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Problem-Solving and Data Analysis / Percentages Difficulty: Hard

The positive number $a$ is $230 percent sign$ of the number $b$ , and $a$ is $60 percent sign$ of the number $c$ . If $c$ is $p %$ of $b$ , which of the following is closest to the value of $p$ ?

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Explanation

Choice D is correct. It's given that $a$ is $230 percent sign$ of $b$ . It follows that $a = \frac{230}{100} b$ . It's also given that $a$ is $60 percent sign$ of $c$ . It follows that $a = \frac{60}{100} c$ . Since $a = \frac{230}{100} b$ and $a = \frac{60}{100} c$ , it follows that $\frac{230}{100} b = \frac{60}{100} c$ . Multiplying each side of this equation by $StartFraction 100 Over 60 EndFraction$ yields $\frac{23}{6} b = c$ . If $c$ is $p %$ of $b$ , then $c = \frac{p}{100} b$ . It follows that $\frac{23}{6} = \frac{p}{100}$ . Multiplying each side of this equation by $100$ yields $\frac{2,300}{6} = p$ . It follows that the value of $p$ is approximately $383.33$ . Therefore, of the given choices, $383$ is closest to the value of $p$ .

Choice A is incorrect. This is closest to the value of $p$ if $b$ is $230 percent sign$ of $a$ , rather than if $a$ is $230 percent sign$ of $b$ , and if $b$ is $p %$ of $c$ , rather than if $c$ is $p %$ of $b$ .

Choice B is incorrect. This is closest to the value of $p$ if $a$ is $230 percent sign$ greater than $b$ , rather than $230 percent sign$ of $b$ .

Choice C is incorrect. This is closest to the value of $p$ if $c$ is $p %$ greater than $b$ , rather than $p %$ of $b$ .