### Numerator and Denominator

In a fraction n/d, n is the numerator and d is the denominator. The denominator of a fraction can never be 0, because division by 0 is not defined.

**Equivalent Fractions**

Two fractions are said to be equivalent if they represent the same number. For example, 8/36 and 14/63 are equivalent since they both represent the number 2/9. In each case, the fraction is reduced to the lowest terms by dividing both numerator and denominator by their greatest common divisor (gcd). The gcd of 8 and 36 is 4 and the gcd of 14 and 63 is 7.

### Types of Fractions

**Proper fractions:** Fractions where the numerator is less than the denominator.

**Improper fractions:** Fractions where the numerator is either greater than or equal to the denominator.

**Mixed fractions:** Another way of formatting improper fractions with a whole number and a proper fraction.

**Reciprocal:** The flip-flop of a fraction. The numerator and denominator switch places.

### Addition and Subtraction of Fractions

Two fractions with the **same denominator** can be added or subtracted by performing the required operation with the numerators, leaving the denominators the same. For example,

**3/5 + 4/5 = (3+4)/5 = 7/5** and **5/7 - 2/7 = (5-2)/7 = 3/7**

If two fractions **do not have the same denominator**, express them as equivalent fractions with the same denominator. For example, to add 3/5 and 4/7, multiply the numerator and denominator of the first fraction by 7 and the numerator and denominator of the second fraction by 5, obtaining 21/35 and 20/35, respectively.

**3/5 + 4/7 = 21/35 + 20/35 = 41/35**

### Multiplication and Division of Fractions

To multiply two fractions, simply multiply the two numerators and multiply the two denominators. For example,

**2/3 × 4/7 = 8/21**

To divide by a fraction, invert the divisor (find its reciprocal) and multiply. For example

**2/3 ÷ 4/7 = 2/3 × 7/4 = 14/12 = 7/6**

### Mixed Numbers

A number that consists of a whole number and a fraction is a mixed number. For example, 7½ is a mixed number which means 7 + ½.

To change a mixed number into a fraction, multiply the whole number by the denominator of the fraction and add this number to the numerator of the fraction; then put the result over the denominator of the fraction. For example

**7½ = (2×7+1)/2 = 15/2**

### Tips

- To compare two fractions, cross-multiply them. The larger product will be on the same side as the larger fraction.
- Taking the square root of a fraction between 0 and 1 makes it larger. This is not true for fractions greater than one.
- Squaring a fraction between 0 and 1 makes it smaller.