In a certain office, one-third of the workers are women. One-half of these women are married and one-third of the married women have children. Of the men, three-fourths are married and two-thirds of the married men have children. What fraction of all the workers in the office are without children?

Introduction / Context:
This arithmetic reasoning question is about fractions and population breakdown in an office. The workers are divided into women and men, and within each group, some are married and some of the married workers have children. The aim is to find what fraction of the total workers do not have children at all. Such questions test comfort with fractions, proportional reasoning, and careful reading of layered conditions.

Given Data / Assumptions:

• One-third of the workers are women.
• One-half of the women are married.
• One-third of the married women have children.
• Three-fourths of the men are married.
• Two-thirds of the married men have children.
• All workers belong either to the group of men or to the group of women.

Concept / Approach:
To keep the calculation simple, treat the total number of workers as 1 unit (or 100 percent). Express all given pieces of information as fractions of this total. First, compute the fractions of women with children and women without children. Then do the same for men with children and men without children. Finally, add the fractions of women without children and men without children to obtain the fraction of all workers who are without children.

Step-by-Step Solution:
Step 1: Let the total number of workers be 1. Then women = 1/3 and men = 2/3 of the workers. Step 2: Married women = (1/2) * (1/3) = 1/6 of all workers. Step 3: Women with children = (1/3) * (1/6) = 1/18 of all workers. Step 4: Women without children = total women - women with children = 1/3 - 1/18 = 6/18 - 1/18 = 5/18. Step 5: Married men = (3/4) * (2/3) = 6/12 = 1/2 of all workers. Step 6: Men with children = (2/3) * (1/2) = 1/3 of all workers. Step 7: Men without children = total men - men with children = 2/3 - 1/3 = 1/3. Step 8: Total workers without children = women without children + men without children = 5/18 + 1/3 = 5/18 + 6/18 = 11/18.

Verification / Alternative check:
To verify, imagine a convenient total, for example 18 workers. Then women = 6 and men = 12. Married women = 3, women with children = 1, so 5 women have no children. Married men = 9, men with children = 6, so 6 men have no children. Total without children = 5 + 6 = 11 out of 18 workers, again 11/18, which confirms the fraction is correct.

Why Other Options Are Wrong:
Option A 5/18 counts only the women without children and ignores men. Option B 4/9 is less than 1/2 and does not match the computed combined fraction. Option D 17/36 is smaller than 11/18 and arises from incorrect subtraction or mixing of fractions. None of these match both the direct fraction calculation and the verification by assuming 18 workers.

Common Pitfalls:
A very common error is to treat percentages or fractions inside the women and men groups as if they were applied directly to the whole office, without first isolating the fraction of men and women. Another frequent mistake is to confuse fractions of married workers with fractions of all workers. Always anchor each fraction either to the whole office or to the subgroup, and convert all of them to fractions of the total before adding.

Final Answer:
The fraction of workers who are without children is 11/18.