With five numbers there are 5 ways to group 4 of them together. If you take the averages of each group of 4 numbers you get...
3428.5
3425.75
3398
3120.25
342.5
What are the 5 numbers?
Solution:
The numbers are 1, 12, 123, 1234, and 12345
Let the five numbers be A, B, C, D and E, with A < B < C < D < E.
Clearly, A+B+C+D must then be the lowest group sum followed by A+B+C+E,
while B+C+D+E must be the highest group sum, preceded by A+C+D+E.
That means the middle number must be A+B+D+E.
Matching these up in the problem, we obtain five equations:
(A+B+C+D)/4 = 342.5 .... (i)
(A+B+C+E)/4 = 3120.25 .... (ii)
(A+B+D+E)/4 = 3398 ...... (iii)
(A+C+D+E)/4 = 3425.75 .... (iv)
(B+C+D+E)/4 = 3428.5 ..... (v)
Adding (i) thru (v), we obtain:
A+B+C+D+E = 13715 (vi)
Multiplying (i) thru (v) by 4 in turn and separately subtracting these from (vi), we obtain:
E = 12345
D = 1234
C = 123
B = 12
A = 1
Thus, the required numbers (in ascending order) are 1, 12, 123, 1234 and 12345
Correctly solved by:
| 1. Kamal Lohia |
Holy Angel School, Hisar, Haryana, India |
| 2. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 3. Seth Cohen | Concord, New Hampshire, USA |
| 4. Davit Banana | Istanbul, Turkey |
| 5. Eloise Michael |
Smith Academy, Hatfield, Massachussetts, USA |
| 6. Pam Slone | Thacker, West Virginia, USA |
| 7. Kelly Stubblefield | Mobile, Alabama, USA |