For all non-zero real numbers x and y, the custom operation \star is defined by x \star y … — GMAT

For all non-zero real numbers $x$ and $y$ , the custom operation $\star$ is defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . If $p$ and $q$ are positive numbers such that $p + q = 5$ and $p \star q = \frac{5}{6}$ , what is the value of $(p + 1) \star q$ ?

$\frac{3}{2}$ Answer
B
$-\frac{3}{2}$
C
$\frac{24}{5}$
D
$0$
E
$\frac{21}{10}$

Answer

\frac{3}{2}

By simplifying

p \star q = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

, we substitute

p + q = 5

to get

\frac{5(p - q)}{pq} = \frac{5}{6}

, which reduces to

6(p - q) = pq

. Substituting

q = 5 - p

yields the quadratic equation

p^2 + 7p - 30 = 0

. Since

p > 0

, we find

p = 3

and

q = 2

. Evaluating

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

Step-by-Step Solution

Express the custom operation using algebraic fractions and factor the numerator

p \star q = \frac{p}{q} - \frac{q}{p} = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

Finding a common denominator converts the expression into a form where known identities can be substituted.

Substitute

p + q = 5

and

p \star q = \frac{5}{6}

into the simplified expression

\frac{5(p - q)}{pq} = \frac{5}{6} \implies \frac{p - q}{pq} = \frac{1}{6} \implies 6(p - q) = pq

Simplifying by dividing both sides by 5 establishes a direct relation between the difference and product of p and q.

Substitute

q = 5 - p

to form and solve a quadratic equation for

p

6(2p - 5) = p(5 - p) \implies 12p - 30 = 5p - p^2 \implies p^2 + 7p - 30 = 0 \implies (p + 10)(p - 3) = 0

Since p and q are positive numbers, p must be positive, giving p = 3 and q = 2.

Evaluate

(p + 1) \star q

with

p = 3

and

q = 2

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

Applying the custom operator definition to inputs 4 and 2 yields the final value.

Key Concept

Custom Operator Evaluation and Quadratic System Solving

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