A mother can complete a certain job alone in x hours. Her daughter takes exactly twice as long as her mother to do the same job on her own. Working together, the mother and the daughter can finish the job in 6 hours. How many hours does the mother alone take to complete the job?

Introduction / Context:
This is a classic work rate problem involving two people with different individual times but a known combined time. The mother is faster, the daughter is slower, and together they complete the job in a given time. We are asked to find how long the mother alone would take to complete the job.

Given Data / Assumptions:

• Mother alone takes x hours to complete the job.
• Daughter alone takes 2x hours to complete the same job.
• Together, they can complete the job in 6 hours.
• Work rates of both remain constant and add when they work together.

Concept / Approach:
If a person takes t hours to finish a job, the work rate is 1 / t jobs per hour. For two people working together, their combined rate is the sum of their individual rates. Here, we write the combined rate of mother and daughter in terms of x and then set it equal to the known combined rate 1 / 6. Solving that equation gives x, the mother's time alone.

Step-by-Step Solution:
Mother's time alone = x hours, so her rate = 1 / x jobs per hour. Daughter's time alone = 2x hours, so her rate = 1 / (2x) jobs per hour. Together their rate = 1 / x + 1 / (2x) = (2 + 1) / (2x) = 3 / (2x). It is given that together they finish the work in 6 hours, so combined rate = 1 / 6. Therefore, 3 / (2x) = 1 / 6. Cross multiply: 3 * 6 = 2x. So, 18 = 2x, which gives x = 9. Thus the mother alone will take 9 hours to complete the job.

Verification / Alternative check:
If the mother takes 9 hours, her rate is 1 / 9; the daughter then takes 18 hours, with rate 1 / 18. The combined rate is 1 / 9 + 1 / 18 = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6, meaning they finish in 6 hours as stated in the problem. This confirms that x = 9 hours is consistent.

Why Other Options Are Wrong:
3 hours would make the mother much too fast and the combined time far less than 6 hours. 6 hours would imply mother and daughter have equal speeds and contradicts the condition that the daughter takes twice as long. 12 hours and 15 hours would both give combined rates that do not correspond to a 6 hour completion time.

Common Pitfalls:
A common mistake is to average the times directly (for example, taking (x + 2x) / 2) instead of working with rates. Another error is forgetting that the daughter takes twice the time of the mother, not twice the rate. Always set up the problem in terms of rates (reciprocals of time) when dealing with combined work scenarios.

Final Answer:
The mother alone takes 9 hours to complete the job.