In an office, one-third of the workers are women, half of the women are married, and one-third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men?
1 : 3
Solving the Office Ratio Problem
Let's break down the problem step by step to find the ratio of married women to married men in the office. We are given information about the proportions of women and men, and within those groups, the proportions who are married. The information about having children is extra and not needed to solve for the ratio of married women to married men.
Assumptions and Initial Proportions
To make calculations clear, let's assume the total number of workers in the office is \(T\). We can use this variable throughout the calculation or assume a specific number that is easy to work with (like a multiple of 3 and 4, for example, 12 or 24), but using \(T\) keeps it general.
- One-third of the workers are women.
- The rest are men.
Calculating the Number of Women and Men
Based on the above, we can find the number of women and men in terms of \(T\):
- Number of women = \(\frac{1}{3}\) of Total Workers = \(\frac{1}{3}T\)
- Number of men = Total Workers - Number of women = \(T - \frac{1}{3}T = \frac{2}{3}T\)
Calculating the Number of Married Women and Married Men
Now, let's find the number of married individuals in each group based on the given proportions:
- Half of the women are married.
- Three-fourth of the men are married.
Using these proportions:
- Number of married women = \(\frac{1}{2}\) of Women = \(\frac{1}{2} \times \left(\frac{1}{3}T\right) = \frac{1}{6}T\)
- Number of married men = \(\frac{3}{4}\) of Men = \(\frac{3}{4} \times \left(\frac{2}{3}T\right) = \frac{6}{12}T = \frac{1}{2}T\)
Finding the Ratio of Married Women to Married Men
The question asks for the ratio of married women to married men. We have calculated the number of married women as \(\frac{1}{6}T\) and the number of married men as \(\frac{1}{2}T\). The ratio is:
Ratio = \(\frac{\text{Number of Married Women}}{\text{Number of Married Men}}\)
Ratio = \(\frac{\frac{1}{6}T}{\frac{1}{2}T}\)
Since \(T\) appears in both the numerator and the denominator, it cancels out, provided \(T \neq 0\) (which must be true as there are workers). So the ratio simplifies to:
Ratio = \(\frac{\frac{1}{6}}{\frac{1}{2}}\)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
Ratio = \(\frac{1}{6} \times \frac{2}{1} = \frac{1 \times 2}{6 \times 1} = \frac{2}{6}\)
Simplifying the fraction \(\frac{2}{6}\) by dividing both the numerator and denominator by their greatest common divisor, 2:
Ratio = \(\frac{2 \div 2}{6 \div 2} = \frac{1}{3}\)
So, the ratio of married women to married men is \(1:3\).
Let's summarize the proportions calculated:
| Group | Proportion of Total Workers | Proportion of Group that is Married | Proportion of Total Workers who are Married in this Group |
|---|---|---|---|
| Women | \(\frac{1}{3}\) | \(\frac{1}{2}\) | \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\) (Married Women) |
| Men | \(\frac{2}{3}\) | \(\frac{3}{4}\) | \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) (Married Men) |
Ratio of Married Women to Married Men = \(\frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3}\)
The ratio is \(1:3\).
Revision Table: Key Information
| Category | Proportion / Ratio |
|---|---|
| Workers who are Women | \(\frac{1}{3}\) |
| Workers who are Men | \(\frac{2}{3}\) |
| Women who are Married | \(\frac{1}{2}\) of Women |
| Men who are Married | \(\frac{3}{4}\) of Men |
| Married Women (as proportion of Total Workers) | \(\frac{1}{6}\) |
| Married Men (as proportion of Total Workers) | \(\frac{1}{2}\) |
| Ratio: Married Women : Married Men | \(1:3\) |
Additional Information: Extraneous Data in Ratio Problems
This problem includes information about the proportion of married women and married men who have children (\(1/3\) of married women have children, \(1/3\) of married men have children). However, the question specifically asks for the ratio of married women to married men. The number or proportion of individuals with children does not affect the total number of married women or married men. Therefore, this information is extraneous or irrelevant to solving the specific question asked. In many quantitative aptitude problems, you might encounter extra data points. It's important to identify exactly what the question is asking for and use only the necessary information.
Ratio problems often involve simplifying fractions or proportions. Remember that a ratio \(a:b\) is equivalent to the fraction \(\frac{a}{b}\). When finding the ratio of two quantities calculated as proportions of a total (like \(\frac{1}{6}T\) and \(\frac{1}{2}T\)), the total variable \(T\) cancels out, meaning the ratio is independent of the total number of workers, as long as there is a non-zero number of workers.
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