Mary purchased a book whose price plus a 4% sales tax equaled the total amount she paid. She gave the cashier a 10 bill and received correct change of less than 3.00. Which of the following statements must be true? Select all that apply.

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GRE Quantitative Reasoning QUESTION #8192

Question 1 (Select all that apply)

Mary purchased a book whose price plus a 4% sales tax equaled the total amount she paid. She gave the cashier a $10 bill and received correct change of less than $3.00. Which of the following statements must be true? Select all that apply.

The price of the book was less than $9.50
The price of the book was greater than $6.90
The sales tax was less than $0.45✔️
The sales tax was greater than $0.50

Correct Answer Explanation

Let the price of the book $= p$. Total paid $= 1.04p$.

She paid $10 and received change less than $3.00, meaning total paid was more than $7.00:

$1.04p > 7.00 \Rightarrow p > 6.73$

She also received some change, so total paid $< \$10.00$:

$1.04p < 10.00 \Rightarrow p < 9.615$

So $6.73 < p < 9.615$. Now check each statement:

Statement A: $p < \$9.50$ — Not necessarily, since $p$ can be up to $9.615$. ✗
Statement B: $p > \$6.90$ — Not necessarily, $p$ can be $6.80$ (giving change $\approx \$2.93 < \$3.00$). ✗
Statement C: Sales tax $= 0.04p < 0.04 \times 9.615 \approx \$0.385 < \$0.45$. This is always true within the valid range. ✓
Statement D: Tax $> \$0.50$ would require $p > 12.50$, which is impossible here. ✗

Only Statement C must be true.

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