### Mixed Numbers

**Mixed numbers** are numbers that are an integer plus a proper fraction.

The number 4\dfrac{2}{3} is the integer 4, plus the fraction \dfrac{2}{3}

Any mixed number can be written as a fraction, and any fraction greater than 1 can be written as a mixed number.

*Proper fractions* have a value between 0 and 1.

*Improper fractions* have a value greater than 1. The numerator is greater than the denominator. An improper fraction is another way to write a mixed number.

To express a mixed number as an improper fraction, write the integer as a fraction, then add the fractions.

##### Example

Convert 4\dfrac{2}{3} into an improper fraction.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The remainder will be the numerator of the fraction.

### Equivalent Fractions

A fraction that has a common factor in both the numerator and denominator is equal to the fraction with the common factor canceled. The fraction 6/10 is equivalent to the fraction 3/5 since they are equal with the common factor 2 in both numerator and denominator of 6/10.

Multiplying the numerator and denominator of a fraction by the same (non-zero) number also gives a new fraction which is equivalent to the original fraction. The fractions 3/5 = 6/10 = 9/15 = 12/20 are equivalent.

\dfrac{3 × 2}{5 × 2} = \dfrac{6}{10} = \dfrac{6 × 2}{10 × 2} = \dfrac{12}{20}

A fraction with a negative numerator or denominator is equivalent to a negative fraction.

-\dfrac{3}{5} = \dfrac{-3}{5} = \dfrac{3}{-5}

If both numerator and denominator are negative, the fraction is positive.

\dfrac{-3}{-5} = \dfrac{3}{5}

To simplify fractions, one method is to use the GCF (greatest common factor)

Divide the numerator and denominator by the GCF to reduce the fraction.

##### Example one

Reduce 275/525 to lowest possible terms.

Another method is to find the prime factors and cancel.

##### Example two

Reduce 220/594 to simplest terms.

### Multiplying Fractions

To multiply fractions, multiply the numerators, then multiply the denominators, and then reduce the fraction. There is a shortcut that will make fraction multiplication less tedious.

To multiply mixed numbers, first change the numbers to improper fractions.

#### Dividing Fractions

When dividing fractions, use the reciprocal. Informally, the reciprocal of a fraction is the fraction flipped upside down.

The reciprocal of 2/3 is 3/2. The reciprocal of 5/4 is 4/5.

To divide fractions, change the divisor to its *reciprocal*, then multiply the fractions. (Remember that the divisor is the second number.) When multiplying, cancel out any common factors that appear in both numerators and denominators.

##### Example one

Divide: (5/6) ÷ (7/2)

To divide mixed numbers, first change the numbers to improper fractions.

##### Example two

Divide: \dfrac{3}{5} ÷ 2\dfrac{1}{10}

### Complex Fractions

A **complex fraction** is a fraction that has a fraction in the numerator and/or denominator. In other words, it is a fraction divided by a fraction. Complex fractions contain variable expressions. To simplify, use the reciprocal of the divisor, then multiply.

##### Example one

Simplify: (*x*/*y*)/(2*x*/3)

### Adding and Subtracting Fractions

To add or subtract fractions that have the same denominator, add or subtract the numerators. Check to see that the answer is in simplest terms.

For example, 3/8 – 1/8 = 2/8 = 1/4.

To add or subtract fractions that have different denominators, the first step is to write equivalent fractions that have the same, or a common, denominator.

To write all fractions with the same denominator, a quick choice is to multiply the denominators. For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

But multiplying the denominators may give a rather large denominator. To avoid a large denominator, use the least common denominator (LCD). The LCD is the least common multiple (LCM) of all the denominators.

##### Example one

Subtract: 7/12 – 1/16

A final rule about adding and subtracting fractions is that you can split up the numerator, but you can never split up the denominator.

\dfrac{7 + 14}{7} = \dfrac{7}{7} + \dfrac{14}{7} = 1 + 2 = 3 Splitting the numerator works gives you the same value.

\dfrac{7}{7 + 14} ≠ \dfrac{7}{7} + \dfrac{7}{14} = 1 + \dfrac{1}{2} Splitting the denominator gives you a different value, and is therefore incorrect.

### Cross Multiplication

One method of comparing fractions is to find the common denominator.

##### Example two

Which fraction is greater, 3/4 or 8/11?

An easier way to compare fractions is **cross multiplication**. Multiply the numerator of one fraction by the denominator of the other fraction and compare the products.

##### Example three

Which fraction is greater, 7/30 or 21/91?

Cross multiplication is also used to solve rational equations. Rational equations have fractions set equal with variable expressions in the numerators and denominators. There will be more about rational equations in Chapter 6 Algebra.

##### Example one

Solve: *x*/16 = 5/2