Nine men and twelve boys finish a job in 12 days, while twelve men and twelve boys finish the same job in 10 days. In how many days will ten men and ten boys working together finish the job?

Introduction / Context:
This work rate question compares the combined efficiencies of men and boys in different group sizes. By using two scenarios where the total job is completed in different times by different groups, we can find the individual daily work rates of a man and a boy. Then we can determine how long a third combination of ten men and ten boys will take to finish the same job. These kinds of questions are standard in quantitative aptitude sections and test skills with simultaneous equations and proportional reasoning.

Given Data / Assumptions:

• Nine men and twelve boys complete the job in 12 days.
• Twelve men and twelve boys complete the job in 10 days.
• All men work at the same rate and all boys work at the same rate.
• We are required to find the time taken by ten men and ten boys working together.

Concept / Approach:
We denote the work done in one day by one man as m and by one boy as b. The total work in one day by each group can be expressed in terms of m and b and equated to 1 divided by the given number of days. This yields two linear equations which we can solve for m and b. Once we know these rates, we can calculate the daily work of ten men and ten boys and then invert that daily work rate to find the total number of days required to complete one job.

Step-by-Step Solution:
Step 1: Let one man do m units per day and one boy do b units per day. Step 2: From the first condition, (9m + 12b) times 12 days equals 1 job, so 9m + 12b = 1 / 12. Step 3: From the second condition, (12m + 12b) times 10 days equals 1 job, so 12m + 12b = 1 / 10. Step 4: Solving these equations gives m = 1 / 180 and b = 1 / 360. Step 5: Daily work of ten men and ten boys is 10m + 10b = 10(1/180) + 10(1/360) = 1/18 + 1/36 = 1/12. Step 6: Time taken by ten men and ten boys to finish the job is therefore 1 divided by 1/12 which equals 12 days.

Verification / Alternative check:
We can verify the rates by substituting back into the original conditions. For nine men and twelve boys, daily work is 9(1/180) + 12(1/360) which equals 1/20 + 1/30 = 1/12. In 12 days they complete 12 times 1/12 which equals 1 job. For twelve men and twelve boys, daily work is 12(1/180) + 12(1/360) which equals 2/30 + 1/30 = 1/10, and in 10 days they complete 1 job. This confirms that the solved rates and the derived time of 12 days for ten men and ten boys are correct.

Why Other Options Are Wrong:

• 15 days and 14 days both correspond to daily work rates smaller than 1/12, which would not match the calculated combined rate.
• 11 days would require a slightly higher daily rate than 1/12, which does not arise from the solved values of m and b.
• Only 12 days is exactly consistent with the computed daily work of ten men and ten boys.

Common Pitfalls:
Common mistakes include adding the times instead of the rates or mismanaging the algebra when solving the simultaneous equations. Another pitfall is to assume men and boys contribute equally, which is not supported by the data. Keeping track of units as work per day and carefully solving for m and b avoids these errors and leads to a clear, correct answer.

Final Answer:
Ten men and ten boys working together will finish the job in 12 days.