The mean, the median, the mode of the distribution of the heights of 9 boys from the school basketball team are all equal to 183cm. three of the boys have a height of 183cm and the tallest boy has a height of 187cm. given that the heights of all the boys are integers, find the least possible height of shortest boy

the answer is 172

give me the working

bennykim0905
May 18, 2017

#1
**+1 **

Note that we can have the following distribution of heights :

a b 183 183 183 c d e 187

Note that if all nine boys averaged 183 cm, the total number of cm = 1647

But.....we know that one boy has a height of 187 cm and three of them have a height of 183 cm...so...

187 + 3 (183) = 736 cm

So.....the remaining number of cm to be allotted amongst the other five boys is 1647 - 736 = 911 cm

Now.......assume that boys "d" and "e" can be 186 cm tall, boy "c" can be 185 cm tall and boy "b" can be 182 cm tall......these are the max heights possible under the given constraints and will guaratnee the shortest boy possible...then....the shortest boy "a" is

911 - 2(186) - 185 - 182 = 172 cm tall

And the heights are

172 182 183 183 183 185 186 186 187

Check to see that the mean, median and mode = 183

CPhill
May 18, 2017