Difference between revisions of "2015 AMC 12A Problems/Problem 5"

m (Problem)
(Problem)
 
Line 1: Line 1:
 
==Problem==
 
==Problem==
  
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> 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 <math>\frac{a}{b} - c</math>?
+
Sreshtha needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> 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 <math>\frac{a}{b} - c</math>?
  
 
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\
 
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\

Latest revision as of 05:00, 15 February 2021

Problem

Sreshtha needs to estimate the quantity $\frac{a}{b} - c$, where $a, b,$ and $c$ 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 $\frac{a}{b} - c$?

$\textbf{(A)}\ \text{She rounds all three numbers up.}\\ \qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{down.}\\ \qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{down.} \\ \qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{down.}\\ \qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{down.}$

Solution

To maximize our estimate, we want to maximize $\frac{a}{b}$ and minimize $c$, because both terms are positive values. Therefore we round $c$ down. To maximize $\frac{a}{b}$, round $a$ up and $b$ down. $\Rightarrow \boxed{\textbf{(D)}}$

See Also

2015 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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
All AMC 12 Problems and Solutions
Invalid username
Login to AoPS