CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 8 : Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job. However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be

Video Explanation

Explanatory Answer

Let Rahul, Rakshita and Gurmeet take A, B and C days respectively to finish the job on their own, this means, in one day, the fraction of work done by Rahul, Rakshita and Gurmeet is \( \frac { 1 } { A } , \frac { 1 } { B } , \frac { 1 } { C } \).
Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job.
This means, the fraction of work done by all three of them, together, in one day, is less than \( \frac { 1 } { 7 } \).
\( \frac { 1 } { A } + \frac { 1 } { B } + \frac { 1 } { C } < \frac { 1 } { 7 } \)
However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job.
6 × (the fraction of work done by them together in one day) + 3 × (the fraction of work done by Rakshita in one day) = (one full job)
\( 6 \left( \frac { 1 } { A } + \frac { 1 } { B } + \frac { 1 } { C } \right) + \frac { 3 } { B } = 1 \)
\( \frac { 3 } { B } = 1 - 6 \left( \frac { 1 } { A } + \frac { 1 } { B } + \frac { 1 } { C } \right) \)
But, we know that, \( \frac { 1 } { A } + \frac { 1 } { B } + \frac { 1 } { C } < \frac { 1 } { 7 } \)
\( \frac { 3 } { B } > 1 - \frac { 6 } { 7 } \)
\( \frac { 1 } { B } > \frac { 1 } { 21 } \)
\( B < 21 \)
So, Rakshita should take less than 21 days to finish the job.
The extra information can be used to find the lower bound of B.
On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job.
\( \frac { 1 } { A } + \frac { 1 } { c } > \frac { 1 } { 15 } \)
But, we know that, \( \frac { 1 } { A } + \frac { 1 } { B } + \frac { 1 } { C } < \frac { 1 } { 7 } \)
So, the maximum value of \( \frac { 1 } { B } = \frac { 1 } { 7 } - \frac { 1 } { 15 } = \frac { 8 } { 7 ( 15 ) } \).
The minimum value of B = 13.125

The question is " Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job. However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be "

The answer is '21'

Choice B is the correct answer.