Question

gre 数学题

The total amount that Mary paid for a book was equal to the price of the book plus a sales tax that was 4 percent of the price of the book. Marry paid for the book with $10 bill and received the correct change, which was less than$3.00. which of the following statements must be true?
indicate all such statements.
A. the price of the book was less than$9.50.
B. the price of the book was greater than $6.90.
C.the sales tax was less than $0.45.

Answer 1

书钱+税 = 付的钱 = （1+0.04）*书钱 > 10 - 3 = 7,

7< 付的钱 = 1.04*书钱 < 10.
7/1.04 < 书钱 < 10/1.04 ~= 9.615,
A is not always true.
比如，书钱=9.6刀时，付的钱=9.6*1.04=9.984,找零0.016<3. 符合题意。但不符合A。所以，A不对。

书钱> 7/1.04 ~= 6.73, B is not always true.
比如，书钱 = 6.75刀时，付的钱=6.75*1.04=7.02，付10刀，找零2.98刀<3刀，符合题意。但不符合B。所以，B不对哈。

税 = 0.04*书钱 < 0.04*10/1.04 = 0.4/1.04 ~= 0.38 < 0.45, C is true.

Answer 2

依题 10-3=7<付的钱= 书钱+税钱<10 且 税钱=书钱*0.04
所以 付的钱=书钱+税钱=书钱+书钱*0.04=书钱*(1+0.04)=书钱*1.04
由 7<书钱*1.04<10 得
7/1.04<书钱<10/1.04
解得 6.73<书钱<9.62
当9.50<=书钱<9.62时符合题意，但不符合A，A错
当6.73<书钱<=6.90时符合题意，但不符合B，B错
排除AB，C正确
验证：
因为 税钱=书钱*0.04
所以 书钱=税钱/0.04
由 7/1.04<书钱<10/1.04 得
7/1.04<税钱/0.04<10/1.04
故 7/1.04*0.04<税钱<10/1.04*0.04
解得0.2692<税钱<0.3846<0.45，得证C正确
综上选C

Answer 3

应该选C。

设书价为x，
x乘以104%大于或等于（10-3）
解不等式可得x大于或等于6.73

因此关于书价是多少，我们只能知道它是比6.73大，比10小。不能证明它比9.50小或比6.90大。因此A和B不对。

现在看看C。根据题意，如果这本书是10美元，那么它的税也不过是40美分而已，Marry花10美元还找了零，说明这本书肯定不到10美元，那么它的税肯定低于40美分，就更低于45美分了。因此C是正确的。