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A ratio is an expression used to compare two quantities. A simplified ratio can be taken as is, but if a ratio has not yet been simplified, you should do so to make the quantities easier to compare and understand. In order to simplify a ratio, you divide both terms (both sides of the ratio) by the same number. This process is equivalent to reducing a fraction. Example: 15:21{\displaystyle 15:21} Note that neither number in this example is a prime number. Since that is the case, you'll need to factor both numbers to determine whether or not the two terms have any identical factors that can cancel each other in the simplification process. A factor is a whole number (or expression) that can evenly divide into the term, leaving another whole number (or expression) as the quotient. Both terms in the ratio must share at least one factor (other than the number 1) or the ratio cannot be simplified. Before you can determine if the terms do share a factor, you must discover what the factors of each term are. Example: The number 15 has four factors: 1,3,5,15{\displaystyle 1,3,5,15} 151=15{\displaystyle {
rac {15}{1}}=15} 153=5{\displaystyle {
rac {15}{3}}=5} In a separate space, list all the factors of the ratio's second term. At this point do not consider the factors of the first term; focus only on factoring this second term. Example: The number 21 has four factors: 1, 3, 7, 21 211=21{\displaystyle {
rac {21}{1}}=21} 213=7{\displaystyle {
rac {21}{3}}=7} Look at the factors for both terms of the ratio. Circle, list, or otherwise identify any factors that appear in both lists. If the only shared factor is 1, then the ratio is already in its simplest form, and no further work needs to be done. If the two terms of the ratio have other shared factors, however, sort through them and identify the highest factor common to both lists. This number is the greatest common factor (GCF). Example: Both 15 and 21 share two common factors: 1 and 3 The GCF for the two terms of the original ratio is 3. Since both terms of the original ratio contain the GCF, you can divide each term by that number and come up with whole numbers as a result. Both terms must be divided by the GCF. Example: Both 15 and 21 are divided by 3. 153=5{\displaystyle {
rac {15}{3}}=5} 213=7{\displaystyle {
rac {21}{3}}=7} You are left with two new terms. The new ratio is equivalent in value to the original ratio, meaning that the terms in one ratio are in the same proportion as the terms in the other ratio. Note that the terms of the new ratio should not share any common factors between them (other than 1). If they do, the ratio is not yet in simplest form. Example: 5:7{\displaystyle 5:7} The point of all this is that the simplified ratio 5:7 is easier to work with than the original ratio 15:21.
Look at the ratio. Factor the first term. Factor the second term. Find the greatest common factor. Divide both terms by the greatest common factor. Write down the new simplified ratio.