Question

Twelve men and 5 women can complete a work in 2 days whereas 4 men and 3 women can complete the same work in 5 days. In how many days can 8 men complete the work?

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

When given combined work rates, form simultaneous equations in terms of individual rates, then solve. Once the rate of the required group is found, time = work ÷ rate.

Answer 1

The Correct Option is B

Step 1: Represent rates of work.
Let 1 man’s 1-day work = \(m\) and 1 woman’s 1-day work = \(w\). From the first condition: \[ (12m + 5w) \times 2 = 1 \quad \Rightarrow \quad 12m + 5w = \frac{1}{2}. \] From the second condition: \[ (4m + 3w) \times 5 = 1 \quad \Rightarrow \quad 4m + 3w = \frac{1}{5}. \] Step 2: Solve for \(m\) and \(w\).
Multiply the second equation by 3: \(12m + 9w = \frac{3}{5}\). Subtract the first equation: \[ (12m + 9w) - (12m + 5w) = \frac{3}{5} - \frac{1}{2} \quad \Rightarrow \quad 4w = \frac{6 - 5}{10} = \frac{1}{10}. \] Thus \(w = \frac{1}{40}\). Substitute into \(4m + 3w = \frac{1}{5}\): \[ 4m + 3 \cdot \frac{1}{40} = \frac{1}{5} \quad \Rightarrow \quad 4m + \frac{3}{40} = \frac{8}{40}. \] \[ 4m = \frac{5}{40} \quad \Rightarrow \quad m = \frac{5}{160} = \frac{1}{32}. \] Step 3: Time for 8 men to complete the work.
In 1 day, 8 men do \(8 \times \frac{1}{32} = \frac{1}{4}\) of the work. Thus time required = \(4\) days. \[ \boxed{4} \]