Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2015 CEMC Gauss (Grade 8) Problems/Problem 9

Problem

If $x = 4$ and $y = 2$, which of the following expressions gives the smallest value?

$\textbf{(A)}\ x + y \qquad\textbf{(B)}\ xy \qquad\textbf{(C)}\ x - y \qquad\textbf{(D)}\ x \div y \qquad\textbf{(E)}\ y \div x$

Solution 1

Solving for each of the answer choices, we get:

$x + y = 4 + 2 = 6$

$xy = 4 \times 2 = 8$

$x - y = 4 - 2 = 2$

$x \div y = 4 \div 2 = 2$

$y \div x = 2 \div 4 = 0.5$

The smallest of these is $\boxed {\textbf {(E) } y \div x}$.

~anabel.disher

Solution 2

Without calculating the values, we can note that $x$ and $y$ are positive integers and $x > y + 1$

Thus,

$x + y > 2y + 1 > 1$

$xy > 1$

$x - y > y + 1 - y = 1$, therefore $x - y > 1$

$x \div y > (y + 1) \div y > 1$ since the numerator is greater than the denominator. Therefore, $x \div y > 1$.

$y \div x < y \div (y + 1) < 1$, therefore $x \div y < 1$.

The only answer choice that is less than $1$ is $\boxed {\textbf {(E) } y \div x}$.

~anabel.disher

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_CEMC_Gauss_(Grade_8)_Problems/Problem_9&oldid=248948"