### Equivalent Fractions

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

### Reducing Fractions

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.