Arithmetic

Time & Work

Medium

One day, Rahul started a work at $9$ AM and Gautam joined him two hours later. They then worked together and completed the work at $5$ PM the same day. If both had started at $9$ AM and worked together, the work would have been completed $30$ minutes earlier. Working alone, the time Rahul would have taken, in hours, to complete the work is

$12$

$10$

$11.5$

$12.5$

✅ Correct Option: 2

We need to set up equations based on the given information and solve systematically.

Let us define our variables clearly:

R = Rahul's work rate (fraction of work completed per hour)

G = Gautam's work rate (fraction of work completed per hour)

W = Total work (we can set this as 1 unit for simplicity)

Scenario 1 (What actually happened):

Rahul works alone from 9 AM to 11 AM = 2 hours

Both work together from 11 AM to 5 PM = 6 hours

Total time: Rahul works 8 hours, Gautam works 6 hours

Scenario 2 (If both started together):

Both work together from 9 AM to 4:30 PM = 7.5 hours

They finish 30 minutes earlier than 5 PM

From Scenario 1:

Work done by Rahul + Work done by Gautam = Total work

$R \times 8 + G \times 6 = W \quad \text{...(1)}$

From Scenario 2:

Work done by both together = Total work

$(R + G) \times 7.5 = W \quad \text{...(2)}$

Since both expressions equal W, we can set them equal:

$R \times 8 + G \times 6 = (R + G) \times 7.5$

Expanding the right side:

$8R + 6G = 7.5R + 7.5G$

$8R - 7.5R = 7.5G - 6G$

$0.5R = 1.5G$

$R = 3G$

This tells us that Rahul is 3 times as efficient as Gautam.

Substituting R = 3G back into equation (1):

$3G \times 8 + G \times 6 = W$

$24G + 6G = W$

$30G = W$

$G = \dfrac{W}{30}$

Therefore: $R = 3G = 3 \times \dfrac{W}{30} = \dfrac{W}{10}$

Since Rahul's rate is $\dfrac{W}{10}$ (he completes 1/10 of the work per hour), he would need 10 hours to complete the entire work alone.

Key Insight: This problem demonstrates how comparing different work scenarios helps us find individual efficiencies. The 30-minute difference gave us the crucial relationship between their work rates.

CAT 2017 Slot 2

A tank has an inlet pipe and an outlet pipe. If the outlet pipe is closed then the inlet pipe fills the empty tank in $8$ hours. If the outlet pipe is open then the inlet pipe fills the empty tank in $10$ hours. If only the outlet pipe is open then in how many hours the full tank becomes half-full?

CAT 2020 Slot 3

A contractor agreed to construct a $6$ km road in $200$ days. He employed $140$ persons for the work. After $60$ days, he realized that only $1.5$ km road has been completed. How many additional people would he need to employ in order to finish the work exactly on time?

CAT 2023 Slot 3

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

CAT 2023 Slot 1

The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for $4$ days and $9$ days, respectively, Kamal needs to work for $16$ days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone, is

CAT 2017 Slot 1

A person can complete a job in $120$ days. He works alone on Day $1$ . On Day $2$ , he is joined by another person who also can complete the job in exactly $120$ days. On Day $3$ , they are joined by another person of equal efficiency. Like this, everyday a new person with the same efficiency joins the work. How many days are required to complete the job?

CAT 2019 Slot 2

Anil alone can do a job in $20$ days while Sunil alone can do it in $40$ days. Anil starts the job, and after $3$ days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done $10\%$ of the job, then in how many days was the job done?

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)