Contents:

1.Proper Fractions.

2.Improper Fractions

3.Mixed Numbers

4.Equivalent Fractions

5. Complex fractions

6. Reciprocals.

1.Proper Fractions.

If the absolute value of the numerator is less than the absolute value of the denominator, the fraction is said to be a proper fraction.

Examples:

 are proper fractions.

2.Improper Fractions

Improper fractions have numerators that are larger than or equal to their denominators.

Examples:

 are improper fractions.

3.Mixed Numbers

Mixed numbers have a whole number part and a fraction part.

Examples:

 are mixed numbers also written as . Here we denote mixed numbers in the form .

3.1 Converting Mixed Numbers to Improper Fractions

To change a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator of the fractional part.

Examples:

3.2 Converting Improper Fractions to Mixed Numbers

To change an improper fraction into a mixed number, divide the numerator by the denominator. The remainder is the numerator of the fractional part.

Examples:

= 11 ÷ 4 = 2 r3 = 2 3/4

 = 13 ÷ 2 = 6 r1 = 6 1/2

4.Equivalent Fractions

Equivalent fractions are different fractions which name the same amount.

Examples:

The fractions  are all equivalent fractions.
The fractions  are all equivalent fractions.
We can check if two fractions are equivalent by cross-multiplying their numerators and denominators. This is also called taking the cross-product.

Example:

Check if   are equivalent fractions.
The first cross-product is the product of the first numerator and the second denominator: 3 × 42 = 126.

The second cross-product is the product of the second numerator and the first denominator: 18 × 7 = 126.

Since the cross-products are the same, the fractions are equivalent.

Example:

Check if   are equivalent fractions.
The first cross-product is the product of the first numerator and the second denominator: 2 × 20 = 40.

The second cross-product is the product of the second numerator and the first denominator: 4 × 13 = 52.

Since the cross-products are different, the fractions are not equivalent

5. Complex fractions

A complex fraction (or compound fraction) is a fraction in which the numerator and denominator contain a fraction. For example, ^½⁄_⅓ is a complex fraction. To simplify a complex fraction, divide the numerator by the denominator, as with any other fraction: ^½⁄_⅓ = ³⁄₂.

6. Reciprocals.

The reciprocal of a fraction is obtained by switching its numerator and denominator.

To find the reciprocal of a mixed number, first convert the mixed number to an improper fraction, then switch the numerator and denominator of the improper fraction.

Notice that when you multiply a fraction and its reciprocal, the product is always 1.

Example:

Find the reciprocal of

We switch the numerator and denominator to find the reciprocal:

.

Example:

Find the reciprocal of 

First, convert the mixed number to an improper fraction:

 = 

Next, we switch the numerator and denominator to find the reciprocal: