Pannalal completes one-third of a job in 20 days. Saiprasad completes the remaining two-thirds of the same job in 10 days. In how many days can Pannalal and Saiprasad together complete the entire job if they work simultaneously from the start?

Introduction / Context:
This problem involves two workers, Pannalal and Saiprasad, who complete different fractions of the same job in different times when working separately. The task is to compute how long they will take to complete the entire job if they work together from the beginning. This is a standard time and work problem that checks the understanding of fractional work and combined work rates.

Given Data / Assumptions:

• Pannalal finishes 1/3 of the job in 20 days.
• Saiprasad finishes the remaining 2/3 of the job in 10 days.
• We must find the time taken if both work together from the start on the whole job.
• Total work is taken as 1 complete job.
• Both workers have constant work rates.

Concept / Approach:
We first convert the information about fractions of work completed into daily work rates. From the fact that Pannalal completes 1/3 of the job in 20 days, we can find his daily rate. Similarly, from Saiprasad completing 2/3 of the job in 10 days, we find his daily rate. Adding both rates gives the combined rate when they work together. Finally, time is computed as total work divided by combined rate. This direct approach is efficient and accurate.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. Step 2: Pannalal completes 1/3 of the job in 20 days. Step 3: Therefore, Pannalal daily rate = (1/3) / 20 = 1/60 job per day. Step 4: Saiprasad completes 2/3 of the job in 10 days. Step 5: Saiprasad daily rate = (2/3) / 10 = 2/30 = 1/15 job per day. Step 6: Combined daily rate of Pannalal and Saiprasad = 1/60 + 1/15. Step 7: LCM of 60 and 15 is 60; 1/15 = 4/60. Step 8: Combined rate = (1/60 + 4/60) = 5/60 = 1/12 job per day. Step 9: Time taken to complete 1 job together = 1 / (1/12) = 12 days.

Verification / Alternative check:
To verify, in 12 days Pannalal would do 12 * (1/60) = 12/60 = 1/5 of the job. Saiprasad would do 12 * (1/15) = 12/15 = 4/5 of the job. Adding these gives 1/5 + 4/5 = 1 full job. This confirms that the combined time of 12 days is correct and consistent with their individual rates.

Why Other Options Are Wrong:
If they took 6 or 3 days, the total work done would be too small given their rates. If they took 24 or 15 days, the total work done would exceed one full job. Only 12 days results in a total of exactly one complete job when both work simultaneously.

Common Pitfalls:
A frequent mistake is to average the times 20 and 10 instead of converting to rates based on fractions of work. Another error is forgetting that Pannalal completed only one-third of the job in 20 days, not the entire job, which would incorrectly suggest a different daily rate. Always pay close attention to the fraction of work completed before calculating the rate.

Final Answer:
Pannalal and Saiprasad together will complete the entire job in 12 days.