Question 320596
Let x = # of teeth and y = rpm



So "If a gear with 10 teeth rotates at 240rpm", then {{{x=10}}} and {{{y=240}}}. Because "The speed of a gear varies directly as the number of teeth it contains", this means that {{{y=kx}}}. Plug in {{{x=10}}} and {{{y=240}}} to get {{{240=k(10)}}} and solve for 'k' to get {{{k=240/10=24}}}. So {{{k=24}}} giving us the equation {{{y=24x}}}




Now we want to know the rpm when 18 teeth are involved. So now {{{x=18}}}. Plug this into the equation above to get {{{y=24(18)=432}}} meaning that a 18 tooth gear will spin at 432 rpm. 



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Alternate Solution:


"If a gear with 10 teeth rotates at 240rpm, with what speed would a gear with 18 teeth revolve under the same conditions? " means that {{{10/240=18/x}}}. Basically, the ratio of the number of teeth to the rpm is the same. 



{{{10/240=18/x}}} Start with the given ratio



{{{(10/240)*x=18}}} Multiply both sides by x



{{{10*x=18*240}}} Multiply both sides by 240



{{{10*x=4320}}} Multiply



{{{x=4320/10}}} Divide both sides by 10



{{{x=432}}} Reduce




So our answer is  {{{x=432}}} which means that a gear with 18 teeth will rotate at 432 rpm. So we get the same answer.