A can finish a piece of work in 12 days and B can finish the same work in 24 days. If A and B work together on the job for 4 days, what fraction of the total work remains unfinished?

Introduction / Context:
This problem evaluates understanding of combined work rates and asks for the fraction of work left after a certain number of days of joint work. Instead of computing the total time to finish, we calculate how much work has been completed and then subtract from the whole job.

Given Data / Assumptions:

• A alone can finish the work in 12 days.
• B alone can finish the work in 24 days.
• A and B work together for 4 days and then stop.
• Their work rates are constant for the entire duration.

Concept / Approach:
We again use the idea that a worker who finishes a job in T days has a daily rate of 1 / T of the work. Adding A's and B's rates gives the joint daily work. Multiplying this joint rate by the number of days they actually work gives the fraction of the job completed. The remaining fraction is simply 1 minus this completed fraction.

Step-by-Step Solution:
Step 1: Assume total work = 1 unit.Step 2: Rate of A = 1 / 12 of the work per day.Step 3: Rate of B = 1 / 24 of the work per day.Step 4: Combined rate of A and B together = 1 / 12 + 1 / 24.Step 5: Convert to a common denominator 24. Then 1 / 12 = 2 / 24, so combined rate = 2 / 24 + 1 / 24 = 3 / 24 = 1 / 8 of the work per day.Step 6: In 4 days, work completed = 4 * (1 / 8) = 4 / 8 = 1 / 2 of the total work.Step 7: Therefore fraction of work remaining = 1 - 1 / 2 = 1 / 2.

Verification / Alternative check:
We can verify by imagining that if they continued working together at the same rate, they would need another 4 days to complete the remaining 1 / 2 of the work, because 4 days more at rate 1 / 8 per day equals another 1 / 2. That suggests the total time to finish together is 8 days, which is consistent with the rate calculation since time = 1 divided by rate = 1 / (1 / 8) = 8 days. Thus after 4 days only half the job is completed.

Why Other Options Are Wrong:

• 1/3: This would imply that two thirds of the work is completed, which does not match the computed total work done of 1 / 2.
• 1/4: This suggests three quarters of the work has been completed, which is incorrect for 4 days at rate 1 / 8 per day.
• 1/5: This implies completion of four fifths of the work, which contradicts the rate based calculation.

Common Pitfalls:
Some learners incorrectly add the days (12 and 24) or average them instead of using rates. Others forget to subtract the completed fraction from 1 and mistakenly treat the completed fraction as the remaining fraction. Always calculate work done first, then subtract from 1 to find how much work remains.

Final Answer:
The fraction of the work left unfinished after 4 days is 1/2.