2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

Two scenarios are taken here.

2 women and 5 men complete the embroidery work in 4 days.

3 women and 6 men complete the same work in 3 days.

Setting up Equations:

We can represent the combined work rate of each group per day using fractions.

Let x be the time taken by one woman alone to finish the work, and y be the time taken by one man alone.

Work Rate Equations:

From the first scenario, the combined work rate of 2 women and 5 men is 1/4.

This translates to the equation 5y + 2x = 1/4.

From the second scenario, the combined work rate of 3 women and 6 men is 1/3.

This leads to the equation 6y + 3x = 1/3.

Solving the Equations:

Multiplying the first equation by 3 and the second equation by 4 simplifies the system.

This yields 15y + 6x = 3/4 and 24y + 12x = 4/3.

Subtracting the second equation from the first eliminates y and solves for x, giving x = 1/36.

Individual Time Calculations:

With the value of x, we substitute it back into one of the original equations to solve for y.

Substituting x = 1/36 into 6y + 3x = 1/3 gives y = 1/18.

Therefore, one woman alone takes 18 days to finish the work, and one man alone takes 36 days.

Algebraic Methods for Work Rate Analysis

In mathematics, problems involving work rates are often solved using algebraic equations to determine individual work rates and time taken to complete tasks. When multiple individuals work together on a task, their combined work rate is the sum of their individual work rates. This concept is applied in scenarios where groups of people complete tasks together and the goal is to find out how long each person would take to finish the task alone.

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For example, consider a scenario where a certain number of women and men together can complete a job in a specified time frame. These equations express the combined work rate of the group and can be solved simultaneously to find the values of x and y.

Once the individual work rates are determined, we can calculate the time each person would take to complete the task if they were working alone. It also provides a systematic method for solving complex problems involving multiple variables and work rates, making it a valuable tool in various applications.

Steps to Calculate Work Rate Problems

Follow and know the guidelines and steps to learn how to calculate the work rate problems.

Understand the Information:

Read the problem carefully and understand the quantities involved, such as the number of people working together, the time taken to complete the task, and any specific rates mentioned.

Assign Variables:

Use variables to represent unknown quantities, such as the time taken by one person to complete the task alone (e.g., x days for a woman, y days for a man).

Set up Equations:

Translate the information into equations based on work rates. Use fractions to represent the work done per day by each person or group.

Combine Work Rates:

If multiple people are working together, add their individual work rates to get the combined work rate of the group per day.

Solve the Equations:

Solve the system of equations to find the values of the variables (e.g., x and y) representing the time taken by one person to complete the task alone.

Interpret Results:

Once you have the values for the variables, interpret them in the context of the problem. Determine the time each person would take to complete the task alone based on the calculated work rates.

Check Solutions:

Double-check your calculations and ensure that the solutions make sense in the context of the problem. Verify that the combined work rates match the given information about the time taken to complete the task when working together.