Choose two numbers with a product between 6 and 7

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Quantitative Reasoning

Two distinct numbers $x$ and $y$ are selected from the numbers below. If $6 < xy < 7$ , which two numbers could be $x$ and $y$ ? Indicate both numbers.

Select all answers that apply.

-\sqrt{20}

2^{-2}

3^{2}

-\sqrt{2}

\sqrt{3}

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Answer: A ( $-\sqrt{20}$ ) and D ( $-\sqrt{2}$ )

Because the product must be positive, the two selected numbers must have the same sign. The only two negative numbers are $-\sqrt{2}$ and $-\sqrt{20}$ , and their product is $\sqrt{40}$ . Since $36 < 40 < 49$ , it follows that $6 < \sqrt{40} < 7$ . Among the positive numbers, the possible products are $(2^{-2})$ ${}(\sqrt{3})$ ${}= \frac{\sqrt{3}}{4}$ , $(2^{-2})$ ${}(3^{2})$ ${}= \frac{9}{4}$ , and $(\sqrt{3})$ ${}(3^{2})$ ${}= 9\sqrt{3}$ . None of these products is between 6 and 7. Therefore, the two numbers are $-\sqrt{2}$ and $-\sqrt{20}$ .

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Quantitative Reasoning

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