P and Q can complete a task in 50 days and 20 days respectively when working alone. If they work together from the beginning, how many days will they take to complete 70% of the task?

Introduction / Context:
This question involves two workers P and Q with different individual completion times. Instead of asking for the time to finish the entire task, it asks how long they will take working together to complete 70 percent of the task. This tests your ability to apply the concept of combined work rates to a fraction of the total job, not just the full job.

Given Data / Assumptions:

• P alone can complete the full task in 50 days.
• Q alone can complete the full task in 20 days.
• P and Q work together for the entire period considered.
• We need the time taken to complete 70% (or 0.7) of the full task.
• All rates are constant over time.

Concept / Approach:
We treat the full task as 1 unit of work. P and Q's individual rates are 1 divided by their times. The combined rate is the sum of these rates. To find the time to complete only 70% of the work, we multiply the full completion time by 0.7, or more directly divide 0.7 by the combined rate. This method generalizes to any fraction of work, not just 70% or 100%.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: P's rate = 1/50 of the work per day. Step 3: Q's rate = 1/20 of the work per day. Step 4: Combined rate of P and Q = 1/50 + 1/20. Step 5: Using common denominator 100, 1/50 = 2/100 and 1/20 = 5/100, so combined rate = 7/100 of the work per day. Step 6: We need time to finish 70% of the work = 0.7 units. Step 7: Time = required work / rate = 0.7 / (7/100) = 0.7 × (100/7) = 70/7 = 10 days.

Verification / Alternative check:
If P and Q work for 10 days, total work done = 10 × (7/100) = 70/100 = 0.7 of the full task, which is exactly 70%. Also, the full task would take 1 / (7/100) = 100/7 ≈ 14.29 days when they work together, so finishing 70% in 10 days is consistent with this overall rate. Nothing contradicts the given individual times.

Why Other Options Are Wrong:
12, 14, 18 and 20 days all correspond to more than 70% of the work done at the rate of 7/100 per day. For example, in 14 days they would complete 14 × 7/100 = 0.98 of the work, which is 98%, not 70%. Therefore, none of those options match the required fraction. Only 10 days produces exactly 70% of the job.

Common Pitfalls:
One common mistake is to compute the time for the full task and then forget to scale it down for 70%. Another is to multiply 50 and 20 or average them instead of adding their rates. Some learners also directly take 70% of each individual time, which is not correct because work combines through rates, not times. Always work with rates and required fractions of work for accurate results.

Final Answer:
P and Q together will complete 70% of the task in 10 days.