Time and Work, Pipes and Cisterns

1. Take the total work as the LCM of the days given.

2. Divide the total work by each person's days to get that person's efficiency (units per day).

3. Add efficiencies for people working together, and subtract for those who undo the work.

4. Time = total work / combined efficiency.

Example. A takes 10 days and B takes 15 days. LCM = 30 units. A does 3 units a day, B does 2 units a day, together 5 units a day, so the job takes 30/5 = 6 days.

• An inlet pipe fills the tank (positive work). An outlet pipe or leak empties it (negative work).

• Net rate = sum of the inlet rates - sum of the outlet rates.

• If the net rate is negative, the tank empties, and the time to empty is 1/|net rate|.

• A leak: if a tank fills in x hours normally but y hours with a leak (y larger than x), the leak alone empties a full tank in xy/(y - x) hours.

Example 1. A can do a job in 12 days. A and B together do it in 8 days. How long does B alone take?

• xy/(y - x) = 8 x 12/(12 - 8) = 24 days. (LCM method: with a total of 24, A does 2 and the pair does 3 a day, so B does 1 a day.)

Example 2. A and B can finish a job in 12 days, B and C in 15 days, and C and A in 20 days. How long will A, B and C take together?

• 1/12 + 1/15 + 1/20 = (5 + 4 + 3)/60 = 12/60 = 1/5. This is twice the combined rate of the three.

• A + B + C do 1/10 per day, so they take 10 days.

Example 3. A takes 10 days and B takes 15 days. They start together, but B leaves after 3 days. How long does A take to finish the rest?

• Together they do 1/10 + 1/15 = 1/6 per day, so in 3 days they finish 1/2.

• The remaining 1/2 takes A alone 5 days. Total time is 3 + 5 = 8 days, and A works 5 more days.

Example 4 (alternate days). A takes 10 days and B takes 15 days, working on alternate days and starting with A. How long does the job take?

• In every two days they finish 1/10 + 1/15 = 1/6.

• Six such pairs finish exactly 1 job, so it takes 12 days.

Example 5 (wages). A takes 10 days and B takes 15 days, and together they earn 5000. How should it be split?

• Efficiency ratio = 3 : 2. A gets 3000 and B gets 2000.

Example 6 (pipes). Pipe A fills a tank in 12 hours, pipe B in 18 hours, and pipe C empties it in 36 hours. All are opened together. When is the tank full?

• Net rate = 1/12 + 1/18 - 1/36 = (3 + 2 - 1)/36 = 4/36 = 1/9. The tank fills in 9 hours.

Example 7 (men-days). 10 men finish a job in 12 days. How many men are needed to finish it in 8 days?

• Men x days is constant: 10 x 12/8 = 15 men.

Example 8 (efficiency). A is twice as good a worker as B, and together they finish a job in 12 days. How long does B alone take?

• Let B take x days, so A takes x/2. Then 1/x + 2/x = 3/x = 1/12, so x = 36. B takes 36 days and A takes 18 days.

• Choose the total work as the LCM to avoid fractions.

• "A is 25% more efficient than B" means the efficiency ratio is 5 : 4, so the time ratio is 4 : 5.

• Alternate days: compute the work done in one full cycle first, then see how many cycles are needed and how the last partial cycle ends.

• Dividing money: the wage share is proportional to work done, not simply to days worked.

• Pipes: convert hours and minutes to the same unit and treat every pipe as a signed rate.

• Adding times instead of rates.

• Forgetting to subtract a leak or outlet pipe.

• Using time ratios when efficiency ratios are needed (they are inverse).

• Ignoring who starts in an alternate-day question.