## How to Divide by Fractions:

**Multiplying by the reciprocal**

*Example with a fraction being divided by a fraction*

problem:

Step 1: turn the fraction you are dividing by upside-down (this gives you its’ reciprocal) and flip the division sign into a multiplication sign.

Step 2: Multiply across

5×4=20 ; 8×2=16 So,

Step 3: Simplify

*If you end up with an improper fraction, make sure to keep simplifying*

*Example with a whole number being divided by a fraction*

Problem:

*Before you begin, understand that every whole number has an “invisible” denominator of “1”

For example, , , etc.

Multiply using the reciprocal

Doing so, we get an answer of

The fraction is an **improper fraction** (the numerator is greater than the denominator).

While there is nothing incorrect about this, an improper fraction is typically

simplified further into a **mixed number**.

The **whole number** part of the mixed number is found by dividing the 16 by the 3.

In this case we get **5
**

The

**fractional part**of the mixed number is found by using the remainder of the division,

which in this case is

**1**(16 divided by 3 is 5 remainder 1).

*Example with a mixed number being divided by a fraction*

problem:

To start, we first need to convert to an improper fraction.

To convert to an improper fraction:

multiply the 9 (the denominator), and the 2 (the whole number). To this product, add the 5 (the numerator)

giving 23, to form the new numerator, and use the 9 as the new denominator.

So, as an improper fraction is

The problem here is to divide and

This problem can be solved by multiplying by the **reciprocal** of

To find the reciprocal of

Simply exchange the numerator and denominator, or just “flip” the fraction

upside-down.

The reciprocal of is

The problem here is to multiply and

This problem can be solved by multiplying together the two numerators (the 23 and 5),

giving 115, which will be the numerator in our answer.

Also, we’ll multiply together the two denominators (the 9 and 3),

giving 27, which will be the denominator in our answer.

Doing so, we get an answer of

The fraction is an **improper fraction** (the numerator is greater than the denominator).

While there is nothing incorrect about this, an improper fraction is typically

simplified further into a **mixed number**.

The **whole number** part of the mixed number is found by dividing the 115 by the 27.

In this case we get

The **fractional part** of the mixed number is found by using the remainder of the division,

which in this case is **7** (115 divided by 27 is 4 remainder 7).