Compute the result of dividing 3 by one third, that is, evaluate 3 divided by 1/3.

Introduction / Context:
This problem tests understanding of division involving fractions, particularly the idea that dividing by a fraction is equivalent to multiplying by its reciprocal. Many students remember rules mechanically but still make mistakes when applying them to simple numbers. The calculation 3 divided by 1/3 is a classic example used to reinforce this concept in arithmetic and quantitative aptitude tests.

Given Data / Assumptions:
- The dividend is 3, a whole number.
- The divisor is 1/3, a proper fraction less than 1.
- We are asked to compute 3 divided by 1/3 exactly.
- All operations are in the usual real number system with standard fraction rules.

Concept / Approach:
The key rule is that dividing by a nonzero fraction a / b is the same as multiplying by its reciprocal b / a. In symbols, x divided by (a / b) equals x * (b / a), provided a is not zero. Applying this to the present problem, we replace division by 1/3 with multiplication by 3/1. This leads to a straightforward integer multiplication. Interpreting the operation in words as How many one third portions are there in 3 also gives an intuitive check on the result.

Step-by-Step Solution:
Step 1: Write the expression in fractional form as 3 divided by 1/3. Step 2: Use the rule for dividing by a fraction: 3 divided by 1/3 equals 3 multiplied by the reciprocal of 1/3. Step 3: The reciprocal of 1/3 is 3/1. Step 4: Multiply: 3 * 3/1 = 9/1. Step 5: Simplify 9/1 to 9. Step 6: Therefore the value of 3 divided by 1/3 is 9.

Verification / Alternative check:
Think of 3 as 3 whole units of something, for example 3 metres of rope. Each piece of size 1/3 metre is one third of a metre. The question 3 divided by 1/3 asks how many pieces of length 1/3 metre can be cut from 3 metres. Each metre can be divided into 3 such pieces, so from 3 metres we get 3 * 3 = 9 equal pieces. This real world interpretation matches the calculated result of 9 and confirms that our arithmetic is correct.

Why Other Options Are Wrong:
Option 1: This might come from confusing division and multiplication, but 3 divided by 1/3 is much larger than 3, not smaller.
Option 0: Division by a positive fraction greater than zero cannot produce zero from a positive whole number, so this is impossible here.
Option None of these: Since one of the options matches the correct value 9, this option is not correct.
Option 3/9: This fraction simplifies to 1/3, which is actually the divisor, not the result, indicating a misunderstanding of the operation.

Common Pitfalls:
A common error is to multiply 3 by 1/3 instead of dividing, which gives 1. Another mistake is to invert the wrong fraction or to think that dividing by a fraction always makes the result smaller. In fact, dividing by a fraction less than 1 makes the result larger. Keeping the rule multiply by the reciprocal and treating fractions carefully helps avoid these missteps.

Final Answer:
The value of 3 divided by 1/3 is 9.