the board is going to be cut = \(\frac { 1 }{ 4 } \)

No of pieces Devin split the board = 4 ÷ \(\frac { 1 }{ 4 } \)

= \(\frac { 4 }{ 1 } \) x \(\frac { 4 }{ 1 } \)

= \(\frac { 16 }{ 1 } \)

Hence, the final solution is 16 pieces.

Therefore, Devin will split the board into 16 pieces.