AMC12 2004 A

AMC12 2004 A · Q11

AMC12 2004 A · Q11. It mainly tests Averages (mean), Money / coins.

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?

Paula 的钱包中所有便士、五分币、十分币和二十五分币的平均价值为 $20$ 美分。如果她再多一枚二十五分币，平均价值将变为 $21$ 美分。她钱包里有多少枚十分币？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let the total value, in cents, of the coins Paula has originally be $v$, and the number of coins she has be $n$. Then $\frac{v}{n}=20\Longrightarrow v=20n$ and $\frac{v+25}{n+1}=21$. Substituting yields: $20n+25=21(n+1),$ so $n=4$, $v = 80.$ Then, we see that the only way Paula can satisfy this rule is if she had $3$ quarters and $1$ nickel in her purse. Thus, she has $\boxed{\mathrm{(A)}\ 0}$ dimes.

设 Paula 原来硬币的总价值（单位：美分）为 $v$，硬币枚数为 $n$。则 $\frac{v}{n}=20\Longrightarrow v=20n$，且 $\frac{v+25}{n+1}=21$。代入得：$20n+25=21(n+1),$ 所以 $n=4$，$v = 80.$ 于是可知 Paula 满足条件的唯一方式是钱包里有 $3$ 枚二十五分币和 $1$ 枚五分币。因此她有 $\boxed{\mathrm{(A)}\ 0}$ 枚十分币。

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