Plugging In: DOZEN F for "Must Be" Questions
If the operation $$x\text{&}y$$ is defined by the equation $$x\text{&}y=\frac{xy}{x+y}$$ for all $$x\ne −y$$, then $$m\text{&}m$$ must be equal to
__Alternative Method__:
Variables in the answer choices? __Plug In__ numbers and eliminate! __Plug In__ $$m = 4$$.
>$$\displaystyle m\text{&}m = 4\text{&}4 = \frac{4\cdot 4}{4+4} = \frac{16}{8} = 2$$
Therefore, if $$m=4$$, then the desired result is 2. That's your *goal*. __Plug In__ $$m=4$$ into the answer choices and eliminate all other answer choices that do not match your *goal*.
>$$\displaystyle \frac{m}{2} = \frac{4}{2} = 2$$,
so this answer choice cannot be eliminated for this Plug-In. All other answer choices are eliminated for the same Plug-In, so this is the right answer choice.
Incorrect.
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Incorrect.
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Incorrect.
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Correct.
Use the definition of the operation to calculate $$m\text{&}m$$. Then reduce the fraction to arrive at a definite answer.
>$$\displaystyle m\text{&}m = \frac{m\cdot m}{m+m} = \frac{m^2}{2m} = \frac{m}{2}$$
$$\frac{1}{2}$$
$$1$$
$$\frac{m}{2}$$
$$m$$
$$\frac{m^2}{2}$$
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