The total amount that Mary paid for a book was equal to the price of

Question

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. Mary paid for the book with a $10 bill and received the correct change, which was less than $3.00. Which of the following statements must be true?

I. The price of the book was less than $9.50.
II. The price of the book was greater than $6.90.
III. The sales tax was less than $0.45.

A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

Archit3110 · Answer

although the question can be solved ; but isnt gmat type  Bunuel  because the calculation involved here does require calculator to get the values... 
IMO C would be correct
as max change can be 2.99$ and 10-2.99 ; 7.01 ; 7.01/1.04 ; 6.74 so 7.01-6.74 =0.26

CrackverbalGMAT · Answer

This is a slightly difficult question and something you can expect in the 650 to 700 range. It’s a lengthy word problem and it’s also a ‘Must Be’ kind of question. 
In a ‘must be’ kind of question, the idea is to take simple cases and make each statement false, once. If a statement is false once, it cannot be always true. So, that statement can be eliminated.

Alternatively, analyzing the statements and drawing up equations/inequalities to prove a statement false, can also be done.

Let the price of the book be $x. Then, sales tax = 4% of $x which comes out to be  \frac{1}{25} x. Therefore, the total price of the book = $ \frac{26}{25} x.

We know that the cost of the book is less than $10. Therefore, 
  \frac{26}{25}  x < 10.

On solving the above inequality, x <  \frac{125}{13} , which translates to x < 9.60. This means that the price of the book is $9.51. So, statement I is not necessarily true. Options A, D and E can be eliminated.

We know that the change obtained is less than $3. Therefore,

10 –  \frac{26}{25}  x < 3

On solving the above inequality, we get, x > 6.70. This means statement II is also not necessarily true. So, option B can be eliminated. 
The only option left, C, has to be the answer.

From the above discussion, we know the upper bound of the cost of the book is $9.60. 4% of $9.60 is approximately $0.38. In case the cost of the book is less than $9.60, the sales tax will be lesser than $0.38. If it is lesser than $0.38, it will also be lesser than $0.45. Therefore, statement III HAS to be true and so, option C, IMHO, is the answer.

In questions like these, it is important to break down the question stem and form equations/inequalities and analyse. Trying values from the word go, especially in this problem is not a good idea because this question involves inequalities; and with inequalities, you have to deal with a range and so, considering a few values doesn’t ensure compliance always.

Hope this helps!

gadde22 · Answer

It is clearly evident that I & II statements must not be true. So C should be the Answer. Bunuel sir , could you please change the OA to C, which is E now.

Posted from my mobile device

Bunuel · Answer

____________________________
The OA is C. Edited. Thank you!

ScottTargetTestPrep · Answer

Letting b = the price of the book and 1.04b = the total cost of the book plus sales tax, we can create the inequality:

10 - 1.04b < 3

7 < 1.04b

6.73 < b

This is the minimum price of the book.

The maximum price of the book is such that its price plus the cost of sales tax is less than $10.00:

1.04b < 10.00

b < 9.62

Thus, we see that the price of the book was greater than $6.73 and less than $9.62. Since I and II are not true, III must be true.

Answer: C

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The total amount that Mary paid for a book was equal to the price of

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