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Problem-Solving and Data Analysis / Percentages Difficulty: Hard

The expression $0.35 x$ represents the result of decreasing a positive quantity $x$ by what percent?

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Explanation

Choice D is correct. Let $n %$ represent the percent by which the positive quantity $x$ is decreased to result in $0.35 x$ . The value of $n$ can be found by solving the equation $x - (\frac{n}{100}) x = 0.35 x$ . Since $x$ is a common factor of each of the terms on the left-hand side of this equation, the equation can be rewritten as $x (1 - \frac{n}{100}) = 0.35 x$ . Dividing each side of this equation by $x$ yields $1 - \frac{n}{100} = 0.35$ . Multiplying each side of this equation by $100$ yields $100 - n = 35$ . Subtracting $100$ from each side of this equation yields $- n = - 65$ . Dividing each side of this equation by $negative 1$ yields $n equals 65$ . Therefore, the expression $0.35 x$ represents the result of decreasing the positive quantity $x$ by $65 percent sign$ .

Choice A is incorrect. Decreasing the quantity $x$ by $3.5 percent sign$ yields $x minus 0.035 x$ , or $0.965 x$ , not $0.35 x$ .

Choice B is incorrect. Decreasing the quantity $x$ by $35 percent sign$ yields $x minus 0.35 x$ , or $0.65 x$ , not $0.35 x$ .

Choice C is incorrect. Decreasing the quantity $x$ by $6.5 percent sign$ yields $x minus 0.065 x$ , or $0.935 x$ , not $0.35 x$ .