Question

A piece of work can be completed by 10 men and 6 women in 18 days. Men works 9 hours per day while women works \(7.5\) hours per day. Per hour efficiency of a woman is \(\frac{2}{3}\) rd of man's efficiency. In how many days 10 men and 9 women complete the work? (a) 16 days (b) 20 days (c) 30 days (d) 25 days

Short answer

Answer: The provided choices do not contain the correct answer, as it takes approximately 17 days for 10 men and 9 women to complete the work.

Step-by-step solution

Determine the amount of work completed by one man and one woman in one day

Men work 9 hours per day and women work 7.5 hours per day. Given that the efficiency of a woman is \(\frac{2}{3}\) rd of a man, we can represent their efficiencies as \(m\) for men and \(\frac{2}{3}m\) for women. The amount of work completed in one day by 10 men and 6 women is: $$(10m \times 9) + (6\frac{2}{3}m \times 7.5)= 162m$$

Calculate the total work

We know it takes 18 days to complete the work with 10 men and 6 women. Now, to find the total work, we multiply the work completed in one day (step 1) by the number of days needed to complete the work: $$162m \times 18 = 2916m$$

Calculate the daily work completed by 10 men and 9 women

Now, let's calculate the amount of work completed in one day by 10 men and 9 women working together. We already know men's efficiency is \(m\) and women's efficiency is \(\frac{2}{3}m\). Therefore: $$(10m \times 9) + (9\frac{2}{3}m \times 7.5)= 135m+ 45m = 180m$$

Calculate the number of days for 10 men and 9 women to complete the work

We know the total amount of work is \(2916m\) and the daily work completed is \(180m\). To find the number of days needed by 10 men and 9 women to complete the work, we simply divide the total work by their daily work: $$\frac{2916m}{180m} = 16.2$$ The number of days can't be in decimals, therefore, we have to round it up to the nearest integer, which is 17 days. However, there is no 17 days in the given choices, so, we'd conclude that the provided choices don't have the correct answer.

Key concepts

Quantitative Aptitude

The term 'quantitative aptitude' refers to one's capacity to handle numerical problems with accuracy and speed. This skill is essential in competitive exams and various fields where numerical problem-solving is involved. For example, work and time calculations require a solid understanding of basic mathematical concepts, such as addition, subtraction, multiplication, division, and ratios.

In the given exercise, quantitative aptitude comes into play when calculating work rates and total work done by different groups of workers. Students need to understand how to proportionally assign work based on the hours spent and efficiency levels, which is a practical application of quantitative reasoning.

Efficiency Calculation

Efficiency calculation in work and time problems involves determining how effectively a person or machine can complete a specific task. It is often expressed in terms of work done per unit of time. In the exercise, the efficiency of a woman is given as \( \frac{2}{3} \) of a man's efficiency.

Understanding these relative efficiencies is crucial as they help us quantify how much work each person can contribute. By setting a man's efficiency as \( m \) hours of work per hour, the calculation becomes clearer and allows for straightforward multiplication to determine the total work done by any number of men or women in a given timeframe.

Man-Hours Approach

The man-hours approach is a method frequently used in work and time problems to represent the amount of work done as a function of the number of workers and the number of hours they work. It provides a clear view of the total labor input required to complete a task.

In the text, the total work done by the initial group of 10 men and 6 women over 18 days is represented through man-hours. This method simplifies complex scenarios by converting them into uniform units of man-hours, enabling easier calculation of work done and simplifying the understanding of collective work output.

Work Completion Rate

Work completion rate reflects how fast a task can be completed given a certain level of work efficiency and the number of workers. It's the inversely proportional counterpart to the concept of time taken to complete a task.

The current exercise demonstrates this rate when expressing the daily work output of 10 men and 9 women, and then using this to calculate the total number of days required to complete the work. The final step in such problems typically involves understanding that the completion rate is the ratio of total work to the daily work output, and solving for the unknown time or workforce needed.