Difference between revisions of "1970 AHSME Problems/Problem 25"
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== Solution == | == Solution == | ||
| − | <math>\fbox{E}</math> | + | This question is trying to convert the floor function, which is more commonly notated as <math>\lfloor x \rfloor</math>, into the ceiling function, which is <math>\lceil x \rceil</math>. The identity is <math>\lceil x \rceil = -\lfloor -x \rfloor</math>, which can be verified graphically, or proven using the definition of floor and ceiling functions. |
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| + | However, for this problem, some test values will eliminate answers. If <math>W = 2.5</math> ounces, the cost will be <math>18</math> cents. Plugging in <math>W = 2.5</math> into the five options gives answers of <math>15, 12, 6, 18, 18</math>. This leaves options <math>D</math> and <math>E</math> as viable. If <math>W = 2</math> ounces, the cost is <math>12</math> cents. Option <math>D</math> remains <math>18</math> cents, while option <math>E</math> gives <math>12</math> cents, the correct answer. Thus, the answer is <math>\fbox{E}</math>. | ||
== See also == | == See also == | ||
Latest revision as of 06:54, 15 July 2019
Problem
For every real number 
, let ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
be the greatest integer which is less than or equal to . If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing 
ounces is always
![$\text{(A) } 6W\quad \text{(B) } 6[W]\quad \text{(C) } 6([W]-1)\quad \text{(D) } 6([W]+1)\quad \text{(E) } -6[-W]$](http://latex.artofproblemsolving.com/d/d/8/dd87b2bfea27ac57c7b6152f364f45de184af6ac.png)
Solution
This question is trying to convert the floor function, which is more commonly notated as 
, into the ceiling function, which is 
. The identity is 
, which can be verified graphically, or proven using the definition of floor and ceiling functions.
However, for this problem, some test values will eliminate answers. If 
ounces, the cost will be 
cents. Plugging in into the five options gives answers of 
. This leaves options 
and 
as viable. If 
ounces, the cost is 
cents. Option remains cents, while option gives cents, the correct answer. Thus, the answer is 
.
See also
| 1970 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Problem 26 | |
| 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 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
