A takes 5 days more than B to do a certain job and 9 days more than C; A and B together can do the job in the same time as C. How many days A would take to do it?

### Concept & Logic When multiple individuals' completion times are expressed relative to one person, define that person's time as a variable (e.g., $t$). Then, build a rational equation based on the condition that combining efficiencies adds up to a known total efficiency. $$ \text{Rate of A} + \text{Rate of B} = \text{Rate of C} $$ ### Step-by-Step Solution 1. **Define variables based on the text:** Let A's time to complete the job be $t$ days. A takes 5 days more than B $\implies$ B's time = $t - 5$ days. A takes 9 days more than C $\implies$ C's time = $t - 9$ days. 2. **Set up the work equation:** A and B together take the same time as C. Thus, their combined daily work rate equals C's daily work rate. $\frac{1}{t} + \frac{1}{t - 5} = \frac{1}{t - 9}$ 3. **Solve the rational equation:** Find a common denominator for the left side: $\frac{(t - 5) + t}{t(t - 5)} = \frac{1}{t - 9}$ $\frac{2t - 5}{t^2 - 5t} = \frac{1}{t - 9}$ Cross-multiply: $(2t - 5)(t - 9) = t^2 - 5t$ Expand the left side: $2t^2 - 18t - 5t + 45 = t^2 - 5t$ $2t^2 - 23t + 45 = t^2 - 5t$ Bring all terms to one side to form a quadratic equation: $t^2 - 18t + 45 = 0$ 4. **Factor the quadratic equation:** We need two numbers that multiply to 45 and add to -18. These are -15 and -3. $(t - 15)(t - 3) = 0$ So, $t = 15$ or $t = 3$. If $t = 3$, B's time would be $3 - 5 = -2$ days (impossible). Therefore, $t = 15$. A takes 15 days. ### Exam Strategy & Shortcut For quadratic equations derived from work problems, checking the options is often the fastest route. Try Option (c) 15: If A = 15, then B = 10, and C = 6. Check if A + B = C: $\frac{1}{15} + \frac{1}{10} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}$. Since $\frac{1}{6}$ is exactly C's rate, 15 is the correct answer. ### Common Pitfall A common mistake is setting up the algebraic relations backwards (e.g., making B's time $t+5$). Always double-check "who takes more time" to ensure the subtractions/additions go in the right direction. ### Final Answer Therefore, the correct answer is **15**.