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Directions: For the following question, select all the answer choices that apply.
Q. 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? 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.
Correct answer is option 'C'. Can you explain this answer?
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Directions: For the following question, select all the answer choices...
Statement: 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.

Analysis:
To solve this problem, we need to consider the given information and determine which statements must be true.

Solution:

Let's assume the price of the book is "x" dollars.
According to the statement, the total amount that Mary paid for the book was equal to the price of the book plus a sales tax that was 4 percent of the price of the book.

So, the total amount Mary paid = x + (4% of x) = x + 0.04x = 1.04x

Mary paid for the book with a $10 bill and received the correct change, which was less than $3.00.
This means that the change Mary received must be less than $3.00.

The change Mary received = $10 - (total amount paid) = $10 - (1.04x)

Now, we know that the change must be less than $3.00. So, we can write the inequality:

$10 - (1.04x) < />

Simplifying the inequality:

$7.00 < />

Dividing both sides of the inequality by 1.04:

$6.73 < />

So, the price of the book (x) must be greater than $6.73.

Conclusion:

Based on our analysis, we can conclude that the following statements must be true:

a) The price of the book was less than $9.50. (This statement is not true as we found that the price of the book must be greater than $6.73)
b) The price of the book was greater than $6.90. (This statement is not necessarily true as we found that the price of the book must be greater than $6.73)
c) The sales tax was less than $0.45. (This statement is true as the sales tax is 4% of the price of the book, which is less than $0.45)

Therefore, the correct answer is option 'C'.
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Community Answer
Directions: For the following question, select all the answer choices...
For this problem you may find it helpful to translate the given information into an algebraic expression. Since the price of the book is unknown, you can call it x dollars, and then the total amount that Mary paid is x dollars plus 4% of x dollars, or 1.04x dollars. The problem states that Mary received some change from a $10 bill, so 1.04x dollars must be less than $10. Since the change was less than $3.00, the total amount Mary paid for the book must have been greater than $7.00. You can express this information algebraically by the inequality
7.00 < 1.04x="" />< />
Solving the inequality for x by dividing by 1.04, and rounding, you get
6.73 < x="" />< />
So you see that x, the price of the book, must be between $6.73 and $9.62, and each price in between is possible. With this information, you can quickly examine the first two statements. Choice A is not necessarily true because the price could be as high as $9.61, and Choice B is not necessarily true because the price could be as low as $6.74.
To examine Choice C, you could compute the tax for the greatest possible price, which would be 4% of 9.61, or (0.04)(9.61) ≈ 0.38. Since this greatest possible tax is less than $0.45, Choice C must be true.
You can also quickly see that Choice C must be true if you note that 4% of $10.00 would only be $0.40, and since the price must be less than $10.00, the tax must be less than $0.40. The correct answer consists of Choice C.
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