Directions: Read the information carefully and answer the questions. A, B, C and D together can do a work 'X' in 14 days. The ratio of efficiency of A to that of C is $3 : 2$. C is $50\%$ less efficient than that of B and B is $33\frac{1}{3}\%$ more efficient than D. A and B together can complete work 'X' in 'x' days, while A and D can do the same work together in 'y' days. 'j' men can complete a work in $(x + 4)$ day, while '$(y - 8)$' men can complete the same work in 'k' days. If '$(y + 2)$' men can complete the same work in $(0.5k + 3.5)$ days, then find in how many days $(k - j)$ men can complete the same work.

### Concept & Man-Days Work Equivalence This problem builds on the previously found variables and uses the standard Man-Days formula where total work is conserved across different group sizes. $$ M_1 \times D_1 = M_2 \times D_2 = \text{Total Work} $$ ### Step-by-Step Solution 1. **Recall 'x' and 'y' from the directions:** As derived in the previous question based on the efficiency ratios: $x = 24$ $y = 28$ 2. **Formulate the work equations:** - Case 1: 'j' men complete work in $(x + 4) = 24 + 4 = 28$ days. $\text{Total Work} = 28j$ - Case 2: '$(y - 8)$' men complete work in 'k' days. $(28 - 8) = 20$ men. $\text{Total Work} = 20k$ - Case 3: '$(y + 2)$' men complete work in $(0.5k + 3.5)$ days. $(28 + 2) = 30$ men. $\text{Total Work} = 30(0.5k + 3.5) = 15k + 105$ 3. **Solve for 'k':** Since the work is the same in all cases, equate Case 2 and Case 3: $20k = 15k + 105$ $5k = 105 \implies k = 21$ 4. **Solve for 'j':** Equate Case 1 and Case 2: $28j = 20k$ $28j = 20(21)$ $28j = 420 \implies j = 15$ 5. **Calculate the final requirement:** - Total Work = $420$ man-days. - We need the time for $(k - j)$ men to complete it. - $(k - j) = 21 - 15 = 6$ men. - Number of days = $\frac{\text{Total Work}}{\text{Number of Men}} = \frac{420}{6} = 70$ days. ### Exam Strategy & Shortcut You can solve for $k$ directly by only comparing the cases with $y$ values, since $y$ is a known constant. Equating $20k = 30(0.5k + 3.5)$ bypasses the unknown $j$ entirely until you actually need it to find the total work. ### Common Pitfall A common mistake here is messing up the algebraic expansion of $30(0.5k + 3.5)$. Often, students will multiply the $30$ by $0.5k$ but forget to multiply it by the $3.5$ term. ### Final Answer Therefore, the correct answer is **70 days**.