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Rounding Necessarily Increases the Answer

Source: 2015 AMC 12A #5

February 4, 2015
AMC

Problem Statement

Amelia needs to estimate the quantity ab−c\tfrac ab-c, where aa, bb, and cc are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of ab−c\tfrac ab-c?
<spanclass=′latex−bold′>(A)</span>She rounds all three numbers up.<span class='latex-bold'>(A) </span>\text{She rounds all three numbers up.}
<spanclass=′latex−bold′>(B)</span>She rounds a and b up, and she rounds c down.<span class='latex-bold'>(B) </span>\text{She rounds }a\text{ and }b\text{ up, and she rounds }c\text{ down.}
<spanclass=′latex−bold′>(C)</span>She rounds a and c up, and she rounds b down.<span class='latex-bold'>(C) </span>\text{She rounds }a\text{ and }c\text{ up, and she rounds }b\text{ down.}
<spanclass=′latex−bold′>(D)</span>She rounds a up, and she rounds b and c down.<span class='latex-bold'>(D) </span>\text{She rounds }a\text{ up, and she rounds }b\text{ and }c\text{ down.}
<spanclass=′latex−bold′>(E)</span>She rounds c up, and she rounds a and b down.<span class='latex-bold'>(E) </span>\text{She rounds }c\text{ up, and she rounds }a\text{ and }b\text{ down.}
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