AMC8 2004

AMC8 2004 · Q20

AMC8 2004 · Q20. It mainly tests Linear equations, Fractions.

Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?

房间里的人中有三分之二坐在四分之三的椅子上。其余人站着。如果有6把空椅子，房间里有多少人？

(A) 12 12

(B) 18 18

(D) 27 27

(E) 36 36

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) Because the 6 empty chairs are $\frac{1}{4}$ of the chairs in the room, there are $6 \times 4 = 24$ chairs in all. The number of seated people is $\left(\frac{3}{4}\right)24 = 18$, and this is $\frac{2}{3}$ of the people present. It follows that \[ \frac{18}{\text{people present}}=\frac{2}{3}. \] So there are 27 people in the room.

（D）因为 6 把空椅子是房间里椅子总数的 $\frac{1}{4}$，所以一共有 $6 \times 4 = 24$ 把椅子。坐着的人数是 $\left(\frac{3}{4}\right)24 = 18$，而这又是在场人数的 $\frac{2}{3}$。因此 \[ \frac{18}{\text{在场人数}}=\frac{2}{3}. \] 所以房间里有 27 个人。

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