Pradeep completes one-fourth of a job in 14 days when working alone, and Saquib completes the remaining three-fourths of the job in 56 days when working alone. If Pradeep and Saquib now work together from the start at their usual constant rates, in how many days will they complete the entire job?

Introduction / Context:
This time and work problem gives you information about how long Pradeep and Saquib each take to complete different fractions of the same job when working alone. You are then asked to find how long it will take them to complete the job together if they start at the same time. This tests your understanding of converting partial-job times into full-job rates and then combining those rates.

Given Data / Assumptions:

Pradeep completes 1/4 of the job in 14 days working alone.
Saquib completes the remaining 3/4 of the job in 56 days working alone.
Both work at constant individual rates throughout their work.
We want the time taken if Pradeep and Saquib work together from the beginning on the whole job.

Concept / Approach:
We first compute Pradeep's daily work rate and Saquib's daily work rate relative to the full job. Then we add these rates to obtain their combined daily rate. Finally, we divide the total work (1 unit) by the combined rate to determine how many days they need to complete the job together.

Step-by-Step Solution:
Let the total work be 1 unit. Pradeep completes 1/4 of the job in 14 days, so his rate p satisfies 14p = 1/4. Thus, p = (1/4) / 14 = 1/56 of the job per day. Saquib completes 3/4 of the job in 56 days, so his rate s satisfies 56s = 3/4. Thus, s = (3/4) / 56 = 3/224 of the job per day. Combined daily rate of Pradeep and Saquib = p + s = 1/56 + 3/224. Convert 1/56 to denominator 224: 1/56 = 4/224. So combined rate = 4/224 + 3/224 = 7/224. Simplify 7/224 to 1/32. Therefore, they together complete 1/32 of the job per day. Time taken together = total work / combined rate = 1 / (1/32) = 32 days.

Verification / Alternative check:
If they finish 1/32 of the job per day, then in 32 days they complete exactly the whole job. Also, you can check that Pradeep alone would finish the job in 56 days, since his rate is 1/56, and Saquib alone would finish in 224/3 days (around 74.67 days). Since the combined time 32 days is less than both individual times, it is logical and confirms the correctness of the result.

Why Other Options Are Wrong:
Option 64 days would correspond to a combined rate of 1/64, which is smaller than both individual rates, clearly impossible.
Option 16 days would require a combined rate of 1/16, which is higher than the sum 1/56 + 3/224, and does not match the calculations.
Option 8 days would be even faster and completely inconsistent with the given data, as neither worker is that fast alone or even combined.
Option 40 days does not match the exact combined rate of 1/32 and would correspond to only 40 * (1/32) = 1.25 of the job if 1/32 were the rate, which is impossible.

Common Pitfalls:
One common mistake is to treat the given partial times as full-job times without converting, which leads to incorrect rates. Another pitfall is manging fractions incorrectly, especially when changing denominators or simplifying. Always convert partial job information into daily rates in terms of the full job, then add the rates and finally invert to obtain the total time together.

Final Answer:
Thus, working together from the start, Pradeep and Saquib can complete the entire job in 32 days, so the correct option is 32 days.