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Problem-Solving and Data Analysis Difficulty: Hard

Data set A consists of the heights of $75$ objects and has a mean of $25$ meters. Data set B consists of the heights of $50$ objects and has a mean of $65$ meters. Data set C consists of the heights of the $125$ objects from data sets A and B. What is the mean, in meters, of data set C?

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Explanation

The correct answer is $41$ . The mean of a data set is computed by dividing the sum of the values in the data set by the number of values in the data set. It’s given that data set A consists of the heights of $75$ objects and has a mean of $25$ meters. This can be represented by the equation $\frac{x}{75} = 25$ , where $x$ represents the sum of the heights of the objects, in meters, in data set A. Multiplying both sides of this equation by $75$ yields $x equals 75 left parenthesis 25 right parenthesis$ , or $x equals 1,875$ meters. Therefore, the sum of the heights of the objects in data set A is $1,875$ meters. It’s also given that data set B consists of the heights of $50$ objects and has a mean of $65$ meters. This can be represented by the equation $\frac{y}{50} = 65$ , where $y$ represents the sum of the heights of the objects, in meters, in data set B. Multiplying both sides of this equation by $50$ yields $y equals 50 left parenthesis 65 right parenthesis$ , or $y equals 3,250$ meters. Therefore, the sum of the heights of the objects in data set B is $3,250$ meters. Since it’s given that data set C consists of the heights of the $125$ objects from data sets A and B, it follows that the mean of data set C is the sum of the heights of the objects, in meters, in data sets A and B divided by the number of objects represented in data sets A and B, or $\frac{1,875 + 3,250}{125}$ , which is equivalent to $41$ meters. Therefore, the mean, in meters, of data set C is $41$ .