Question

Two fractions were written as decimals, then added. the sum is 0.45. what might the decimals be? what might the fractions be? show how you know.

Answer 1

Since the sum 0.45 as a fraction is 45/100 = 9/20 the number of possible fractions is infinite.
For example:
0.45 = 0.40 + 0.05 = 2/5 + 1/20 = 8/20 + 1/20 = 9/20
0.45 = 0.20 + 0.25 = 1/5 + 1/4 = 4/20 + 5/20 = 9/20
0.45 = 0.17 + 0.28 = 17/100 + 7/25 = 17/100 + 28/100 = 45/100
0.45 = 0.5 + -0.05 = 1/2 + -1/20 = 10/20 + -1/20 = 9/20

In general when a/b + c/d = 9/20 then bd must be a multiple of 20
a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd = n(9/20)

Let’s try 1/3 as the first fraction
1/3 + c/d = 9/20
c/d = 9/20 - 1/3 = 27/60 - 20/60 = 7/60
1/3 + 7/60 = 0.3333… + 0.11666… = 0.45

Here is an extreme example
3/7 + c/d = 9/20
c/d = 9/20 - 3/7 = 63/140 - 60/140 = 3/140
So 3/7 + 3/140 = 9/20 = 0.45
The decimals needed are recurring and hard to find!