Propertiesof Proportions
LearningObjectives
- Provetheoremsaboutproportions.
- Recognizetrue proportions.
- Use proportionstheoremsin problemsolving.
Introduction
The CrossMultiplicationTheoremis the basic,definingpropertyof proportions.Wheneveryou are in doubt
aboutwhethera proportionis true, you can alwayscheckit by crossmultiplication.Additionally, thereare
also a numberof “sub-theorems”aboutproportionsthat are usefulto applyfor solvingproblems.In each
casethe sub-theoremis easyto proveusingcrossmultiplication.
Propertiesof Proportions
Technicallyspeaking,the theoremsin this lessonare not calledsub-theorems.The formalterm iscorollary.
The wordcorollaryis ratherlooselydefinedin mathematics.Basically, a corollaryis a theoremthat follows
quickly, easily, and directlyfrom anothertheorem—inthis casefrom the CrossmultiplicationTheorem.
The corollariesin this sectionare not absolutelyessential—youcouldalwaysgo backto usingcrossmulti-
plication.But theremay be timeswhenthe corollariesmakethingsquickeror easier, so it’s goodto have
themif and whenthey are needed.
CrossMultiplicationCorollaries
Beloware threecorollariesthat are immediateresultsof the CrossMultiplicationTheoremand the funda-
mentallaws of algebra.
Corollaries1, 2, and 3 of The CrossMultiplicationTheoremIf , ,, and , and , then ....1..2.
.3..In words.
- A true proportionis also true if you “swap”the “means.”
- A true proportionis also true if you “swap”the “extremes.”
- A true proportionis also true if you “flip” it upsidedown.
Example 1