Depth · one concept from first principles · step 2 of 4
Gears P and Q, whose numbers of teeth are in the ratio , mesh with each other, and gears Q and R, whose numbers of teeth are in the ratio , mesh with each other, so all three gears turn together. Gear P makes turns per minute and gear R makes turns per minute. Write an equation that gives in terms of .
Q is 3 parts in the first ratio and 2 parts in the second. Rescale both so Q is 6 parts.
A shared middle term has to be written with the same number before the ratios can be joined.
6.RP.A.3Organize Information In More WaysLet the gears have 4a, 6a and 15a teeth for some natural number a.
Only the ratio is known, so a common multiplier a stands in for the actual size.
6.RP.A.3Convert To AlgebraEach minute 4a × x teeth pass on P; Q hands the same count on to R, where it is 15a × y.
Q's own speed does not matter; it just hands the teeth count from P to R.
8.F.B.4Analyze The UnitsIn one minute the same number of teeth pass on P as on R, so .
Where two gears mesh, each tooth that goes by on one gear pushes exactly one tooth by on the other, so the two counts match.
P turns times a minute and each turn carries all of its teeth past, so teeth go by; R likewise moves .
P's count equals Q's count and Q's count equals R's count, so P's count equals R's count.
Divide both sides by 15a.
R has more teeth than P, so R turns more slowly; the factor 4/15 is less than 1.
8.F.B.4Convert To AlgebraFor meshed gears, multiply teeth by turns: that product is the same on every gear in the chain.
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