AMC8 2015

AMC8 2015 · Q20

AMC8 2015 · Q20. It mainly tests Systems of equations, Parity (odd/even).

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?

拉尔夫去商店买了12双袜子，总共24美元。他买的一些袜子每双1美元，一些每双3美元，一些每双4美元。如果他至少买了每种各一双，拉尔夫买了多少双1美元的袜子？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): If Ralph buys 6 pairs of \$1 socks, then the other 6 pairs of socks would cost at least \$19 making the total cost more than \$24. Buying fewer than 6 pairs of \$1 socks would make Ralph’s cost even higher. If he bought 8 pairs of \$1 socks, then the other 4 pairs would cost less than \$16 making the total cost less than \$24. Buying more than 8 pairs of \ $1 socks would make his total cost even lower. So Ralph bought 7 pairs of \$1 socks, 3 pairs of \$3 socks, and 2 pairs of \$4 socks.

答案（D）：如果 Ralph 买 6 双 $1 的袜子，那么其余 6 双袜子至少要花 $19，使得总花费超过 $24。若买少于 6 双 $1 的袜子，Ralph 的花费会更高。若他买 8 双 $1 的袜子，那么其余 4 双会少于 $16，从而总花费低于 $24。若买多于 8 双 $1 的袜子，总花费会更低。因此 Ralph 买了 7 双 $1 的袜子、3 双 $3 的袜子，以及 2 双 $4 的袜子。

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