A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done in 23 days ?

### Concept & Percentage Efficiency When combining the efforts of two workers where one is given as a percentage of the other, map their efficiencies to simple units (like 100 and 130) to easily find total work and combined rates. $$ \text{Total Work} = \text{Individual Efficiency} \times \text{Individual Time} $$ ### Step-by-Step Solution 1. **Establish the Efficiency Ratio:** Let B's efficiency be 100 units/day. A is 30% more efficient, so A's efficiency is 130 units/day. Efficiency ratio of A : B = 130 : 100 = 13 : 10. 2. **Calculate Total Work:** A alone completes the work in 23 days. Total Work = A's Efficiency $\times$ A's Time Total Work = $13 \times 23 = 299$ units. 3. **Calculate Combined Time:** Combined efficiency of A and B = $13 + 10 = 23$ units/day. Time taken working together = $\frac{\text{Total Work}}{\text{Combined Efficiency}}$ Time taken = $\frac{299}{23} = 13$ days. ### Exam Strategy & Shortcut By using the unit method: A = 13 units/day, B = 10 units/day. Combined = 23 units/day. Total work based on A = $13 \times 23$. Time together = $\frac{13 \times 23}{23}$. The 23s cancel out immediately, leaving exactly 13 days. You don't even need to calculate $13 \times 23 = 299$. ### Common Pitfall A common error is unnecessarily calculating the total work to 299 and then doing long division, which wastes time and increases the risk of arithmetic errors compared to simply canceling out the factors. ### Final Answer Therefore, the correct answer is **13 days**.