Time and Work Question 1 - CAT 2020
John takes twice as much time as Jack to finish a job. Jack and Jim together take one-third of the time to finish the job than John takes working alone. Moreover, in order to finish the job, John takes three days more than that taken by three of them working together. In how many days will Jim finish the job working alone?
Step-by-Step Explanation
Let us establish the individual work completion timeframes in terms of a variable.
- Analyze the Time and Efficiency Profiles:
Let the time taken by Jack to complete the job be t days.
Since John takes twice as much time, John takes 2t days.
Now convert these into daily efficiencies (work done per day):
Jack's Efficiency = 1/t
John's Efficiency = 1/(2t)
- Determine Jim's Efficiency:
Jack and Jim together take 1/3 of John's standalone time, which means their combined time is:
2t/3 days
Therefore, their combined daily efficiency is the inverse of their combined time:
Jack + Jim's Combined Efficiency = 3/(2t)
To isolate Jim's daily efficiency, subtract Jack's rate from the combined rate:
Jim's Efficiency
= 3/(2t) − 1/t = 3/(2t) − 2/(2t) = 1/(2t)
Since Jim's daily efficiency is 1/(2t), Jim takes exactly 2t days to complete the job alone.
- Analyze the Combined Team Work Frame:
Combine the rates of all three workers (John, Jack, and Jim) to find their joint daily output:
Total Combined Efficiency = John + Jack + Jim
= 1/(2t) + 1/t + 1/(2t) = 1/(2t) + 2/(2t) + 1/(2t) = 4/(2t) = 2/t
The time taken by all three working together is the inverse of this total efficiency: t/2 days
- Solve for the Time Variable (t):
We are given that John takes 3 days more than the three of them working together:
2t = t/2 + 3
2t − t/2 = 3
⟹ 3t/2 = 3
⟹ t = 2 days
- Calculate Jim's Standalone Completion Time:
Jim's time = 2t
= 2 × 2
= 4 days
Correct Answer: 4
💡 FundaMakers Tip:
Avoid long-winded equations by converting time statements into simple efficiency ratios early on. Knowing that Jack and Jim take 1/3 of John's time means their combined efficiency is exactly 3 times John's efficiency. Since Jack is twice as fast as John, Jim must cover the remaining balance, making Jim's efficiency exactly equal to John's. Recognizing this identity lets you skip the variable tracking entirely.