Question

Ralph went to the store and bought $12$ pairs of socks for a total of $\$24$ . Some of the socks he bought cost $\$1$ a pair, some of the socks he bought cost $\$3$ a pair, and some of the socks he bought cost $\$4$ a pair. If he bought at least one pair of each type, how many pairs of $\$1$ socks did Ralph buy?

$4$

$5$

$6$

$7$

$8$

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Written solution:

Let $a$ , $b$ , and $c$ be the numbers of $\$1$ , $\$3$ , and $\$4$ pairs, respectively. Then $a+b+c=12,\qquad a+3b+4c=24.$

Subtracting gives $2b+3c=12$ . Since at least one pair of each type was bought, $b>0$ and $c>0$ . Also $3c<12$ , so $c<4$ . Modulo $2$ , the equation gives $c$ even, so $c=2$ .

Then $2b+6=12$ , so $b=3$ , and $a=12-3-2=7$ .

Thus, D is the correct answer.

2015 AMC 8 Problem 20

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Written solution:

Problem 20 in Other Years