# Mixed Numbers and Improper Fractions

A **mixed number** is a combination of a whole number and a fraction. For example, if you have two whole apples and one half apple, you could describe this as 2 + ^{1}/2 apples, or 2^{1}/2 apples.

### Writing Mixed Numbers as Fractions

This mixed number can also be expressed as a fraction. Each whole apple contains two half apples. Your two whole apples are also four half apples. Four half apples plus one half apple is five half apples. So you have ^{5}/2 apples.

To put this another way: **to turn a mixed number into a fraction, multiply the whole number by the denominator (the bottom part), and add the result to the numerator (the top part).**

2^{1}/2 = ?

Multiply the whole number by the denominator.

The whole number is 2.

The denominator is 2.

2 x 2 = 4.

Add the result to the numerator:

The numerator is 1.

4 + 1 = 5

The numerator is 5. The denominator remains 2.

2^{1}/2 =^{5}/2

### Another Example

Let's try another example:

5^{2}/3 = ?

Multiply the whole number by the denominator.

The whole number is 5.

The denominator is 3.

5 x 3 = 15.

Add the result to the numerator:

The numerator is 2.

15 + 2 = 17

The numerator is 17. The denominator remains 3.

5^{2}/3 =^{17}/3

### Proper and Improper Fractions

A fraction in which the numerator is smaller than the denominator, like ^{1}/3 or ^{2}/5 is called a **proper fraction.** A fraction in which the numerator is larger than or equal to the denominator, like ^{5}/2, ^{17}/3, or ^{6}/6 is called an **improper fraction.** (To put it another way, a fraction with a value less than 1 is a proper fraction. A fraction with a value greater than or equal to 1 is an improper fraction.)

As we have shown above, mixed numbers can be written as improper fractions. Similarly, improper fractions can be written as mixed numbers.

### Writing Improper Fractions as Mixed Numbers

**To write an improper fraction as a mixed number, divide the numerator (top part) by the denominator (bottom part). The quotient is the whole number, and the remainder is the numerator.**

How would you express^{17}/4 as a mixed number?

Divide the numerator by the denominator:

17 Ã· 4 = 4, with a remainder of 1

The quotient, 4, is the whole number. The remainder, 1, is the numerator. The denominator remains 4.^{17}/4 = 4^{1}/4

### Two More Examples

Let's try another couple of examples:

^{14}/9 = ?

Divide the numerator by the denominator:

14 Ã· 9 = 1, with a remainder of 5

The quotient, 1, is the whole number. The remainder, 5, is the numerator. The denominator remains 9.^{14}/9 = 1^{5}/9

If there is no remainder, just take the quotient as the whole number:

^{20}/5 = ?

Divide the numerator by the denominator:

20 Ã· 5 = 4

The quotient, 4, is the whole number. There is no remainder.^{20}/5 = 4