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Difference between revisions of "2017 AMC 8 Problems/Problem 9"

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==Solution==
 
==Solution==
  
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The 6 green marbles are part of <math>1 - \frac13 - \frac14 = \frac5{12}</math> of the total marbles. If <math>6 \implies \frac13</math> of the total number of marbles, then there would be 18 marbles. Since a fourth of 18 is not a whole number, we cannot have 18 marbles. Then in <math>6 \implies \frac14</math> of the total number of marbles, it works, because 24/4 = 6, and 24/3 = 8. So we have that 24 - 6 - 6 - 8 = 10 - 8 = 4 marbles, or D.
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The <math>6</math> green marbles and yellow marbles form <math>1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}</math> of the total marbles. Now suppose the total number of marbles is <math>x</math>. We know the number of yellow marbles is <math>\frac{5}{12}x - 6</math> and a positive integer. Therefore, <math>12</math> must divide <math>x</math>. Trying the smallest multiples of <math>12</math> for <math>x</math>, we see that when <math>x = 12</math>, we get there are <math>-1</math> yellow marbles, which is impossible. However when <math>x = 24</math>, there are <math>\frac{5}{12} \cdot 24 - 6 = \textbf{(D) } 4</math> yellow marbles, which must be the smallest possible.
  
 
==See Also==
 
==See Also==

Revision as of 16:10, 22 November 2017

Problem 9

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Solution

The $6$ green marbles and yellow marbles form $1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}$ of the total marbles. Now suppose the total number of marbles is $x$. We know the number of yellow marbles is $\frac{5}{12}x - 6$ and a positive integer. Therefore, $12$ must divide $x$. Trying the smallest multiples of $12$ for $x$, we see that when $x = 12$, we get there are $-1$ yellow marbles, which is impossible. However when $x = 24$, there are $\frac{5}{12} \cdot 24 - 6 = \textbf{(D) } 4$ yellow marbles, which must be the smallest possible.

See Also

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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