How do you get 9/10?

Question

So I have to get to 9/10 using unit fractions and all the denominators have to me different. You can only use addition.

Example:

1/2 + 1/5 + 1/10=8/10

Answer 1

This may seem like working backwards (because it is):

   Q:  How do you add   1/x + 1/y   ?

   A:  Find the common denominator (z, the LCM [least common multiple of x and y]) and multiply --

               y/xy + x/xy = (x + y)/xy

 

We want to find values of x and y that add to, say, 900 when their LCM=1000.  [note: I picked 1000 because the LCM has to be pretty large]

 

Let's pick some numbers whose LCM=1000:

     2, 4, 5, 8, 10, 20, 25, 50, 100, 200, 500 

 

O.K., now select one of each so they add to 900:

      500 + 200 + 100 + 50 + 25 + 10 + 8 + 5 + 2

 

Then include the denominator:

        500/1000 + 200/1000 + 100/1000 + 50/1000 + 25/1000 + 10/1000 + 8/1000 + 5/1000 + 2/1000

 reduce those:

     1/2 + 1/5 + 1/10 + 1/20 + 1/40 + 1/100 + 1/125 + 1/200 + 1/500 = 900/1000 = 9/10