Question

X, Y, and Z are three software experts, who work on upgrading the software in a number of identical systems. X takes a day off after every 3 days of work, Y takes a day off after every 4 days of work and Z takes a day off after every 5 days of work.
Starting afresh after a common day off,
i) X and Y working together can complete one new upgrade job in 6 days
ii) Z and X working together can complete two new upgrade jobs in 8 days
iii) Y and Z working together can complete three new upgrade jobs in 12 days If X, Y and Z together start afresh on a new upgrade job (after a common day off), exactly how many days will be required to complete this job?

• A. 3 days
• B. 4 days
• C. 2 days
• D. 3.5 days
• E. 2.5 days

Answer 1

Answer: (E)

To solve this problem, we need to determine how many new upgrade jobs X, Y, and Z can complete together in a single day and use this to find out how many days they will take to complete one job. Here's how we do that: 

1. Determine each expert's work rate considering their days off:
   • X works 3 days and takes 1 day off; thus, he effectively works 75% of the time. Y is effective for 80%, and Z for 80%.
2. Given conditions:
   • X + Y complete 1 job in 6 days, so their joint work rate is 1/6 jobs per day.
   • Z + X complete 2 jobs in 8 days, so their joint work rate is 2/8 = 1/4 jobs per day.
   • Y + Z complete 3 jobs in 12 days, so their joint work rate is 3/12 = 1/4 jobs per day.
3. Using these equations:
   • X + Y = 1/6
   • Z + X = 1/4
   • Y + Z = 1/4
4. Solving these equations simultaneously:
   • Add (X + Y), (Z + X), and (Y + Z):
   • 2X + 2Y + 2Z = 1/6 + 1/4 + 1/4 = 2/3
   • Divide throughout by 2: X + Y + Z = 1/3
5. This means X, Y, and Z together can complete one job in 3 days if they work continuously. Considering their actual working day percentages:
   • X's effective work rate: 3/4 of the time
   • Y's effective work rate: 4/5 of the time
   • Z's effective work rate: 4/5 of the time
6. Total effective work rate:
   • X, Y, Z combined = 1/3 job per day on full efficiency days.
   • Effective working rate considering days off = (3/4)*(1/3) + (4/5)*(1/3) + (4/5)*(1/3) = 1/4 + 1/5 + 1/5 = 0.2667+0.2+0.2 = 0.6667 per day.
7. Therefore, to complete 1 job, the number of days required = 1/0.6667 ≈ 2.5 days.

Hence, X, Y, and Z working together will complete one job in 2.5 days.