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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$\frac{12}{n} - \frac{2}{t} = - \frac{2}{w}$

The given equation relates the variables $n$ , $t$ , and $w$ , where $n greater than 0$ , $t greater than 0$ , and $w > t$ . Which expression is equivalent to $n$ ?

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Explanation

Choice D is correct. Adding $StartFraction 2 Over t EndFraction$ to each side of the given equation yields $\frac{12}{n} = - \frac{2}{w} + \frac{2}{t}$ . The fractions on the right side of this equation have a common denominator of $t w$ ; therefore, the equation can be written as $\frac{12}{n} = \frac{2 w}{t w} - \frac{2 t}{t w}$ , or $\frac{12}{n} = \frac{2 w - 2 t}{t w}$ , which is equivalent to $\frac{12}{n} = \frac{2 (w - t)}{t w}$ . Dividing each side of this equation by $2$ yields $\frac{6}{n} = \frac{w - t}{t w}$ . Since $n$ , $t$ , $w$ , and $w - t$ are all positive quantities, taking the reciprocal of each side of the equation $\frac{6}{n} = \frac{w - t}{t w}$ yields an equivalent equation: $\frac{n}{6} = \frac{t w}{w - t}$ . Multiplying each side of this equation by $6$ yields $n = \frac{6 t w}{w - t}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is equivalent to $StartFraction 1 Over n EndFraction$ rather than $n$ .