Question

Convert 11.42424242 … to a rational expression in the form of a over b, where b ≠ 0.

Answer 1

this is a little complicated to explain, so please let me know if any step confuses you

let's re-write the original expression as

11 + 0.424242...

let's let x = the decimal part = 0.424242...
we want an expression that puts the repeating part of the decimal on the left side (this is a little easier to demonstrate than explain)

if x = 0.424242...
then 100x = 42.424242... correct? we just multiplied the original decimal by 100

let's now subtract 100x - x = 99x = 42 (notice how the decimal part "cancels" out with subtraction)

if 99x = 42, then x = 42/99. so now that's the original decimal converted to a fraction.

so going back to the original expression:

11 + 0.424242 = 11 + 42/99

from there, since they want a rational expression in the form a/b, simply convert this mixed number into an improper fraction