AMC10 2015 A

AMC10 2015 A · Q13

AMC10 2015 A · Q13. It mainly tests Money / coins, Basic counting (rules of product/sum).

Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?

Claudia 有 12 枚硬币，每枚硬币要么是 5 分硬币，要么是 10 分硬币。用她的一枚或多枚硬币进行组合，恰好可以得到 17 种不同的金额。Claudia 有多少枚 10 分硬币？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): If Claudia only has 10-cent coins, then she can make 12 different values. Otherwise, suppose that the number of 10-cent coins is $d$ and thus the number of 5-cent coins is $12-d$. Then she can make any value that is a multiple of 5 from 5 to $10d+5(12-d)=5(d+12)$. Therefore $d+12=17$, and $d=5$.

答案（C）：如果 Claudia 只有 10 美分硬币，那么她可以凑出 12 种不同的金额。否则，设 10 美分硬币的数量为 $d$，则 5 美分硬币的数量为 $12-d$。那么她可以凑出从 5 到 $10d+5(12-d)=5(d+12)$ 之间任意一个 5 的倍数金额。因此 $d+12=17$，且 $d=5$。

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