1970 AHSME Problems/Problem 25

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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
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