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As is true of any ratio, an algebraic ratio compares two quantities, although in this case variables (letters) have been introduced into one or both terms. You will need to simplify numerical terms (as shown above) as well as any variables when finding a ratio's simplified form. Example: 18x2:72x{\displaystyle 18x^{2}:72x} Remember that factors can be whole numbers which divide evenly into a given quantity. Look at the numerical values in both terms of the ratio. Write out all factors for both numerical terms in separate lists. Example: To solve this problem, you will need to find the factors of 18 and 72. The factors of 18 are: 1, 2, 3, 6, 9, 18 The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Go through both factor lists and circle, underline, or otherwise identify all of the factors shared by both lists. From this new selection of numbers, identify the highest number. This value is the greatest factor common to both of the numerical terms. Note, however, that this value represents only part of the greatest common factor within the ratio. (We still have the variables to deal with.) Example: Both 18 and 72 share several factors: 1, 2, 3, 6, 9, and 18. Of these factors, 18 is the greatest. You should be able to evenly divide both numerical terms by the GCF. Do so now, and write down the whole numbers that you get as a result. These numbers will be part of the final simplified ratio. Example: Both 18 and 72 are now divided by the factor 18. 1818=1{\displaystyle {
rac {18}{18}}=1} 7218=4{\displaystyle {
rac {72}{18}}=4} Look at the variable in both terms of the ratio. If the same variable appears in both terms, it can be factored out. If there are exponents (powers) applied to the variable in both terms, deal with them now. If the exponents are the same in both terms, they cancel each other completely. If the exponents are not the same, subtract the smaller exponent from the larger. This completely cancels the variable with the smaller exponent and leaves the other variable with a diminished exponent. Understand that by subtracting one power from the other, you are essentially dividing the larger variable amount by the smaller one. Example: When examined separately, the ratio of variables was: x2:x{\displaystyle x^{2}:x} You can factor out an x{\displaystyle x} from both terms. The power of the first x{\displaystyle x} is 2, and the power of the second x{\displaystyle x} is 1. As such, one x{\displaystyle x} can be factored out from both terms. The first term will be left with one x{\displaystyle x}, and the second term will be left with no x{\displaystyle x}. x(x:1){\displaystyle x(x:1)} x:1{\displaystyle x:1} Combine the GCF of the numerical values with the GCF of the variables to find the full GCF. This GCF is the term that must be factored out of both terms of the ratio. Example: The greatest common factor in this example is 18x{\displaystyle 18x}. 18x(x:4){\displaystyle 18x\cdot (x:4)} After you remove the GCF, the remaining ratio is the simplified form of the original ratio. This new ratio is proportionally equivalent to the original ratio. Note again that the two terms of the final ratio must not share any common factors (except 1). Example: x:4{\displaystyle x:4}
Look at the ratio. Factor both terms. Find the greatest common factor. Divide both sides by the greatest common factor. Factor out the variable if possible. Note all of the greatest common factor. Write the simplified ratio.