Efficient Workforce Planning: A Mathematical Challenge

In the world of productivity and project management, efficient workforce planning is crucial. This involves understanding the productivity ratios between different types of workers. Let's explore a practical problem that can be solved using simple mathematical principles. The problem at hand is related to the completion of a particular work, with different individuals (men, women, and boys) working at varying productivity rates. Our task is to determine how many women are needed to complete the same work in a shorter time frame.

Understanding the Productivity Ratios

The problem is as follows: 3 men, 4 women, and 6 boys can complete a work in 6 days. It is given that a woman does triple the work a man does, and a boy does half the work a man does. This information allows us to establish the productivity ratios between these individuals.

Step 1: Define Work Rates

Let's denote the work done by one man in one day as (m).

A woman does triple the work of a man, so a woman's work rate is (3m). A boy does half the work of a man, so a boy's work rate is (frac{m}{2}).

Step 2: Calculate Total Work Contribution

Next, we calculate the total work done by the group in one day.

Work done by 3 men in one day: [3m] Work done by 4 women in one day: [4 times 3m 12m] Work done by 6 boys in one day: [6 times frac{m}{2} 3m]

Step 3: Total Work Done in One Day

We find the total work done by the entire group in one day.

[text{Total work in one day} 3m 12m 3m 18m]

Step 4: Total Work for 6 Days

Since they complete the work in 6 days, the total amount of work (W) can be calculated as:

[W text{Total work per day} times text{Number of days} 18m times 6 108m]

Step 5: Determine Work Done by Women Alone in 4 Days

Now, we need to find out how many women alone can complete this work in 4 days. Let (x) be the number of women required.

The total work done by (x) women in 4 days is given by:

[text{Total work by } x text{ women in 4 days} x times 3m times 4 12xm]

Step 6: Set Up the Equation

We need this to equal the total work (W).

[12xm 108m]

Dividing both sides by (m) (assuming (m eq 0)):

[12x 108]

[x frac{108}{12} 9]

Conclusion

Thus, 9 women alone will be able to complete the work in 4 days. This problem illustrates the importance of understanding productivity ratios and how to apply them in workforce planning to achieve optimal results.

Alternative Solution Insight

Another way to solve this problem involves understanding the productivity rates differently. Let us define:

Boy x units per day Man 2x units per day Woman 4x units per day

3M 4W 6B in 1 day 6x 16x 6x 28x units per day

5 days 14 units

14 / 7 2 units per day

2 / 4x 5 women required (ANSWER)

This solution highlights the importance of simplifying and cross-verifying the mathematical approach.