Difference between revisions of "1970 AHSME Problems/Problem 25"

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== Solution ==
 
== Solution ==
<math>\fbox{E}</math>
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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 05:54, 15 July 2019

Problem

For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. 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 $W$ ounces is always

$\text{(A) } 6W\quad \text{(B) } 6[W]\quad \text{(C) } 6([W]-1)\quad \text{(D) } 6([W]+1)\quad \text{(E) } -6[-W]$

Solution

This question is trying to convert the floor function, which is more commonly notated as $\lfloor x \rfloor$, into the ceiling function, which is $\lceil x \rceil$. The identity is $\lceil x \rceil = -\lfloor -x \rfloor$, 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 $W = 2.5$ ounces, the cost will be $18$ cents. Plugging in $W = 2.5$ into the five options gives answers of $15, 12, 6, 18, 18$. This leaves options $D$ and $E$ as viable. If $W = 2$ ounces, the cost is $12$ cents. Option $D$ remains $18$ cents, while option $E$ gives $12$ cents, the correct answer. Thus, the answer is $\fbox{E}$.

See also

1970 AHSC (ProblemsAnswer KeyResources)
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

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