Find the rational number whose decimal expansion is 0.3344444

The fraction 301/900
Note: I'm assuming the 4's continue on forever



Work Shown:

I'm going to assume that the 4s go on forever. I'll represent this with three dots after the last 4 like so
0.334444...

Now let x = 0.334444...

Multiply both sides by 100
x = 0.334444...
100x = 100*(0.334444...)
100x = 33.444444...

And repeat with 1000
x = 0.334444...
1000x = 1000*(0.334444...)
1000x = 334.444444...

Then subtract and solve for x. Notice how the decimal parts line up and cancel
1000x-100x = (334.444444...) - (33.444444...)
1000x-100x = (334+0.444444...) - (33+0.444444...)
1000x-100x = 334+0.444444... - 33 - 0.444444...
1000x-100x = (334-33)+(0.444444... - 0.444444...)
900x = 334 - 33
900x = 301
x = 301/900

1. 0.3 •4 is the same as 4 groups of 0.32. 7•0.8 is the same as 7 groups of 0.83. it increases by 0.5 every time, so 0.5x4 it increases by 0.9 every time, so 0.9x5. 0.9•12=10.86. 0.47•5=2.35i can finish the next ones in a separate post