A research laboratory conducted 6 experimental trials to measure the duration, in millisec… — GRE General Test

A research laboratory conducted $6$ experimental trials to measure the duration, in milliseconds, of a specific chemical reaction. The durations recorded for $5$ of the trials were $240$ , $215$ , $260$ , $225$ , and $245$ . If the median duration of all $6$ trials was $235$ milliseconds, what was the duration, in milliseconds, of the 6th trial?

Answer: 230 milliseconds

Answer

230

For a dataset with 6 numbers, the median is the arithmetic mean of the 3rd and 4th numbers in ascending order. Arranging the 5 given numbers gives

215, 225, 240, 245, 260

. Since the target median is

235

, the sum of the two middle numbers must be

235 \times 2 = 470

. Placing

x = 230

into the dataset yields the ordered set

215, 225, 230, 240, 245, 260

, where the 3rd and 4th numbers are

230

and

240

. Their mean is

(230 + 240) / 2 = 235

, which matches the given median.

Step-by-Step Solution

Order the 5 given numbers from least to greatest

The sorted list of known values is

215, 225, 240, 245, 260

Calculating or using median requires ordering the data points.

Express the median condition for an even number of data points (

n = 6

)

\text{Median} = \frac{\text{3rd value} + \text{4th value}}{2} = 235

, so

\text{3rd value} + \text{4th value} = 470

For an even number of values, the median is the average of the two middle numbers.

Determine the position and value of the unknown 6th trial

x

x \le 225

, the 3rd and 4th values would be

225

and

240

(median

232.5

). If

x \ge 245

, the 3rd and 4th values would be

240

and

245

(median

242.5

). Thus,

x

must lie between

225

and

240

Analyzing boundary conditions places

x

as the 3rd value and

240

as the 4th value.

Solve for

x

x + 240 = 470 \implies x = 230

The sum of the two middle values must equal

470

to yield a median of

235

Key Concept

Finding a missing value in a dataset given the median of an even number of observations.

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