5. Negative rate of completion

Negative rate of completion questions are typically pipes, drains/leaks and tanks based. For instance, Pipe A and Pipe B can fill a tank in $2$ hours and $4$ hours while drain C can empty a tank in $6$ hours. If all three are open at the same time, how long does it take to fill an empty tank?

In such questions the efficiency of a leak, or something undoing the work, will be negative.

Example 19

Tap X can fill a tank in $4$ hours and tap Y can fill the same tank in $6$ hours. A drain in the bottom of the tank can empty a full tank in $12$ hours. If the two taps and the drain are opened at the same time, how long does it take an empty tank to be filled?

Solution

Portion of tank filled by taps X and Y in $1$ hour are $\dfrac{1}{4}$ and $\dfrac{1}{6}$ respectively.

Portion of tank emptied by the drain in $1$ hour is $\dfrac{1}{12}$.

Portion of tank filled in $1$ hour $= \dfrac{1}{4} + \dfrac{1}{6} - \dfrac{1}{12} = \dfrac{1}{3}$

Time taken to fill the tank $= 3$ hours

Alternatively

Let the tank's capacity be LCM$(4, 6, 12) = 12$ units.

Taps X and Y fill the tank at the rate of $\dfrac{12}{4}$= 3 units/hr and $\dfrac{12}{6}$ = 2 units/hr respectively.

The drain empties the tank at the rate of $\dfrac{12}{12}$ = 1 unit/hr

When all are open, rate of filling $= 3 + 2 - 1$ = 4 units/hr

Time taken to fill the tank $= \dfrac{12}{4} = 3$ hours

Answer: $3$ hours

Example 20

Pipes A and B take $5$ hours and $8$ hours respectively to fill an empty tank. Both the pipes are turned on at the same time. Exactly after two hours, a leak develops due to the pressure in the tank. It, therefore, takes $2$ more hours for the tank to be filled. If the water flowing out of the leak was at a steady rate, how long would it take a filled tank to be emptied by the leak?

Solution

Let the leak take $x$ hours to empty the tank.

Portion of the tank filled by the $2$ pipes in $2$ hours is $2\left( \dfrac{1}{5} + \dfrac{1}{8} \right) = \dfrac{13}{20}$

As it takes $2$ more hours to fill the remainder of the tank, which is $1 - \dfrac{26}{40} = \dfrac{7}{20}$.

$2 \left( \dfrac{1}{5} + \dfrac{1}{8} - \dfrac{1}{x} \right) = \dfrac{7}{20} $

⇒$ \dfrac{13}{40} - \dfrac{7}{40} = \dfrac{1}{x}$

⇒ $ x = \dfrac{20}{3}$ = 6 hrs 40 mins

Alternatively

Let the tank's capacity be LCM$(5, 8) = 40$ units.

Pipes A and B fill the tank at the rate of $\dfrac{40}{5}$ = 8 units/hr and $\dfrac{40}{8}$ = 5 units/hr respectively.

Volume filled by A and B in $2$ hours $= 2 \times (8 + 5) = 26$ units

Let the leak empty the tank at the rate of $\bm{x}$ units/hr.
Volume of tank that is empty $= 40 - 26$ = 14 units

Volume filled by A, B and the leak in the next $2$ hours $= 2 \times (8 + 5 - x) = 14$

⇒ $ 26 - 2\text{x} = 14$
⇒ $ x $ = 6 units/hr

Time taken by the leak to empty the tank $= \dfrac{40}{6} = \dfrac{20}{3}$ = 6 hrs 40 mins

Answer: $6$ hrs $40$ mins

Example 21

A pipe usually takes $3$ hours to fill an empty tank with water. However, on a particular day, it took $7.5$ hours to fill the tank due to a leak it had developed. Then, how many hours would the leak take to empty a full tank?

Solution

Let the time taken by the leak to empty a full tank be $t$.

$\dfrac{1}{3} - \dfrac{1}{t} = \dfrac{1}{7.5}$ ⇒ $\dfrac{1}{t} = \dfrac{1}{3} - \dfrac{2}{15}$

⇒$ t = 15$

Answer: $5$

6. People or objects working alternately

These questions involve objects or group of objects with different efficiencies taking turns and working. In these questions you need to calculate the portion of work completed in a cycle and then find out the point where a single person or object's work would render the work complete. The Parts Method is best to use here.

