Men and Boys Work Problem

Video Explanation

Question

2 men and 7 boys can do a piece of work in 4 days. 4 men and 4 boys can do the same work in 3 days. In how many days will one man and one boy do the work?

Solution

Step 1: Let Variables

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

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

Total work = 1 unit

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Step 2: Form Equations

First condition:

\[ (2x + 7y)\times 4 = 1 \Rightarrow 2x + 7y = \frac{1}{4} \quad (1) \]

Second condition:

\[ (4x + 4y)\times 3 = 1 \Rightarrow 4x + 4y = \frac{1}{3} \quad (2) \]

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Step 3: Solve Linear Equations

Simplify (2):

\[ x + y = \frac{1}{12} \quad (3) \]

Substitute \(x = \frac{1}{12} – y\) into (1):

\[ 2\left(\frac{1}{12} – y\right) + 7y = \frac{1}{4} \]

\[ \frac{1}{6} – 2y + 7y = \frac{1}{4} \]

\[ \frac{1}{6} + 5y = \frac{1}{4} \]

\[ 5y = \frac{1}{4} – \frac{1}{6} = \frac{1}{12} \]

\[ y = \frac{1}{60} \]

Then:

\[ x = \frac{1}{12} – \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \]

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Step 4: Find Required Time

Work done by one man + one boy in 1 day:

\[ x + y = \frac{1}{15} + \frac{1}{60} = \frac{5}{60} = \frac{1}{12} \]

Time taken:

\[ \text{Time} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

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Conclusion

\[ \text{One man and one boy will take } 12 \text{ days} \]

Verification

Check values satisfy both equations ✔