AMC10 2004 B

AMC10 2004 B · Q13

AMC10 2004 B · Q13. It mainly tests Rounding & estimation, Digit properties (sum of digits, divisibility tests).

In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack?

在美国，硬币的厚度如下：美分1.55毫米；镍币1.95毫米；角币1.35毫米；25美分币1.75毫米。如果一摞这些硬币正好高14毫米，这摞硬币有多少个？

(A) 7 7

(B) 8 8

(D) 10 10

(E) 11 11

Answer

Correct choice: (B)

正确答案：（B）

Solution

The height in millimeters of any stack with an odd number of coins has a 5 in the hundredth place. The height of any two coins has an odd digit in the tenth place and a zero in the hundredth place. Therefore any stack with zeros in both its tenth and hundredth places must consist of a number of coins that is a multiple of 4. The highest stack of 4 coins has a height of $4(1.95) = 7.8$ mm, and the shortest stack of 12 coins has a height of $12(1.35) = 16.2$ mm, so no number other than 8 can work. Note that a stack of 8 quarters has a height of $8(1.75) = 14$ mm.

任何奇数个硬币的堆叠高度，小数点后两位有5。任何两个硬币的高度，小数点后一位是奇数，百位是0。因此，小数点后两位均为0的堆叠必须是4的倍数硬币数。4个最厚的硬币高度为$4(1.95) = 7.8$毫米，12个最薄的硬币高度为$12(1.35) = 16.2$毫米，所以除了8个没有其他可能。注意8个25美分币高度为$8(1.75) = 14$毫米。

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