To find the difference between one-third and 0.333 without a calculator, let's break it down step by step:

Step 1: Represent one-third as a decimal

One-third is represented as a repeating decimal: $\frac{1}{3} = 0.33333\ldots$ This is an infinite repeating decimal, where the digit 3 repeats indefinitely.

Step 2: Compare with the given decimal

The decimal provided in the question is 0.333, which has only three decimal places and is not the same as $\frac{1}{3}$. Therefore, we need to subtract 0.333 from the exact value of $\frac{1}{3}$.

Step 3: Subtract 0.333 from $\frac{1}{3}$

We calculate the difference between $\frac{1}{3}$ and 0.333. The repeating decimal for $\frac{1}{3}$ can be written as $0.33333\ldots$.

Now, subtract the two values: $\frac{1}{3} - 0.333 = 0.33333\ldots - 0.333 = 0.00033\ldots$ The difference is approximately 0.00033 (with more 3's repeating).

Step 4: Convert the difference to a fraction

The difference 0.00033 is equivalent to $\frac{3}{10000}$.

Step 5: Match with the given choices

Looking at the answer choices:

• Option D gives $\frac{3}{10000}$, which matches our calculation.

Conclusion:

The correct answer is D: $\frac{3}{10000}$.

Would you like further clarification on any step or have additional questions? Here are 5 related questions to deepen your understanding:

1. How do you convert a repeating decimal to a fraction?
2. Why is $\frac{1}{3}$ different from 0.333 exactly?
3. What is the general process for finding the difference between two decimals?
4. Can you express 0.00033 as a fraction in another way?
5. How would the answer change if more decimal places were given for 0.3333?

Tip: When dealing with repeating decimals, remember that they can never be fully represented by a finite decimal, so be careful with rounding or truncating values!