Example 22

When working alone, A, B and C take $12, 15$ and $24$ days to complete a project. A works alone on the project on the first day, B works alone on the second day and C works alone on the third day. This pattern continues with each of the individuals working alone every $3$ days. In how many days is the work completed?

Solution

Portion of work completed every $3$ days $= \dfrac{1}{12} + \dfrac{1}{15} + \dfrac{1}{24} = \dfrac{10 + 8 + 5}{120} = \dfrac{23}{120}$

Let's look at the work as involving $120$ parts. A, B and C can finish $10, 8$ and $5$ parts in a day respectively and in $3$ days $23$ parts of work get completed.

In 5 such cycles involving $15$ days, $5 \times 23 = 115$ parts get completed and $5$ are remaining.

It's A's turn now, who can complete $10$ parts in a day. Since there are $5$ parts left, A can complete in $\dfrac{5}{10} = 0.5$ day. Therefore, the work gets completed in $15.5$ days.

Answer: $15.5$

Example 23

A and C are pipes that take $10$ minutes and $60$ minutes to fill a tank. B and D are leaks that take $40$ and $24$ minutes to empty a tank. At any given time only a tap or a drain is open. A is turned on for the first minute, B is then turned on for the second minute, C for the third, D for the fourth, A for fifth, B for the sixth and so on. This is repeated till the tank is full. How long does it take for the tank to be filled?

Solution

In one cycle of $4$ minutes, portion of tank filled $= \dfrac{1}{10} - \dfrac{1}{40} + \dfrac{1}{60} - \dfrac{1}{24} = \dfrac{12 - 3 + 2 - 5}{120} = \dfrac{6}{120}$

Once again, the denominator can be looked at as the total work (i.e., 120 parts). In one cycle of $4$ minutes, $6$ parts are filled. Parts of tank filled by Pipe A, Leak B, Pipe C and Leak D are $+12, -3, +2$ and $-5$ respectively.

The maximum that a pipe can fill in $1$ minute is Pipe A which is $12$ parts. So, we can find out the number of cycles that fill $120 - 12 = 108$ parts to provide for the possibility of a pipe filling the rest of the tank.

Therefore, in $18$ cycles or $72$ minutes, $18 \times 6 = 108$ parts are filled and $12$ parts are remaining. As it is A's turn for the $73^\text{rd}$ minute and it fills $12$ parts, the work is complete in $73$ minutes.

Answer: $73$ minutes

7. Past Questions

Question 1

There's a lot of work in preparing a birthday dinner. Even after the turkey is in oven, there are still the potatoes and gravy, yams, salad, and cranberries, not to mention setting the table. Three friends, Asit, Arnold, and Afzal, work together to get all of these chores done. The time it takes them to do the work together is six hours less than Asit would have taken working alone, one hour less than Arnold would have taken, and half the time Afzal would have taken working alone. How long did it take them to do these chores working together?
[CAT 2001]

		$20$ minutes
		$30$ minutes
		$40$ minutes
		$50$ minutes

Question 2

A water tank has M inlet pipes and N outlet pipes. An inlet pipe can fill the tank in $8$ hours while an outlet pipe can empty the full tank in $12$ hours. If all pipes are left open simultaneously, it takes $6$ hours to fill the empty tank. What is the relationship between M and N?
[XAT 2016]

		M : N $= 1 : 1$
		M : N $= 2 : 1$
		M : N $= 2 : 3$
		M : N $= 3 : 2$
		None of the above

Question 3

A mother along with her two sons is entrusted with the task of cooking Biryani for a family get together. It takes $30$ minutes for all three of them cooking together to complete $50$ percent of the task. The cooking can also be completed if the two sons start cooking together and the elder son leaves after $1$ hour and the younger son cooks for further $3$ hours. If the mother needs $1$ hour less than the elder son to complete the cooking, how much cooking does the mother complete in an hour?
[IIFT 2013]

		$33.33 \%$
		$50 \%$
		$66.67 \%$
		None of the above