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.