Time and work for CDS

Time and work questions are about rates. A person who finishes a job in 12 days does 1/12 of it each day. Two people working together add their rates, not their days. Nearly every question in this topic, including pipes and cisterns, is that one idea in a different story.

Fractions like 1/12 + 1/18 get messy under time pressure, so most candidates use the total units (LCM) method instead. It turns every rate into a whole number and every question into simple division.

The total-units method

1. Take the total work as the LCM of the given times.
2. Each person's daily work = total units ÷ their time.
3. Add or subtract the daily work as the question requires.
4. Time = units to be done ÷ units done per day.

Example: A takes 12 days and B takes 18. LCM = 36 units. A does 3 units a day and B does 2, so together they do 5 a day and finish in 36 ÷ 5 = 7.2 days.

Why it works: the LCM is just a convenient size for the job. The answer does not depend on how you measure the work, only on the rates.

Efficiency and time

Efficiency (work per day) is inversely proportional to time. If A is twice as efficient as B, A takes half as long. If the times are in the ratio 3 : 4, the efficiencies are in the ratio 4 : 3.

Workers, days and hours

When the same kind of work is done by different groups, the total effort is constant:

• M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂

where M is the number of workers, D the days, H the hours a day and W the amount of work. Drop H or W if the question does not mention them.

Wages

Wages are shared in proportion to the work each person does, not in proportion to their time. If A and B work together for the whole job, their shares are in the ratio of their efficiencies.

Pipes and cisterns

A filling pipe is a worker adding units. An emptying pipe or leak is a worker with a negative rate. Everything else is the same method.

Worked questions

Question 1: A and B together can do a job in 10 days. A alone can do it in 15 days. How long would B take alone?

• LCM of 10 and 15 = 30 units. A + B do 3 a day, A does 2 a day.
• B does 1 unit a day, so B takes 30 days.

Question 2: A and B can do a job in 12 days, B and C in 15 days, and C and A in 20 days. How long would all three take together, and how long would A take alone?

• LCM = 60 units. A + B = 5, B + C = 4, C + A = 3 units a day.
• Adding: 2(A + B + C) = 12, so A + B + C = 6 units a day, and together they take 10 days.
• A = 6 − (B + C) = 6 − 4 = 2 units a day, so A alone takes 30 days.

Question 3: 15 workers working 8 hours a day finish a job in 12 days. How many days will 18 workers take, working 10 hours a day?

• 15 × 12 × 8 = 18 × D × 10.
• 1,440 = 180D, so D = 8 days.

Question 4: Pipe A fills a tank in 20 minutes and pipe B in 30 minutes. Pipe C empties the full tank in 15 minutes. If all three are opened together, how long will the tank take to fill?

• LCM = 60 units. A fills 3, B fills 2 and C empties 4 units a minute.
• Net = 3 + 2 − 4 = 1 unit a minute, so the tank fills in 60 minutes.

Question 5: A can finish a job in 8 days and B in 12 days. They work on alternate days, starting with A. In how many days will the job be finished?

• LCM = 24 units. A does 3 and B does 2, so each two-day cycle does 5 units.
• After 4 cycles (8 days), 20 units are done and 4 remain.
• Day 9: A does 3, leaving 1. Day 10: B needs only 1 of their 2 units, which takes half a day.
• Total = 9.5 days.

Question 6: A can do a job in 12 days and B in 18 days. A works for 4 days and leaves. How long does B take to finish the rest?

• LCM = 36 units. A does 3 a day, so in 4 days A does 12 units.
• 24 units remain. B does 2 a day, so B takes 12 days.

Practice set

1. A can do a job in 20 days and B in 30 days. How long will they take together?
2. Two pipes can fill a tank in 12 hours and 18 hours. How long will they take together?
3. 12 workers can finish a job in 15 days. How long will 20 workers take?
4. A can do a job in 12 days and B in 18 days. Together they are paid ₹4,500 for it. Find each person's share.
5. A is twice as efficient as B, and together they finish a job in 14 days. How long would A take alone?
6. A pipe can fill a tank in 6 hours. Because of a leak, it takes 8 hours. How long would the leak take to empty the full tank?
7. 10 workers can finish a job in 20 days. After 5 days, 5 more workers join. In how many days in total is the job finished?
8. A, B and C can finish a job in 10, 12 and 15 days. How long will all three take together?

Answers

1. 12 days. LCM 60: 3 + 2 = 5 units a day, and 60 ÷ 5 = 12.
2. 7.2 hours. LCM 36: 3 + 2 = 5 units an hour, and 36 ÷ 5 = 7.2.
3. 9 days. 12 × 15 = 180 worker-days, and 180 ÷ 20 = 9.
4. A ₹2,700, B ₹1,800. Efficiencies are 3 : 2 (LCM 36), so the shares are 3/5 and 2/5 of 4,500.
5. 21 days. A does 2 units and B does 1, so the job is 3 × 14 = 42 units. A alone takes 42 ÷ 2.
6. 24 hours. LCM 24: filling 4 units an hour, net 3, so the leak removes 1 unit an hour.
7. 15 days. The job is 200 worker-days. After 5 days, 150 remain for 15 workers, which takes 10 more days.
8. 4 days. LCM 60: 6 + 5 + 4 = 15 units a day, and 60 ÷ 15 = 4.

What to do next

• Solve every question this week with the total-units method, even the easy ones
• Practise three alternate-day questions and check the final day each time
• Move to speed, time and distance, which uses the same rate idea
• Time yourself on the time and work questions from the last five CDS papers