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2015 CEMC Gauss (Grade 8) Problems/Problem 9

Revision as of 12:35, 23 May 2025 by Anabel.disher (talk | contribs) (Created page with "== Problem== If <math>x = 4</math> and <math>y = 2</math>, which of the following expressions gives the smallest value? <math> \textbf{(A)}\ x + y \qquad\textbf{(B)}\ xy \qqu...")
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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

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