Equal Averages

If you work out the mean, median, mode and range of 2,5,5,6,7, you'll notice something interesting...

The mean, mode, median, and range of this set are all 5.
 

Can you find other sets of five positive whole numbers where Mean = Median = Mode = Range?

 

Charlie found that 50,100,100,100,150 had Mean = Median = Mode = Range = 100

Alison found that 40,100,100,120,140 also had Mean = Median = Mode = Range = 100

How many sets of five positive whole numbers are there with Mean = Median = Mode = Range = 100?

 

Comments

n = any number
first number 2 * n
second number 5 * n
third number 5 * n
fourth number 6 * n
fifth number 7 * n

To get SOME of the answers.

2n, 5n, 5n, 6n, 7n provides a "family" of solutions that satisfy the criteria.

Can you now find some different "families" that also satisfy the criteria?

Case 1: 50, 100, 100, 100, 150

Case 2: X1 < X2 < 100 < 100 < X3

50<X1<X2<100
and 150<X3<200
and X3X1=100
and 2X3+X2=400

There are 16 possibilities:

X1 = 51   X2 = 98   X3 = 151
X1 = 52   X2 = 96   X 3= 152
.
.
.
X1 = 66   X2 = 68   X3 = 166


Case 3: X1 < 100 < 100 < X2 < X3

0<X1<50
and 100<X2<X3<150
and X3X1=100
and 2X3+X2=400

There are 16 possibilities:

X1 = 34   X2 = 132   X3 = 134
X1 = 35   X2 = 130   X 3= 135
.
.
.
X1 = 49   X2 = 102   X3 = 149


Therefore there are 33 possibilities altogether.

 

From Penndale Middle School (Lansdale, PA) – Chi Alpha Mu (Math Club)

Knowing that MEAN = MEDIAN = MODE = RANGE = 100,

For any set of five positive whole numbers: A, B, C, D, E

C = 100 (to account for the median = 100)
E = 100 + A (to account for the range = 100)
B or D = 100 (to account for the mode = 100)

Let D = 100, then (A + B + 100 + 100 + (100+A)) / 5 = 100
Simplified, B = 200 - 2A

With this it was found that:

34 ≤ A ≤ 66

Therefore there are 66 – 33 = 33 sets of five positive whole numbers with MEAN = MEDIAN = MODE = RANGE = 100