nine numbers are written in ascending order the middle number is also the average of nine numbers the average of 5 larger numbers is 68 and the average of 5 smaller numbers are 44 the sum of all numbers
Answers
Step-by-step explanation:
There are many solutions below-
1. Let the 9 numbers be x1,x2,....,x9 (in ascending order)
Let A=x1+x2+x3+x4
Let B=x5
Let C=x6+x7+x8+x9
Then (A+B)/5=44,
(B+C)/5=68, and
(A+B+C)/9=B.
Rearranging:
A-8B+C=0.
A+B=220,
B+C=340,
Gaussian elimination:
A-8B+C=0
9B-C=220
10C=2840
Therefore C=284, B=56, and A=164.
We conclude that the sum of the 9 numbers is A+B+C=504, choice (c).
2. Let S the unknown sum of the nine integers a1, a2, . . . a9. Then:
a5 = S/9. Also we have:
a1 + a2 + a3 + a4 + a5 = 5*44 = 220 (1)
a5 + a6 + a7 + a8 + a9 = 5*68 = 340 (2)
Add (1) and (2):
a1 + a2 + a3 + a4 + 2*a5 + a6 + a7 + a* + a9 = 560
S + S/9 = 560, 10S/9 = 560, S = 56*9, S = 504.
So the right answer is (c)
3. The average of all 9 will be the number halfway between the two other averages. (68 + 44) / 2 = 56.
To figure the total of all the numbers, multiply 9 times the average:
9 x 56 = 504
P.S. One sequence that exhibits this property is:
32, 38, 44, 50, 56, 62, 68, 74, 80
The answer is c) 504