To determine the simplest fraction form of 0.333, we need to recognize that this decimal is a repeating decimal. Specifically, 0.333 is equivalent to 0.3333... (with the digit 3 repeating indefinitely).
A well-known property of repeating decimals is that they can be converted into fractions. The repeating decimal 0.3333... can be expressed as the fraction [tex]\( \frac{1}{3} \)[/tex].
Here's a step-by-step exposition of why this is the case:
1. Let [tex]\( x = 0.3333... \)[/tex]
2. To eliminate the repeating part, multiply both sides of the equation by 10:
[tex]\[
10x = 3.3333...
\][/tex]
3. Subtract the original equation from the new equation to isolate the repeating decimal:
[tex]\[
10x - x = 3.3333... - 0.3333...
\][/tex]
[tex]\[
9x = 3
\][/tex]
4. Divide both sides of the equation by 9 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{9}
\][/tex]
5. Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
\frac{3}{9} = \frac{1}{3}
\][/tex]
Therefore, the decimal 0.333 (repeating) is equivalent to [tex]\( \frac{1}{3} \)[/tex].
Hence, the best answer for the question is:
D. [tex]\( \frac{1}{3} \)[/tex]
