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Nine numbers are written in increasing order. The average of the nine numbers is the middle number, the average of the five largest values is 68, and the average of the five smallest values is 44. What is the sum of the nine numbers?

 Jul 17, 2020
 #1
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Since you already have the average of 5 largest values, or 68, and the average of the 5 smallest values, or 44, then all have to do is to take the average of the two averages! That is [68 +44] / 2 * 9 =504 - the sum of the 9 numbers in the sequence.

 Jul 17, 2020
 #2
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(x1 + x2 +  x3 +  x4 + x5  +x6+x7+x8+x9)/9 = x5      given

  x1+ x2+ x3+ x4+  x5 + x6 + x7 + x8 + x9  =  9 x5    (we'll use this shortly)

 

(x1 + x2 + x3 + x4 + x5)/5 = 44        5x44 = 220 = (x1+x2+x3+x4+x5)

(x5 + x6 + x7 + x8+ x9)/5 = 68                     340 = (x5+x6+x7+x8+x9)     Add these together   

    

 

    x1+ x2+ x3+ x4+  2x5 + x6 + x7 + x8 + x9  = 560   subtract x5 from both sides (use the red definition above for x5)

 

  x1+ x2+ x3+ x4+  2x5 + x6 + x7 + x8 + x9    - 9x5

                                 x5 = 560 - 9x5

                                10x5 = 560         x5 = 56  

 

9 * x5 = 9 x 56 = 504 total     (as Guest found (easier)  above!)

 Jul 17, 2020
edited by ElectricPavlov  Jul 17, 2020

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