Let a, b, and c be integers such that -5 \le a \le 3, -4 \le b \le 6, and -3 \le c \le 4. If a b 0, what is the maximum possible value of the expression \frac{a - c}{b}?

Let a, b, and c be integers such that -5 \le a \le 3, -4 \le b \le 6, and -3 \le c \le 4. … — GMAT

Let $a$ , $b$ , and $c$ be integers such that $-5 \le a \le 3$ , $-4 \le b \le 6$ , and $-3 \le c \le 4$ . If $a b < 0$ and $b c^2 > 0$ , what is the maximum possible value of the expression $\frac{a - c}{b}$ ?

Answer: 2

Answer

The maximum possible value of the expression is 2.

From

b c^2 > 0

, since

c^2 > 0

for any non-zero real number

c

, it follows that

b > 0

and

c \neq 0

. Given

a b < 0

and

b > 0

a

must be strictly negative (

a < 0

). To maximize

\frac{a - c}{b}

where

b > 0

, we maximize the numerator

(a - c)

and minimize the denominator

b

. The maximum negative integer for

a

-1

, and the minimum integer for

c

-3

, giving a maximum numerator of

-1 - (-3) = 2

. The minimum positive integer for

b

1

. Hence, the maximum value is

\frac{2}{1} = 2

Step-by-Step Solution

Analyze the sign of

b c^2 > 0

b > 0

and

c \neq 0

The square of any non-zero real number is strictly positive (

c^2 > 0

), so

b

must be positive for the product

b c^2

to be positive.

Analyze the sign of

a b < 0

a < 0

Since

b > 0

a

must be negative for their product to be less than zero.

Determine candidate integer sets

a \in \{-5, -4, -3, -2, -1\}

b \in \{1, 2, 3, 4, 5, 6\}

c \in \{-3, -2, -1, 1, 2, 3, 4\}

Filter the given intervals by the sign restrictions (

a < 0

b > 0

c \neq 0

Optimize numerator

(a - c)

and denominator

b

a = -1

c = -3

b = 1

To maximize a quotient with a positive denominator, make the numerator as large as possible (

a - c = -1 - (-3) = 2

) and the denominator as small as possible (

b = 1

Evaluate the maximum value

\frac{-1 - (-3)}{1} = \frac{2}{1} = 2

Key Concept

Deducing sign constraints from products and powers of variables and optimizing algebraic quotients under sign and domain restrictions.

Estimated Time:1m 30s

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