Final answer:
Person A works on the project for 3 days before leaving. A, B, and C combined work rate is 13/60 of the project per day. B and C's combined rate after A leaves is 7/60 of the project per day.
Step-by-step explanation:
To find out after how many days person A leaves the project when A, B, and C work together and complete the project in 6 days, we first determine their individual work rates. Person A finishes the project in 10 days, B in 15 days, and C in 20 days. Their work rates are thus 1/10, 1/15, and 1/20 of the project per day, respectively.
Next, we find the combined work rate for all three: (1/10) + (1/15) + (1/20). To add these fractions, they must have the same denominator. The least common multiple of 10, 15, and 20 is 60. Therefore, their combined work rates would be (6/60) + (4/60) + (3/60), which equals 13/60 of the project per day.
Let's assume A works for x days before getting sick. In those x days, they complete x(13/60) of the project together. After A leaves, B and C finish the project in (6 - x) days, with a work rate of (1/15) + (1/20) = (4/60) + (3/60) = 7/60 of the project per day. So, (6 - x)(7/60) is the part of the project they complete.
Since the whole project is completed in 6 days, we have the equation: x(13/60) + (6 - x)(7/60) = 1. Multiplying both sides by 60 to clear the denominators and simplifying the equation, we get: 13x + 7(6 - x) = 60. Solving for x gives us 13x + 42 - 7x = 60, which simplifies to 6x = 18, and therefore x = 3.
Person A leaves the project after 3 days, and B and C work for the remaining 3 days to complete it.