Benjamin Banneker, an 18th century mathematician who lived on his parents' Patapsco River farm in Baltimore County, was the first person ever to build a wooden clock in the United States. By taking apart a pocket watch, Banneker figured out that the ratio of the gears was the key to making it all work. Banneker's clock struck every hour for more than forty years, keeping perfect time.
Suppose you have these three gears:
Think of them meshing like this:
If Gear C moves clockwise three revolutions,
Will Gear B move clockwise or counter-clockwise?
How many revolutions will it make?
Will Gear A move clockwise or counter-clockwise?
How many revolutions will it make?
Solution to Problem:
First, you must find how many teeth each gear has.
To work, gears must go different directons. That means that gear B
will go counter clockwise and gear A will go clockwise.
Compare the number of teeth in gears C and B. Gear C had
12 teeth, gear B had 6. To find the number of revolutions, you divide the
number of teeth in C by the number of teeth in B.
12 / 6 = 2. So, gear B makes 2 revolutions for each revolution that gear C makes.
Hence, gear B will make 6 revolutions for the three that gear C makes.
Now compare the number of teeth in gears B and A. Gear B had
6 teeth, but it has to be multiplied by 2 because it is going to
revolve 2 times. A had 3 teeth. To find the number of revolutions, you
divide the number of teeth in B * 2 by the number of teeth in A.
So, gear A will make 4 revolutions for each revolution that gear C makes;
therefore, when gear C makes three revolutions, gear A will make 4 revolutions.
gear B made 3 * (12 / 6) = 6 revolutions in a counter clockwise direction;
gear A made 3 * (12 / 3) = 12 revolutions in a clockwise direction.
From the picture, you can see that
- A has 3 teeth.
- B has 6 teeth.
- C has 12 teeth.
(# of revolutions are shown in the table)
Gear C Gear B Gear A
-------- -------- --------
1 2 4
2 4 8
3 6 12
I. If C made one revolution, then:
- B made 1 * (12 / 6) = 2 revolutions;
- A made 1 * (12 / 3) = 4 revolutions.
II. If C made 2 revolutions, then:
- B made 2 * (12 / 6) = 4 revolutions;
- A made 2 * (12 / 3) = 8 revolutions.
III. If C made n revolutions, then:
- B made n * (12 / 6) = 2n revolutions;
- A made n * (12 / 3) = 4n revolutions.
Correctly solved by:
1. William Funk
San Antonio, Texas
2. Richard Johnson
La Jolla, California
3. Jeffrey Gaither
Winchester, Virginia
4. John Funk
Ventura, California
5. Walt Arrison
Philadelphia, Pennsylvania
6. Rick Jones
Kennett Square, Pennsylvania
7. Matt Stillwagon
Winchester, Virginia
8. Josh Payne
Winchester, Virginia
9. James Alarie
University of Michigan -- Flint
Flint, Michigan
10. Michael Rodriguez
Great Falls, Montana
11. Tina Zahel
Winchester, Virginia
12. Emily Butler
Columbus, Georgia
13. Ben Reames
Columbus, Georgia
14. David & Judy Dixon
Bennettsville, South Carolina
15. Daniel Wilberger
Winchester, Virginia
16. Le Van Hot
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17. Michael Sullivan
Columbus, Ohio
18. Gbenga Kuforiji
Columbus, Georgia
19. Steve Muller
Clearbrook, Virginia