PP-T1-PS-188

[iamsmall]

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000
II. At least one of the homes was sold for more than $130,000 and less than $150,000
III. At least oe of the homes was sold for less than $130,000

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

Answer:A

請問解題思路是?

[ipas3]

由題意, 13萬(13W)為中位數, 表示有七棟house的價格低於13W, 七棟高於13W, 平均值是15W, 15棟房子總價225W.
為方便解釋起見將低於中位數的房子設為pool A, 高於中位數的房子設為pool B
(pool A)-13W-(poow B)

I. At least one of the homes was sold for more than $165,000
想法: 16.5W高於平均, 表示某些房子價格低於平均, 才會有房子要賣高於平均價 (若大家都等於平均值就不會高高低低了)
但若房價都盡可能接近平均, 是否還會有房價必須高於16.5W的呢?
假設讓pool A的房價盡量貼近平均, 全部都是13W好了(因為中位數的關係不可以高過13W), 那麼前八棟房子(pool A+中位數)總價為13W*8=104W
因此pool B房屋總價為225W-104W=121W, 平均為17.2W. 因為在pool B中至少有一棟房價高過17.2W, 所以 (I)成立

II. At least one of the homes was sold for more than $130,000 and less than $150,000
這點應該蠻好排除的, more than $130,000 and less than $150,000是在pool A, 既然是講平均數, 一定可以透過調整數值來排除某些數字必落在特定區間. 所以(II)不一定對

III. At least oe of the homes was sold for less than $130,000
這點同(I), 若全部房價都不低於13W, 題目命題還是可以成立, 所以(III)不一定對

故答案(A)

為了解釋有點囉嗦~希望有幫助