Meghana alone would take 32 hours more than Meghana and Ganesh working together to complete a certain job. If Ganesh worked alone, he would take 12 1/2 hours more than their combined time to finish the same job. How many hours will it take for Meghana and Ganesh to complete the job when they work together from start to end?

Introduction / Context:
This problem tests the standard time and work concept where two people can do a job together faster than either of them can do it alone. The question gives how much extra time Meghana and Ganesh would each take when working alone compared with their combined time, and asks us to find the time required when they work together from start to finish.

Given Data / Assumptions:

• The combined time of Meghana and Ganesh working together is T hours.
• Meghana alone would take T + 32 hours to complete the job.
• Ganesh alone would take T + 12.5 hours (that is, 12 1/2 hours) to complete the job.
• Both work at constant rates and their work rates are additive when they work together.

Concept / Approach:
When people work together, their combined work rate is the sum of their individual work rates. If someone finishes a job in t hours, the work rate is 1 / t jobs per hour. Here, we convert each time into a rate, add the rates for Meghana and Ganesh, and equate this sum to the joint rate 1 / T. This gives us an equation in T that we can solve to get the required joint time.

Step-by-Step Solution:
Let T be the number of hours Meghana and Ganesh take together to finish the job. Meghana's time alone = T + 32, so her rate = 1 / (T + 32). Ganesh's time alone = T + 12.5, so his rate = 1 / (T + 12.5). Their combined rate when both work together is 1 / T. Therefore, we have: 1 / T = 1 / (T + 32) + 1 / (T + 12.5). Multiply through by T (T + 32) (T + 12.5) to clear denominators and simplify the resulting quadratic equation. Solving the equation gives T = 20 as the meaningful positive solution. Thus, Meghana and Ganesh together can complete the work in 20 hours.

Verification / Alternative check:
If T = 20, Meghana alone would take 52 hours and Ganesh alone would take 32.5 hours. Their work rates are 1 / 52 and 1 / 32.5 respectively. Adding these two rates gives a value that is very close to 1 / 20, confirming that the pair working together need about 20 hours for one complete job. This matches the equation and confirms consistency.

Why Other Options Are Wrong:
18 hours is too small and would make the required single person times inconsistent with the given extra time conditions. 22 hours does not satisfy the rate equation formed from the information about extra 32 and 12.5 hours. 20.5 hours is close but still not an exact solution of the equation, so it is not correct. 24 hours is much larger than the real joint time and would not be compatible with the given solo times.

Common Pitfalls:
Many learners forget to convert the mixed fraction 12 1/2 hours into 12.5 hours and make arithmetic mistakes. Another common mistake is to add times directly instead of adding work rates, which is incorrect. Always remember that rates add, not times. Also, take care to correctly form and solve the quadratic equation without dropping terms.

Final Answer:
Meghana and Ganesh together will complete the job in 20 hours.