Let r and s be rational numbers such that 0 < r < \frac{1}{2} and 1 < s < 2. Which of the … — GRE General Test

Let $r$ and $s$ be rational numbers such that $0 < r < \frac{1}{2}$ and $1 < s < 2$ . Which of the following inequalities MUST be true? Select all that apply.

$r \cdot s < 1$ Answer
$\frac{s}{r} > 2$ Answer
C
s - r > 1
D
$\frac{r}{s} > \frac{1}{4}$
E
$r + s > \frac{3}{2}$

Answer

The correct inequalities that must be true are

r \cdot s < 1

and

\frac{s}{r} > 2

The inequality stating that the product of

r

and

s

is less than

1

is guaranteed because multiplying the maximum possible bounds (

\frac{1}{2}

and

2

) gives

1

. The inequality stating that the ratio of

s

r

is greater than

2

is guaranteed because the numerator is strictly greater than

1

and the denominator is strictly less than

\frac{1}{2}

Step-by-Step Solution

Analyze

r \cdot s < 1

Since

r < \frac{1}{2}

and

s < 2

, and both are positive rational numbers, multiplying their upper limits gives

r \cdot s < \left(\frac{1}{2}\right)(2) = 1

. This statement MUST be true.

Properties of inequality multiplication for positive rational numbers.

Analyze

\frac{s}{r} > 2

Since

s > 1

and

r < \frac{1}{2}

, taking the ratio gives

\frac{s}{r} > \frac{1}{1/2} = 2

. This statement MUST be true.

Dividing a larger positive number by a fraction less than

1/2

amplifies the quotient.

Test counterexamples for remaining options

For

s - r > 1

, pick

s = 1.2

and

r = 0.4 \Rightarrow s - r = 0.8 \le 1

(False). For

\frac{r}{s} > \frac{1}{4}

, pick

r = 0.1

and

s = 1.6 \Rightarrow \frac{r}{s} = \frac{1}{16} < \frac{1}{4}

(False). For

r + s > \frac{3}{2}

, pick

r = 0.1

and

s = 1.1 \Rightarrow r + s = 1.2 < 1.5

(False).

A single valid counterexample disproves a 'must be true' statement.

Key Concept

Properties of Rational Numbers and Inequalities

Question

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Step-by-Step Solution

Key Concept

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