math question?

Question

choose any three counting numbers less than ten write all the 2-digit numbers possible with these numbers do not repeat a digit find the sum of the two digit numbers divide by the sum of the three original numbers why is the quotient always 22 who ever figureS it out wins 20 POINTS

EXPLAIN HOW YOU GOT IT

dandf can you show me an example

Answer 1

Take a, b and c to be three different numbers (if they can be the same, this doesn't work).

There are six possible ways to arrange these digits in pairs to form numbers:
'ab' = 10a + b
'ac' = 10a + c
'ba' = 10b + a
'bc' = 10b + c
'ca' = 10c + a
'cb' = 10c + b

The sum is:
= 10a + b + 10a + c + 10b + a + 10b + c + 10c + a + 10c + b

Rearranging to group the like terms:
= 10a + 10a + a + a + 10b + 10b + b + b + 10c + 10c + c + c
= (10a + 10a + a + a) + (10b + 10b + b + b) + (10c + 10c + c + c)
= 22a + 22b + 22c

Distributing out the common 22:
= 22 ( a + b + c )

So if you divide by the sum of the original 3 numbers ( a + b + c ) you get 22! Basic algebra to the rescue!

For example, let's say the digits were 1, 3 and 7.

There are six possible ways to arrange these digits in pairs to form numbers:
'13' = 10 + 3
'17' = 10 + 7
'31' = 30 + 1
'37' = 30 + 7
'71' = 70 + 1
'73' = 70 + 3

Rearranging to group the like terms:
= 10 + 10 + 1 + 1 + 30 + 30 + 3 + 3 + 70 + 70 + 7 + 7
= 1(22) + 3(22) + 7(22)

Distributing out the common 22:
= 22 ( 1 + 3 + 7 )

So when you divide by the sum (11 = 1 + 3 + 7), you get 22.

Stated another way, when you create the 6 different combinations, each digit will appear in the tens digit place 2 times. And in the ones digit place 2 times. So when you divide, you end up with 2 x 10 + 2 x 1 = 22.

(By the way, you can't grant more than the standard 10 points, but I'll be happy to take that nonetheless.)

Answer 2

Possible combinations for three numbers (non-repeating):
ab (10a+b),ac (10a+c),ca (10c+a),ba(10b+a),bc(10b+c),cb (10c+b)
Adding:
10a+b+10a+c+10c+a+10b+c+10b+a+10c+b
22a+22b+22c
22 (a+b+c)
Divide result with (a+b+c), always 22....
Example:
2,3,4
Possible combinations:
23,24,42,32,34,43
Total: 23+24+42+32+34+43 = 198
Divide by (2+3+4)=9 : 198/9 = 22

Answer 3

It fails if the three numbers are all 1.

Answer 4

Man I'm out of here! Sorry but u r on your own