Nine girls working 7 hours every day can complete a certain piece of work in 15 days. If only six girls are available but they work for 9 hours each day, in how many days will they complete the same amount of work?

Introduction / Context:
This problem is a classic time and work question involving different numbers of workers and different daily working hours. The total amount of work is the same in both situations, and we are asked to find how long a smaller group working longer hours will take to finish the same task.

Given Data / Assumptions:

• Nine girls working 7 hours per day can complete the work in 15 days.
• We want to find the number of days required when six girls work 9 hours per day on the same work.
• All girls work at the same constant rate and productivity does not change with group size.

Concept / Approach:
Total work is the product of the number of workers, hours per day, days, and the work rate per worker. Because the nature of the work does not change, the same total work is performed in each scenario. Instead of computing absolute work units, we can equate the total girl hours from both cases and solve for the unknown number of days in the second scenario.

Step-by-Step Solution:
Let the total work be W and the work rate of each girl be r units per hour. In the first scenario, total work W = 9 girls * 7 hours per day * 15 days * r. So W = 9 * 7 * 15 * r = 945 r. In the second scenario, six girls work 9 hours per day for D days, so W = 6 * 9 * D * r. Equate the work: 6 * 9 * D * r = 945 r. Cancel r and simplify: 54 D = 945. Therefore, D = 945 / 54 = 17.5 days. Thus six girls working 9 hours per day need 17.5 days to complete the task.

Verification / Alternative check:
A quick reasonableness check is that we reduced the number of girls from nine to six (a factor of 2/3) but increased daily hours from seven to nine (a factor of 9/7). The combined change in daily girl hours is (6 * 9) / (9 * 7) = 54 / 63 = 6 / 7, so daily work capacity is 6/7 of what it was initially. Therefore, the number of days should increase by a factor of 7/6. Initial days are 15, so new days are 15 * 7 / 6 = 105 / 6 = 17.5, matching our earlier calculation.

Why Other Options Are Wrong:
14 days is too small and would imply higher overall daily capacity, which conflicts with fewer girls even with longer hours. 19.5 days is greater than 17.5 and does not agree with the ratio 7/6 derived from comparing daily girl hours. 21 days also overestimates the needed time and contradicts the calculated ratio. 12.5 days is unrealistically small and clearly inconsistent with having fewer workers.

Common Pitfalls:
A frequent mistake is to compare only numbers of workers and ignore the change in working hours, or to simply add or subtract days. Another error is to assume the work rate per group is linear in days without accounting for hours per day. Always convert everything into total worker hours before equating the two scenarios.

Final Answer:
Six girls working 9 hours a day will complete the work in 17.5 days.