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Question 468163: When two gears are mashed the revolution per minutes (rpm) are inversely proportional to the number of teeth, as show below. In one machine the rpm ratio of the larger gear to the small gear is 2:7 The small gear has 100 teeth In second machine the larger gear revolves at 400 rpm and has 301 teeth. The smaller gear revolves at 700 rpm.
Find the number of teeth on the larger gear of the first machine?
Find the number of teeth on the small gear of the first machine?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Assign the following variables:
LR = Large gear rpm
SR = Small gear rpm
LT = Large gear teeth
ST = Small gear teeth
If the ratio was directly proportional, you would get:
LR / SR = LT / ST
Since the ratio is inversely proportional, you will get:
LR / SR = ST / LT
With your first machine, you have the following information.
LR = 2x
SR = 7x
The reason for the x is because you only know the ratio and you do not know the actual rpm.
Multiply the ratio by x and you get the actual rpm.
This will not mess up the calculations, because those x's are going to cancel out in the equation anyway.
With your first machine, you also have the following information:
LT = ?????
ST = 100
The inverse proportional ratio states that:
LR / SR = ST / LT
Replacing the numbers that we know about, this equation becomes:
2x / 7x = 100 / LT
Since there is an x in the numerator and an x in the denominator, the x will cancel out, so we can just get rid of it.
Our equation becomes:
2 / 7 = 100 / LT
We cross multiply to get:
2 * LT = 7 * 100
We divide both sides of the equation by 2 to get:
LT = (7 * 100) / 2
This gets us:
LT = 350
Our numbers for the first machine are now:
LR = 2x
SR = 7x
LT = 350
ST = 100
The large gear has 2x revolutions per minute with 350 teeth.
The small gear has 7x revolutions per minute with 100 teeth.
More Teeth gets you less RPM which is how inverse proportionality works.
SECOND MACHINE
With the second machine, you have the following information:
LR = 400
SR = 700
LT = 301
ST = ?????
We use the same inverse proportional ratio of:
LR / SR = ST /LT
Plugging in the numbers that we know gets us:
400 / 700 = ST / 301
Cross Multiply to get:
400 * 301 = 700 * ST
Divide both sides of the equation by 700 gets us:
ST = (400 * 301) / 700
This gets us:
ST = 172
The numbers for the second machine are now:
LR = 400
SR = 700
LT = 301
ST = 172
The large gear has 400 rpm with 301 teeth.
The small gear has 700 rpm with 172 teeth.
Once again, the gear with the larger number of teeth revolves slower than the gear with the smaller number of teeth.
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