Equivalent Fractions

Two fractions which represent the same numerical values are said to be equivalent fractions.

For example,

are all fractions equal to the decimal value 0.75. Even though each fraction in this list has a different numerator and denominator combination than the others, still each of these fractions represents the same numerical value – hence they are equivalent fractions.

Two fractions are equivalent when you can convert the one with the smaller (or simpler) numerator and denominator into the other fraction by multiplying both its numerator and denominator by the same number. Thus

 is equivalent to

because if we multiply both the numerator and denominator of 3 / 4 by 5, we get 15 / 20:

The importance of recognizing two fractions are equivalent is that you can replace a fraction by another equivalent fraction if it is convenient or useful to do so, and you will not have changed the numerical value of the expression with which you are working. This provides a powerful tool for doing mathematics with fractions.

Two fractions are also equivalent if we can convert the fraction with the larger numerator and denominator into the other fraction by dividing both numerator and denominator by the same number. Thus

 is equivalent to

because

That is, when we divide both the numerator and the denominator of 15 / 20 by 5, we end up with 3 / 4.