AMC10 2000 A

AMC10 2000 A · Q17

AMC10 2000 A · Q17. It mainly tests Money / coins, Logic puzzles.

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

鲍里斯有一个神奇的找零机。当他投入一个25美分硬币时，机器返回五个5美分硬币；投入一个5美分硬币时，返回五个1美分硬币；投入一个1美分硬币时，返回五个25美分硬币。鲍里斯开始只有一个1美分硬币。经过反复使用机器后，他可能拥有下列哪种金额？

(A) $3.63 $3.63

(B) $5.13 $5.13

(D) $7.45 $7.45

(E) $9.07 $9.07

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Neither of the exchanges quarter → five nickels nor nickel → five pennies changes the total value of Boris’s coins. The exchange penny → five quarters increase the total value of Boris’s coins by \$1.24. Hence, Boris must have \$0.01 + \$1.24$n$ after $n$ uses of the last exchange. Only option D is of this form: $745 = 1 + 124 \cdot 6$. In cents, option A is 115 more than a multiple of 124, B is 17 more than a multiple of 124, C is 10 more than a multiple of 124, and E is 39 more than a multiple of 124.

答案（D）：无论是把 1 枚 25 美分硬币换成 5 枚 5 美分硬币，还是把 1 枚 5 美分硬币换成 5 枚 1 美分硬币，都不会改变 Boris 硬币的总价值。把 1 枚 1 美分硬币换成 5 枚 25 美分硬币，会使 Boris 硬币的总价值增加 \$1.24。因此，进行最后一种交换 $n$ 次后，Boris 的总金额应为 \$0.01 + \$1.24$n$。只有选项 D 具有这种形式：$745 = 1 + 124 \cdot 6$。按“美分”为单位看，选项 A 比某个 124 的倍数多 115，B 多 17，C 多 10，E 多 39。

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