Efficiency Calculation for Work among Men and Boys

In this article, we will explore a problem where we need to determine how long it will take for a specific combination of men and boys to complete a work project. This problem deals with the concept of work rate and how it can be used to solve real-world problems. We will follow a step-by-step approach to find the solution, utilizing equations and efficient problem-solving techniques.

Problem Statement

Given that 3 men and 4 boys can complete a work in 8 days, whereas 4 men and 4 boys can complete the same work in 6 days, we need to find how many days will it take for 2 men and 4 boys to complete the work.

Step-by-Step Solution

Let's denote the work rate of one man as M and the work rate of one boy as B.

Step 1: Set Up Equations

From the first scenario:

$$3M 4B times 8 1$$

This simplifies to:

$$3M 4B frac{1}{8}$$

From the second scenario:

$$4M 4B times 6 1$$

This simplifies to:

$$4M 4B frac{1}{6}$$

Lets express these equations in a more manageable form:

$$3M 4B frac{1}{8} quad text{Equation 1}$$ $$M B frac{1}{24} quad text{Equation 2}$$

Step 2: Solve the Equations

From Equation 2, we can express B in terms of M as:

$$B frac{1}{24} - M$$

Substitute B in Equation 1:

$$3M 4left(frac{1}{24} - Mright) frac{1}{8}$$

This simplifies to:

$$3M frac{1}{6} - 4M frac{1}{8}$$

$$-M frac{1}{6} frac{1}{8}$$

Multiplying by 24 to clear the denominators:

$$-24M 4 3$$

$$-24M -1$$

$$M frac{1}{24}$$

Step 3: Find B

Substituting M back into the expression for B:

$$B frac{1}{24} - frac{1}{24} 0$$

This result is incorrect. Let's re-evaluate. Rearrange Equation 2:

$$M B frac{1}{24}$$

Express B in terms of M:

$$B frac{1}{24} - M$$

Substitute into Equation 1:

$$3M 4left(frac{1}{24} - Mright) frac{1}{8}$$

This simplifies to:

$$3M frac{1}{6} - 4M frac{1}{8}$$

$$-M frac{1}{6} frac{1}{8}$$

Multiplying by 24 to clear the denominators:

$$-24M 4 3$$

$$-24M -1$$

$$M frac{1}{24}$$

$$B frac{1}{24} - frac{1}{24} 0$$

The earlier result was indeed correct, and B 0. Substitute back:

Final Calculation

Now we need to find how many days 2 men and 4 boys will take:

$$2M 4B 2 cdot frac{1}{24} 4 cdot 0 frac{1}{12}$$

This means 2 men and 4 boys can complete (frac{1}{12}) of the work in one day.

Conclusion

To find the total time taken to complete 1 work:

$$text{Days} frac{1}{text{Work Rate}} frac{1}{frac{1}{12}} 12 text{ days}$$

Thus, 2 men and 4 boys will finish the work in 12 days.