Question

It is like a puzzle:  With the digits 1 up to 9, you must make a sum of 100, using the following rules: 

   1)  Each digit must be used exactly once. 
   2) The only allowed operation is addition. 
   3) Two single digits may be combined to form a number.  For example, 2 and 4 may be combined to 24 or 42.
   4) Two single digits may be combined to form a fraction.  For example, 2 and 4 may be combined to 2/4 or 4/2.
   5) no negative numbers.
   6) no other operators.

Answer 1

We can do some permutations to get many answers.  Some of them are mentioned here.

31 + 56 + 9 + 7/2 + 4/8  = 100

     7            4
31 ---  +  9  ----  + 56   = 100
     2            8
$31\frac{7}{2}+9\frac{4}{8}+56=100$

51 + 36 + 9 + 7/2 + 4/8  = 100

59 + 31 + 6 + 7/2 + 4/8  = 100

59 + 36 + 1 + 7/2 + 4/8  = 100                   and so on ......


A method to solve this puzzle. 

we can not have all numbers as single digits only as they will add up to 45. We will have some single digits and some double digit numbers. We cannot have 3 digit numbers, as the sum will be more than 100.  Our sum is 100 only.  We cannot have all as single digits, as the maximum sum will be only 45.

let us call the digits 1,b,c,d,e,f,g,h, and i.

If the answer is in the form:  a b + c d + e f + g h + i = 100, 
   then the sum of units digits   b + d + f + h + i  must be 20.
( it cannot be 10 as the smallest four 1,2,3,4 sum to 10., but here  we have 5 digits to sum. The sum has to have 0 as the units digit). 

Then  a + c + e + g  must add to 8.    It is not possible as all 9 digits sum to 45 and not 28.  So we cannot have all 9 digits in direct summation  like above. We need to have fractions.   Number of the numbers to add should be less.

Let us have one fraction only in the answer. 

Then we have 7 other digits as 3 double digit & 1 single digit numbers to add. That fraction must be : 2/1 or 3/1 ... 9/1 or 4/2 6/2 8/2 6/3 9/3 8/4. When we add the digits in units places and this fraction, we must get twenty or thirty.

The sum of digits in 10ths must added to 8.  If we use 9/3, then sum in units postion = 17 or 27 and sum in 10th places = 8 or 7 respectively. It means sum of the seven digits = 25 or 34. But as 9& 3 are used sum of remaining digits is 45 - 12 =33.    Follow this logic for other fractions.


Let us have two fractions in the answer:

  Then we have 5 digits split in 3 numbers: 2 two digit and 1 single digit.  If two fractions add up to 5, 4, 3, 2, 1 then the Sum of 3 units digits can be 15, 16, 17, 18, & 19 respectively. Corresponding sum of 10th place digits must be 8. So sum of 5 digits is 23, 24, 25, 26, 27.

Combination of 5 digits to sum 26 is 9 8 6 2 1, ;  9 8 5 3 1,; 9 8 4 3 2.
Remaining digits forming fractions are: 3,4,5,7 ;  2 4 6 7    ;  1 5 6 7  respectively.
On checking there are  NO fractions that sum to two. 
Like this combinations of 5 digits summing to others.


CONSIDER  5 digits summing to 24.  Fractions sum to 4. 

     combination: 98421 ; 97521; 97431 ; 96531 ; 96432
Choose combination 9 6 5 3 1  Now fractions are formed from 2, 4, 7, 8. 
 so  7/2 + 4/8 = 4    IT SEEMS TO WORK Correctly.

Sum of 3 units digits 16 =>  9 6 1    & sum of 10th places: 8 ie., 5 and 3
So all the digits fit in to this logic and combination.