If a and b are positive rational numbers such that a < b < 1, which of the following expre… — GRE General Test

If $a$ and $b$ are positive rational numbers such that $a < b < 1$ , which of the following expressions must be strictly greater than $1$ ? Select all that apply.

$\frac{b}{a}$ Answer
$\frac{a+1}{b}$ Answer
C
$\frac{a}{b^2}$
$\frac{b+1}{a+1}$ Answer
E
$\frac{a+b}{2b}$

Answer

The expressions that must be strictly greater than 1 are the ratio of b to a, the fraction (a + 1) over b, and the fraction (b + 1) over (a + 1).

The correct options are those involving ratios where the numerator is demonstrably greater than the denominator. The ratio of b to a compares two positive numbers where b is larger, giving a quotient greater than 1. The fraction with numerator (a + 1) has a value over 1 while denominator b is under 1, ensuring the quotient exceeds 1. The fraction comparing (b + 1) to (a + 1) has a larger numerator because adding 1 preserves the inequality b > a.

Step-by-Step Solution

Analyze the ratio of b to a

Since

b > a > 0

\frac{b}{a} > 1

is always true.

Dividing any positive number by a smaller positive number results in a value greater than 1.

Analyze the fraction (a + 1) over b

Since

a > 0

a + 1 > 1

. Since

b < 1

a + 1 > b

, which implies

\frac{a+1}{b} > 1

The numerator is greater than 1 and the denominator is less than 1, so the fraction exceeds 1.

Test counterexamples for the fraction a over b squared

For

a = \frac{1}{8}

and

b = \frac{1}{2}

\frac{a}{b^2} = \frac{1/8}{1/4} = \frac{1}{2} \le 1

The expression is not guaranteed to be greater than 1 for all valid values of a and b.

Analyze the fraction (b + 1) over (a + 1)

Since

b > a

, adding 1 to both quantities yields

b + 1 > a + 1 > 0

, so

\frac{b+1}{a+1} > 1

Adding the same positive quantity to two numbers maintains their relative order.

Analyze the average fraction (a + b) over 2b

Since

a < b

a + b < 2b

, meaning

\frac{a+b}{2b} < 1

The numerator is the sum of a and b, which is strictly less than twice b.

Key Concept

Properties of Rational Number Inequalities and Fraction Comparison

Estimated Time:1m 30s

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