The heights of 4 athletes on a team are 135 cm, 160 cm, 170 cm, and 175 cm. Laurissa joins the team. On the new team of 5 athletes, the mode height of the players is equal to the median height which is equal to the mean (average) height. How tall is Laurissa?
, 2018
Pick one
Solution
Since the heights of the 4 athletes on the team are all different, then if Laurissa’s height is different than each of these, there is no single mode height.
Therefore, Laurissa’s height must be equal to the height of one of the 4 athletes on the team for there to be a single mode.
If Laurissa’s height is 135 cm, then the median height of the 5 athletes is 160 cm which is not possible, since the median does not equal the mode.
Similarly, if Laurissa’s height is 175 cm, then the median height of the 5 athletes is 170 cm which is not possible.
Therefore, Laurissa’s height must equal 160 cm or 170 cm, since in either case the median height of the 5 athletes will equal Laurissa’s height, which is the mode.
If Laurissa’s height is 170 cm, then the mean height of the 5 athletes is
cm.
If Laurissa’s height is 160 cm, then the mean height of the 5 athletes is
cm.
When Laurissa’s height is 160 cm, the heights of the 5 athletes (measured in cm) are: 135, 160, 160, 170, 175.
In this case, each of the mode, median and mean height of the 5 athletes equals 160 cm.
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