2005 CEMC Pascal Problems/Problem 7

Problem

If $\frac{1}{3}x = 12$ , then $\frac{1}{4}x$ equals

$\text{ (A) }\ 1 \qquad\text{ (B) }\ 16 \qquad\text{ (C) }\ 9 \qquad\text{ (D) }\ 144 \qquad\text{ (E) }\ 64$

Solution 1

We can first find the value of $x$ , and then divide the value by $4$ :

$\frac{1}{3}x = 12$

$x = 12 \times 3 = 36$

$\frac{x}{4} = \frac{36}{4}$

$\frac{1}{4}x = \boxed {\textbf {(C) } 9}$

~anabel.disher

Solution 1.5

Without solving for $x$ , we can notice that multiplying both sides of the equation by $\frac{3}{4}$ will give $\frac{1}{3}x \times \frac{3}{4} = \frac{1}{4}x$ . This gives:

$\frac{1}{4}x = 12 \times \frac{3}{4} = \boxed {\textbf {(C) } 9}$

~anabel.disher

Solution 2 (answer choices)

We can see that for $\frac{1}{4}x$ to be an integer, $x$ must be divisible by $3$ because $\frac{1}{3}x$ is an integer. This means $\frac{1}{4}x$ is also an integer ( $4$ is not a multiple of $3$ ), leaving only answer choices C and D.

D is clearly too large, as $\frac{1}{3} > \frac{1}{4}$ , so $\frac{1}{4}x$ should be less than $\frac{1}{3}x$ , which isn't the case with answer choice D. This means that the answer must be $\boxed {\textbf {(C) } 9}$ .

~anabel.disher

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2005 CEMC Pascal Problems/Problem 7

Contents

Problem

Solution 1

Solution 1.5

Solution 2 (answer choices)