There are four students on the chess team. the shortest student is 55 inches tall. the tallest student is 66 inches tall. the mean height of the four students is 61 inches. the mean absolute deviation of the heights of the four students is less than 4 inches. what could be the heights, in inches, of the two other students? explain how you found your answers and show your work.

59 and 64 using systems of inequalities and equations.  

Step-by-step explanation:

So there are four students.  Let's label them in order of height a,b,c and d.

Now we know a = 55 and d = 66.

Since we know the mean we know (55+b+c+66)/4 = 61 and since we know the mean absolute deviation (gonna call it MAD from now on) we know (|55-61|+|b-61|+|c-6|+|66-61|)/4 <4.  With this we can make a system of equations.  First gonna simplify the MAD  

(6+|b-61|+|c-61|+4)/4<4

(6+4)/4 + (|b-61|+|c-61|)/4<4

10 + (|b-61|+|c-61|) < 16

|b-61|+|c-61| < 6

Now then, I'll take the equation I got from the mean and solve for b

(55+b+c+d)/4 = 61

55+b+c+66 = 244

b+c = 123

b = 123-c

Now I plug this in to the MAD inequality

|b-61|+|c-61| < 6

|(123-c)-61|+|c-61| < 6

|62-c|+|c-61| < 6

Now, to keep it simple let's just deal with integers, so no decimals.  We want the two absolute value terms to equal something less than 6, so that's 0, 1, 2, 3, 4, and 5.  You can pick any, I am going to pick 5.  So we want |62-c|+|c-61| = 5.  Or in other words we want two positive integers to equal 5.  Let's say 2 and 3.  So one of the absolute values will equal 2 and the other will equal 3.  or |62-c| = 2 and |c-61| = 3.  These are easy enough, the answer has to be 64

So now we know c so we'll plug it back into the mean equation to solve for b.

b = 123-c

b = 123-64

b = 59

So there you go.  one of the two is 64 inches and the other is 59.  If you try other combinations some will not work, so there's probably a better method of doing this.  Hopefully this method made sense.  

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