Difference between revisions of "1980 AHSME Problems/Problem 10"
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== Solution == | == Solution == | ||
| − | <math>\ | + | Since the gears are meshed, they must all rotate the same number of teeth per minute. Let <math>k</math> be this rate. Then gears <math>A</math>, <math>B</math>, and <math>C</math> must have angular speeds of <math>\frac{k}{x}</math>, <math>\frac{k}{y}</math>, and <math>\frac{k}{z}</math>, respectively. Thus the ratio of the three gears' angular speeds is <math>\frac{k}{x}:\frac{k}{y}:\frac{k}{z}</math>. Multiplying each term of this ratio by <math>\frac{xyz}{k}</math> yields <math>\boxed{(\textbf{D})\ yz:xz:xy}</math>. |
== See also == | == See also == | ||
Latest revision as of 20:36, 25 July 2025
Problem
The number of teeth in three meshed gears 
, 
, and 
are 
, 
, and 
, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of , , and are in the proportion
Solution
Since the gears are meshed, they must all rotate the same number of teeth per minute. Let 
be this rate. Then gears , , and must have angular speeds of 
, 
, and 
, respectively. Thus the ratio of the three gears' angular speeds is 
. Multiplying each term of this ratio by 
yields 
.
See also
| 1980 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
