4. A repeating decimal corresponds to a rational number. We can think of a purely repeating decimal as a geometric series whose first term is the first repeat, 0.285714, and each subsequent term is multiplied by one tenth to the sixth power, because the pattern is six digits long.
To wit,
[tex]0.\overline{285714} = .285714(1 + 10^{-6} + (10^{-6})^2 + (10^{-6})^3+ ...)[/tex]
The powers of 10 add up to 1000000/999999 so the whole repeating decimal is the rational number 285714/999999.
If a reduced fraction has a denominator with a prime factor other than 2 or 5 we'll get a repeating decimal. The decimal doesn't terminate but the number is still rational.
5.
Since [tex]\frac 2 3 = 0.\overline{6} = 0.\overline{66}[/tex] the two measurements are equal. I'm not sure exactly how or why Chef B's ruler has a tick marked with an infinitely long decimal, but let's not worry about that.
