Men and Women Work Problem

Video Explanation

Question

2 women and 5 men can finish a piece of embroidery in 4 days. 3 women and 6 men can finish it in 3 days. Find the time taken by one woman alone and one man alone.

Solution

Step 1: Let Variables

Let work done by one woman in 1 day = \(x\)

Let work done by one man in 1 day = \(y\)

Total work = 1 unit

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Step 2: Form Equations

First condition:

\[ (2x + 5y)\times 4 = 1 \Rightarrow 2x + 5y = \frac{1}{4} \quad (1) \]

Second condition:

\[ (3x + 6y)\times 3 = 1 \Rightarrow 3x + 6y = \frac{1}{3} \quad (2) \]

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Step 3: Solve Linear Equations

Simplify (2):

\[ x + 2y = \frac{1}{9} \quad (3) \]

From (3):

\[ x = \frac{1}{9} – 2y \]

Substitute into (1):

\[ 2\left(\frac{1}{9} – 2y\right) + 5y = \frac{1}{4} \]

\[ \frac{2}{9} – 4y + 5y = \frac{1}{4} \]

\[ \frac{2}{9} + y = \frac{1}{4} \]

\[ y = \frac{1}{4} – \frac{2}{9} = \frac{1}{36} \]

Then:

\[ x = \frac{1}{9} – \frac{2}{36} = \frac{1}{18} \]

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Step 4: Find Time Taken

For one woman:

\[ \text{Time} = \frac{1}{x} = 18 \text{ days} \]

For one man:

\[ \text{Time} = \frac{1}{y} = 36 \text{ days} \]

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Conclusion

\[ \text{One woman takes } 18 \text{ days, and one man takes } 36 \text{ days} \]

Verification

Values satisfy both equations ✔