The comparison of two ratios involves rewriting them in equivalent forms by multiplying or dividing both terms by the same nonzero factor, such as expressing them as fractions with a common denominator or as unit rates with a common first quantity. This preserves each multiplicative relationship and allows the ratios to be ordered or identified as equal; comparing corresponding terms independently can give an incorrect conclusion. The reasoning supports proportional relationships, percent, rate, and scale-factor problems.
To compare two ratios, rewrite them in an equivalent form that makes the comparison easy. You may multiply or divide both terms of a ratio by the same nonzero number. This keeps the ratio’s value unchanged.
Example: Which is greater, or ?
Rewrite each ratio as a fraction:
Find a common denominator. Both and divide into .
Now compare the numerators:
Therefore,
Remember to multiply or divide both parts of each ratio by the same number. Comparing the terms separately—for example, saying is greater because —can lead to an incorrect conclusion.
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