P can do a piece of work in 5 days less than Q. If both P and Q working together can complete the work in 11 1/9 days, in how many days will Q alone complete the same work?

Introduction / Context:
This is a classic two person time and work problem where there is a given difference in the individual completion times and a given combined completion time. From this information we need to find one person's individual time. The data includes a mixed fraction for the combined time, which must be interpreted correctly. Questions of this type check understanding of reciprocals and the method of expressing rates in terms of unknown times and then solving a quadratic equation that arises naturally.

Given Data / Assumptions:

• P alone takes 5 days less than Q to complete the work.
• P and Q together complete the work in 11 1/9 days, which is 100/9 days.
• All work rates are constant and linear with time.
• We must find the number of days Q alone will take.

Concept / Approach:
Let Q's time to complete the work alone be x days. Then P's time is x minus 5 days. Their daily work rates are the reciprocals of these times. Since together they take 100/9 days, their combined daily rate is 9/100 of the work per day. Setting the sum of the individual rates equal to this combined rate leads to a rational equation which simplifies to a quadratic in x. Solving that quadratic and selecting the physically meaningful positive root gives Q's time.

Step-by-Step Solution:
Step 1: Let Q alone take x days, so P alone takes x - 5 days. Step 2: Daily rate of Q is 1 / x and of P is 1 / (x - 5). Step 3: Together they take 100/9 days, so their combined rate is 1 divided by 100/9 = 9 / 100 per day. Step 4: Set up the equation 1 / x + 1 / (x - 5) = 9 / 100. Step 5: Combine fractions: (2x - 5) / [x(x - 5)] = 9 / 100. Step 6: Cross multiply to get 100(2x - 5) = 9x(x - 5). Step 7: Simplify: 200x - 500 = 9x^2 - 45x, giving 9x^2 - 245x + 500 = 0. Step 8: Solve this quadratic to get x = 25 or x = 20/9. The meaningful positive integer time is x = 25 days.

Verification / Alternative check:
If Q takes 25 days, P takes 25 - 5 = 20 days. P's daily rate is 1/20 and Q's daily rate is 1/25. Their combined daily rate is 1/20 + 1/25 = (5 + 4) / 100 = 9/100 of the work per day. The time taken together is therefore 1 divided by 9/100 which equals 100/9 days, the same as the given 11 1/9 days. This confirms that Q needing 25 days is consistent with all the data in the problem.

Why Other Options Are Wrong:

• 24 days and 19 days would not produce a combined time of 11 1/9 days when used with P's time defined as x - 5.
• 20 days would imply P takes 15 days, and the resulting combined rate would not match 9/100 per day.
• Only 25 days correctly reproduces the given combined time when the implied value for P's time is used.

Common Pitfalls:
A common error is to misread 11 1/9 days and treat it as 11.9 days rather than 100/9 days. Another mistake is to forget that P's time is x minus 5 and accidentally use x plus 5. Finally, algebraic mistakes in solving the quadratic may lead to discarding the correct root. Careful conversion to improper fractions and careful algebra prevent these issues.

Final Answer:
Q alone will complete the work in 25 days.