Efficiency Analysis: Determining the Work Rate of Men and Women

Imagine a scenario where a piece of work can be completed by different groups of workers. This article will delve into how to calculate the work rate of both men and women, and how to determine the number of days required for a larger group to complete the same work.

Introduction to the Problem

A piece of work can be completed by 6 men and 5 women in 6 days or by 3 men and 4 women in 10 days. This article explores how to determine the number of days required to complete the work with a different group: 9 men and 15 women. To solve this, we will determine the work rates of men and women based on the given information and apply these rates to the new scenario.

Step 1: Set Up the Equations

Let's denote: M as the work rate (work done per day) of one man. W as the work rate (work done per day) of one woman.

Calculating the Total Work Done

From the first scenario, we know that 6 men and 5 women can complete the work in 6 days. Therefore, the total work can be expressed as:

[text{Total Work} 6M times 6 5W times 6 36M 30W]

Similarly, from the second scenario, we know that 3 men and 4 women can complete the work in 10 days. Therefore, the total work can also be expressed as:

[text{Total Work} 3M times 10 4W times 10 30M 40W]

Step 2: Setting Up the System of Equations

Since both expressions represent the total work, we can set them equal to each other:

[text{36M} 30W 30M 40W]

Step 3: Solving for M and W

Rearranging the equation to isolate M and W:

[text{36M} - 30M 40W - 30W]

[text{6M} 10W]

[text{M} frac{5}{3}W]

Step 4: Substitute M in One of the Equations

Substitute M frac{5}{3}W into one of the total work equations. Let's choose the first equation:

[text{Total Work} 36 left( frac{5}{3}W right) 30W]

[text{Total Work} 60W 30W]

[text{Total Work} 90W]

Step 5: Calculate the Work Done by 9 Men and 15 Women

Now, let's calculate the work done per day by 9 men and 15 women:

[text{Work per day} 9M 15W]

Substitute M frac{5}{3}W into the equation:

[text{Work per day} 9 left( frac{5}{3}W right) 15W]

[text{Work per day} 15W 15W]

[text{Work per day} 30W]

Step 6: Determine the Number of Days Required

Since the total work is 90W and the work done per day by 9 men and 15 women is 30W, the number of days required can be calculated as:

[text{Number of days} frac{text{Total Work}}{text{Work per day}} frac{90W}{30W} 3]

Therefore, the number of days required for 9 men and 15 women to complete the work is 3.

Conclusion

After careful calculations and analysis, we determined that 9 men and 15 women can complete the work in 3 days. This method can be applied to similar problems to analyze work rates and productivity.