Question

If \(12\) men or \(20\) women can do a work in \(54\) days, then in how many days can \(9\) men and \(12\) women together do the work?

• A. \(38\) days
• B. \(32\) days
• C. \(40\) days
• D. \(35\) days

Hint:

Remember:

• Work and time are inversely proportional to efficiency
• If: \[ a \text{ men} = b \text{ women} \] then efficiencies can be converted easily
• More workers $\Rightarrow$ fewer days

Answer 1

The Correct Option is C

Concept: Work done is calculated using: \[ \text{Work} = \text{Efficiency} \times \text{Time} \] If two groups complete the same work in the same time, then their total efficiencies are equal.

Step 1: Find the relation between efficiency of men and women. Given: \[ 12 \text{ men complete work in } 54 \text{ days} \] Also: \[ 20 \text{ women complete same work in } 54 \text{ days} \] Hence: \[ 12 \text{ men} = 20 \text{ women} \] Dividing by \(4\): \[ 3 \text{ men} = 5 \text{ women} \] Therefore: \[ 1 \text{ man} = \frac{5}{3} \text{ women} \]

Step 2: Convert all workers into women-equivalent. Given: \[ 9 \text{ men and } 12 \text{ women} \] Since: \[ 1 \text{ man} = \frac{5}{3} \text{ women} \] \[ 9 \text{ men} = 9 \times \frac{5}{3} = 15 \text{ women} \] Total equivalent women: \[ 15 + 12 = 27 \text{ women} \]

Step 3: Calculate the required number of days. We know: \[ 20 \text{ women complete work in } 54 \text{ days} \] Using inverse proportion: \[ \text{Days} = \frac{20 \times 54}{27} \] \[ = 40 \] Therefore, \[ \boxed{40\text{ days}} \]