**Fractions**

**Levels 6-7**

The objectives of this section are to

* simplify fractions;

* order fractions;

* convert between improper and mixed fractions;

* add and subtract fractions (including mixed numbers);

* multiply fractions.

**Fractions**

The top number of a fraction is
called the **numerator**.

The bottom number is called the **denominator**.

The fraction is called a **mixed number**.

The fraction is called a top-heavy (or **improper**) fraction its numerator is bigger than its denominator.

** **

**Converting between mixed numbers and
improper fractions**

** **

**Example**: Convert to an improper fraction.

**Solution**: There are 20
fifths in 4 whole numbers. So,

**Example 2**: Convert to an improper fraction.

**Solution**: (the numerator can be found as 6 × 3 + 2 =
20)

**Example 3**: Convert to a mixed number.

**Solution**: 6 goes into 25
four times, remainder 1. So,

**Example 4**: Convert to a mixed number.

**Solution**: 4 goes into 15
three times, remainder 3. So

**Simplifying and ordering fractions**

A fraction can be **simplified** if there is a whole number
that divides exactly into the numerator and denominator.

**Example**:

* * * * *

**Ordering fractions**

**Example**: Which of these
fractions is larger: ?

**Solution**: To decide which
fraction is bigger, we write both fractions with the same denominator.

To find a common denominator we can multiply 5 and 24 together: 5 × 24 = 120.

We therefore write both fractions over 120:

**Adding and subtracting fractions**

Fractions can only be added or subtracted if they have the same denominator.

**Simple cases**

(1)

(2)

Note: It is important to simplify your answers as much as possible. It is usually preferable to give your answers as a mixed number rather than as an improper fraction.

* * * * *

**Adding/ subtracting fractions with different denominators**

If the fractions have different
denominators, the first step must be to find a **common denominator**.

**Example 1**: Find .

**Solution**: Both 8 and 6
divide into 24. So we can use 24 as the
common denominator.

**Example 2**: Find .

**Solution**: Both 3 and 7
divide into 21. So we use 21 as the
common denominator.

* * * * *

**Adding mixed numbers**

Mixed numbers can be added by

(1) adding the whole numbers together

(2) adding the fractions together

(3) combining both answers together.

**Example **

Work out .

**Solution**:

* * * * *

**Subtracting mixed numbers**

When subtracting mixed numbers, it is usually simplest to begin by converting them to improper fractions.

**Example**: Work out

**Solution**:

**Multiplying Fractions**

Multiplying fractions is simple. You just multiply the top and bottom numbers together. But remember to see if your answer can simplify!!

**Example**: Find

**Solution**:

* * * * *

**Multiplying mixed numbers**

Mixed numbers should be converted to improper fractions before multiplying.

**Example 1**:

**Example 2:**

* * * * *

**Multiplying a fraction by a whole number**

A whole number can be written as a fraction with denominator 1. We can then use the rules for multiplying fractions.

**Example 1**: Find .

**Solution**: 10 can be thought
of as the fraction

So,

**Example 2**: Find .

**Solution**: