A and B can do a job together in 7 days. A is $1 \frac{3}{4}$ times as efficient as B. The same job can be done by A alone in :

### Concept & Efficiency Ratio When comparing efficiencies with mixed fractions, convert them to improper fractions to easily determine the ratio of units of work completed per day. $$ \text{Total Work} = \text{Combined Efficiency} \times \text{Combined Time} $$ ### Step-by-Step Solution 1. **Determine the Efficiency Ratio:** A is $1 \frac{3}{4}$ times as efficient as B. Convert this to an improper fraction: $\frac{7}{4}$. Let B's efficiency be 4 units/day. A's efficiency is $\frac{7}{4} \times 4 = 7$ units/day. The efficiency ratio of A : B is 7 : 4. 2. **Calculate Total Work:** Combined efficiency of A and B = $7 + 4 = 11$ units/day. They complete the job together in 7 days. Total Work = Combined efficiency $\times$ Time taken Total Work = $11 \times 7 = 77$ units. 3. **Calculate Time Taken by A Alone:** Time = $\frac{\text{Total Work}}{\text{A's Efficiency}}$ Time for A = $\frac{77}{7} = 11$ days. ### Exam Strategy & Shortcut If A is $\frac{7}{4}$ times as efficient as B, for every 4 units of work B does, A does 7. Total work units per day = 11 parts. Since it takes 7 days to finish 11 parts daily, total work is 77 parts. To find A's time, divide the total work (77 parts) by A's rate (7 parts/day). $\frac{77}{7} = 11$. ### Common Pitfall Students often mix up the ratios when translating "times as efficient as," accidentally assigning 7 to B and 4 to A, leading to the wrong individual times. ### Final Answer Therefore, the correct answer is **11 days**.