Two fractions are equivalent if they express the same part of a whole. For example, 2/3 and 4/6 express the same part of a whole. 12/9 and 4/3 are also equivalent.

Two fractions are equivalent if there is a number by which both the numerator and the denominator of one fraction can be multiplied or divided to yield the other fraction. For example, we can multiply the numerator and denominator of 2/3 by 2 to yield 4/6, and we can divide the numerator and denominator of 12/9 by 3 to yield 4/3.

To find a fraction that is equivalent to another fraction but has a specified (different) denominator, determine what the old denominator must be multiplied by to yield the new denominator. Then multiply the old numerator by that same number. For example, to find a fraction equivalent to 2/9 with a denominator of 45:

1. 9×5 = 45
2. 2×5 = 10
The fraction equivalent to 2/9 is 10/45.

Some fractions, like 6/8, can be written as other fractions with a lower denominator. 6/8 = 3/4 (Note that 6/8 and 3/4 are equivalent by the above definition). Others, like 5/8, cannot be written with a lower denominator. 3/4 and 5/8 are said to be in lowest terms because they cannot be reduced further.

How does one know which fractions can be reduced and which cannot be reduced? In fractions that can be reduced (fractions not in lowest terms), the numerator and the denominator share at least one common factor. In fractions that cannot be reduced (fractions in lowest terms), the numerator and the denominator share no common factors; that is, they are relatively prime.

To write a fraction in lowest terms, factor the numerator and the denominator. Then divide both the numerator and the denominator by the greatest common factor. For instance, take the following steps to factor 36/126:

1. Factor. 36 = 2×2×3×3 and 126 = 2×3×3×7.
2. Find the GCF. The GCF of 36 and 126 is 2×3×3 = 18.
3. Divide. 36/18 = 2 and 126/18 = 7.

The reduced fraction is 2/7.

A common denominator of two numbers is a number that can be divided by the denominators of both numbers. For example, 1/6 and 4/9 have common denominators of 18, 36, 54, 72, etc. The least common denominator, or LCD, is the lowest number that can be divided by the denominators of both numbers. For example, 18 is the least common denominator of 1/6 and 4/9.

The least common denominator of two fractions is the least common multiple of their denominators. 18 is the LCM of 6 and 9.

The least common denominator is a helpful tool in allowing you to take two different fractions (ex. 3/4 and 7/11) and write them as equivalent fractions with the same denominator (ex. 33/44 and 28/44). Such a tool is important in comparing the size of fractions and because fractions can only be added and subtracted from each other when they have the same denominator. The first step in the process is to find the LCD. Then write each fraction as an equivalent fraction with the LCD as a new denominator, using the two steps detailed in the section on equivalent fractions.

Example 1: Write 3/14 and 4/21 as fractions with the same denominator.

I. Find the LCD
II. Write each fraction as an equivalent fraction with the LCD (42) as the new denominator.
Thus, 3/14 = 9/42 and 4/21 = 8/42.

Note: The number by which the numerator must be multiplied in Part II will be the product of the factors of the other denominator that are not factors of its denominator. Here, 3 was multiplied by 3, which is a factor of 21 but not of 14, and 4 was multiplied by 2, which is a factor of 14 but not of 21.

Example 2: Write 2/5, 5/12, and 9/8 as fractions with the same denominator.

I. Find the LCD.
II. Write each fraction as an equivalent fraction with the LCD (120) as the new denominator.
Thus, 2/5 = 48/120, 5/12 = 50/120, and 9/8 = 135/120.

It is very difficult to tell whether 3/14 is greater or less than 4/21 just by looking at the two fractions. Here's where writing them with a common denominator comes in handy. We know that 3/14 = 9/42 and that 4/21 = 8/42. Since 9 is greater than 8, 9/42 is greater than 8/42. Thus 3/14 is greater than 4/21.

To determine which of two fractions is greater, write them as fractions with the same denominator. Then see which new fraction has the larger numerator. This is the greater fraction.