For all real numbers a and b such that a + b \neq 0, the custom operator \star is defined … — GMAT

For all real numbers $a$ and $b$ such that $a + b \neq 0$ , the custom operator $\star$ is defined by $a \star b = \frac{a - b}{a + b}$ . What is the value of $(3 \star 1) \star 2$ ?

A
$-\frac{5}{3}$
$-\frac{3}{5}$ Answer
C
$0$
D
$\frac{3}{5}$
E
$\frac{5}{3}$

Answer

-\frac{3}{5}

Evaluating the expression step-by-step according to parentheses yields

3 \star 1 = \frac{3-1}{3+1} = \frac{1}{2}

. Substituting

\frac{1}{2}

as the first input and

2

as the second input gives

\frac{1/2 - 2}{1/2 + 2} = \frac{-3/2}{5/2} = -\frac{3}{5}

Step-by-Step Solution

Evaluate the inner custom operation inside parentheses:

3 \star 1

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

Follow the order of operations by resolving the grouped expression first using the definition

a \star b = \frac{a - b}{a + b}

with

a = 3

and

b = 1

Substitute the result

\frac{1}{2}

back into the main expression to compute

(\frac{1}{2}) \star 2

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

Apply the definition of the custom operator again, where the left operand is

\frac{1}{2}

and the right operand is

2

Simplify the complex fraction.

\frac{\frac{1}{2} - \frac{4}{2}}{\frac{1}{2} + \frac{4}{2}} = \frac{-\frac{3}{2}}{\frac{5}{2}} = -\frac{3}{5}.

Combine the fractions in the numerator and denominator, then divide.

Key Concept

Custom Operators and Order of Operations

Estimated Time:1m 30s

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