Work Ratio Calculation: Man vs Woman Problem

This problem requires us to determine the ratio of work output between a man and a woman based on information about combined work efforts.

Understanding the Work Scenario

We are given two groups working together:

• Group 1: 9 men and 6 women
• Group 2: 1 man and 2 women

The core information is that the combined work rate of Group 1 is 6 times that of Group 2. Our objective is to find the ratio representing how much work 1 man does compared to 1 woman in the same time period.

Defining Work Rates

Let's denote:

• The work rate of 1 man per hour as '$m$'.
• The work rate of 1 woman per hour as '$w$'.

These variables represent the amount of work each individual completes in one hour.

Formulating the Equation

We can express the total work rate for each group:

• Work rate of Group 1 = $9m + 6w$
• Work rate of Group 2 = $m + 2w$

According to the problem statement, the work rate of Group 1 is 6 times the work rate of Group 2. This gives us the equation:

$$9m + 6w = 6 \times (m + 2w)$$

Solving for the Work Ratio

To find the ratio $m : w$, we solve the equation algebraically:

1. Distribute the multiplier: Apply the factor of 6 to the terms in the parenthesis.

   $$9m + 6w = 6m + 12w$$

2. Isolate variables: Gather terms with '$m$' on one side and terms with '$w$' on the other.

   $$9m - 6m = 12w - 6w$$

3. Simplify: Combine like terms.

   $$3m = 6w$$

4. Determine the ratio: Express the relationship as a ratio '$m : w$' by dividing both sides by $w$ and then by 3.

   $$\frac{m}{w} = \frac{6}{3}$$

   $$\frac{m}{w} = \frac{2}{1}$$

This result indicates that a man's work rate is double that of a woman's.

Conclusion

The ratio of the work done by 1 man to that of 1 woman in a given time is $m : w$, which equals $2 : 1$.