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Problem-Solving and Data Analysis / Ratios, rates, proportional relationships, and units Difficulty: Hard

Objects R and S each travel at a constant speed. The speed of object R is half the speed of object S. Object R travels a distance of $4 x$ inches in $y$ seconds. Which expression represents the time, in seconds, it takes object S to travel a distance of $24 x$ inches?

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Explanation

Choice B is correct. It's given that object R travels a distance of $4 x$ inches in $y$ seconds. This speed can be written as $StartFraction 4 x inches Over y seconds EndFraction$ . It's given that the speed of object R is half the speed of object S. It follows that the speed of object S is twice the speed of object R, which is $2 left parenthesis StartFraction 4 x inches Over y seconds EndFraction right parenthesis$ , or $StartFraction 8 x inches Over y seconds EndFraction$ . Let $n$ represent the time, in seconds, it takes object S to travel a distance of $24 x$ inches. The value of $n$ can be found by solving the equation $\frac{8 x inches}{y seconds} = \frac{24 x inches}{n seconds}$ , which can be written as $\frac{8 x}{y} = \frac{24 x}{n}$ . Multiplying each side of this equation by $n y$ yields $8 x n equals 24 x y$ . Dividing each side of this equation by $8 x$ yields $n equals 3 y$ . Therefore, the expression $3 y$ represents the time, in seconds, it takes object S to travel a distance of $24 x$ inches.

Choice A is incorrect. This expression represents the time, in seconds, it would take object S to travel a distance of $24 x$ inches if the speed of object R were twice, not half, the speed of object S.

Choice C is incorrect. This expression represents the time, in seconds, it takes object S to travel a distance of $128 x$ inches, not $24 x$ inches.

Choice D is incorrect. This expression represents the time, in seconds, it takes object R, not object S, to travel a distance of $24 x$ inches.