Question

What is the rational number equivalent to 1 point 28 with a bar over 28?

Answer 1

bear with me, this is kind of a long explanation, and please feel free to ask me if you don't understand any individual step

starting with 1.28282828...
let's let x = 0.282828... (so x = the repeating decimal part)
we'll re-write 1.282828... as 1 + x

we're going to do a bit of a trick. since there are 2 digits in the repeating decimal part (2 and 8 repeated), we'll multiply

x = 0.2828...
by 100 on both sides to get

100x = 28.282828...

(basically, we want the one repeating unit (28) to be on the left side of the decimal part. you'll see why in a second)

now we have:
100x = 28.282828...
x = 0.282828...

notice what happens if we subtract the top equation - bottom equation

99x = 28 (notice how the decimal part cancels out!)

from there, you can solve for x, thereby getting x as a fraction instead of a repeating decimal. from there, you can simply combine 1 + x into an improper fraction. I'll let you handle it from here.