In this article, let us try to understand the concept of Time and work, some very important Time and Work Formulas, shortcuts and tricks.
You can’t dodge time and work problems for really long. Though this topic can be slightly tricky, however, if your basics are strong and you know how the time and work formulas were derived and which time and work formulas to use in specific kinds of problems you are sorted!
Time and Work Concepts
Time and work problems deal with the simultaneous performance involving the efficiency of an individual or a group and the time taken by them to complete a piece of work. Work is the effort applied to produce a deliverable or accomplish a task.
A certain amount of time (T) is taken to complete a certain work (W). The number of units of work done per unit time is called the rate of work (R). Hence, Work (W) = Rate (R) Time (T)
Whenever some work is done, the total work itself can be taken as one unit. Hence, we assume the total work done as one unit in the problems we encounter in order to simplify the computations. In these cases, R = 1 / T or T = 1 / R. In other words, R and T are inversely proportional as RT = W, which is a fixed quantity.
Time and Work Formulas
Before we plunge into time and work formulas, let’s quickly go through the different terms that make up time and work formulas
- If A can do a piece of work in n days, then A’s one day’s work = 1/n
- If A’s one day’s work = 1/n, then A can finish the work in n days.
- If A is thrice as good a workman B, then
- The ratio of work done by A and B = 3:1
- The ratio of time taken by A and B to finish work = 1:3
- Total work = No of days * Efficiency.
- If a group of people are given salary for a job they do together, their individual salaries are in the ratio of their individual efficiencies if they work for the same number of days. Otherwise, salaries are divided in the ratio of units of work done.
Approach for Time and Work Problems
Now that we know the terms used in time and work concepts, let’s establish the relationship between them. This will help you to tweak time and work formulas as per the need of the questions. Questions on Time and Work Formulas usually fall in one of the following categories:
- The relation between days taken by individuals to complete a given work independently and to complete while working simultaneously or alternately.
- Teams of men, women, children and time taken by the teams to complete work independently or while working simultaneously.
Most questions asked are standard types of questions. Also, the approach to solve questions is a very standard one. There are two approaches, which are exactly the same but seem to be different – an approach using the per day’s work and approaching LCM. The two approaches are best explained through an example.
Time and work Example
If A does a work in 10 days and B does the same work individually in 12 days, in how many days will the work be completed if they work simultaneously?
Working with different efficiencies | Time and Work Formulas
Problems discussed in this section are based on time and work formulas we have discussed above.
Approach 1: Per day’s work
If A can complete the work in ‘x’ days and B can complete the same work in ‘y’ days, when they work together, the time taken to complete the work is given below.
A can complete the work in ‘x’ days. So in one day, he will do 1/x of the work. B can complete the work in ‘y’ days. So in one day, he will do 1/y of the work. Total work done by both in one day = (1/x) + (1/y). Hence, the total time required to do the work = (xy)/(x +y) days.
Since A completes the entire work in 10 days, A does 1/10th of the work in 1 day.
Since B completes the entire work in 12 days, B does 1/12th of the work in 1 day.
Working simultaneously, they do 1/10 + 1/12 = 11/60 of the work in 1 day. Thus total days taken by both working simultaneously = 60/11 days.
Approach 2: LCM Method
In this method, we assume the total amount of work to be completed as a finite divisible value and based on it, we proceed with the calculation. To make the calculation simpler, assume the total amount of work to be completed as the LCM of time taken by different people to complete the same piece of work.
Let the amount of work be 60 units (LCM of 10 and 12). Since A does 60 units in 10 days, he does 6 units every day. Since B does 60 units in 12 days, he does 5 units every day. Working simultaneously, they do 6 + 5 = 11 units each day.
Thus to complete 60 units of work, they will take 60/11 days.
The two approaches are absolutely identical, it is just that in the earlier approach the work was assumed as 1 unit instead of 60 units.
Approach 3: The Chocolate Method
To make calculations more intuitive, the work can be assumed as chocolates to be consumed rather than units of work to be done. We take LCM of all the "number of days" mentioned in the question. This is done so that the work will be a multiple of these "number of days" and thus calculating efficiency will be easier.
For example, if A can do a piece of work in 9 days and B can do the same piece of work in 18 days, in how many days can they complete it working together?
In the chocolate method, we assume work as chocolates.
So, the total amount of chocolate to be consumed = 18 units (LCM of 9 and 18)
This means A can eat 18 chocolates in 9 days.
Chocolate consumed by A in one day = 2 units per day
B, on the other hand, can eat 18 chocolates in 18 days
Chocolate consumed by B in one day = 1 unit per day
Chocolate consumed by A and B in one day = 3 units per day
Time taken by them to consume 18 units of chocolate = 18/3 = 6 days
Work equivalence | Time and Work Formulas
In questions based on man-days concept, the basic assumption is that all men work with equal efficiency unless stated otherwise in the question. The relation between the number of people working (N), the number of days worked (D), the number of hours worked per day (H) and the quantity of work (W) for two different cases is given below:
The relation between the terms: The above equation has been derived based on the following relationships between the variables
- The number of people working is directly proportional to the amount of work done.
- The number of days worked is directly proportional to the amount of work done.
- The number of people working is inversely proportional to the number of days worked.
For example, if 12 men working for 14 days can dig 5 wells, in how many days will 28 men dig 10 wells? We assume that the total work to be done is the same. Therefore, x = 5 days
Division of wages | Time and Work Formulas
The wages paid for any task has to be divided among the workers in the proportion to their contribution towards the completion of the task. In other words, the money earned by completing a piece of work has to be divided in the proportion of the work done by them.
If the workers have worked for the same number of days, the money can be divided in the ratio of their efficiencies. Efficiency is inversely proportional to the time taken to complete a task.
Wages questions are of three types:
- Same efficiency and same number of days.
- Different efficiency but same number of days.
- Different efficiency and different number of days.
Let's look at the question together.
A can complete a task in 10 days and B can complete the same task in 15 days. They start working together but A works for only 1 day. The remaining work is completed by B. If the total wage is 1000, then What is B's share?
Now, the question says that A can complete a task in 10 days and B can do the same in 15 days
A = 10 days , B = 15 days
Remember the chocolate method? Total work assumed will be?
LCM of (10 , 15) = 30
So efficiency, A= 3 chocolates /day and B= 2 chocolates /day
A works for 1 day which means that the work done by him will be: eating 3 chocolates
Work left = 30 - 3 = 27
Now eating 27 chocolates is the work completed by B
The wage is divided in the ratio of the work done.
Ratio of work = 3:27 = 1:9
So the wage will be divided in the same ratio.
Wage of A : Wage of B = 1:9
So B's share = (9/10) x 1000
= Rs. 900
Practice more problems from Time and Work Formulas here.
Pipes and Cisterns | Time and Work Formulas
The concept of people working with different efficiencies is used to solve problems on Pipes and Cisterns also. The only difference is that, in this case, the work done is in terms of filling or emptying a cistern (tank) and the time is the time taken by a pipe or a leak (crack) to fill or empty a cistern respectively. If a pipe is connected with a cistern that fills it, then it is called an inlet pipe. If a pipe is connected with a cistern that empties it, then it is called an outlet pipe.
The work done is filling a cistern is taken as positive and the work done in emptying a cistern is taken as negative.
- Inlet: A pipe connected with a tank or a cistern or reservoir, that fills it, it is known as an inlet.
- Outlet: A pipe connected with a tank or a cistern or a reservoir, emptying it, it is known as an outlet.
- If a pipe can fill a tank in x hours, then the part filled in one hour = 1/x.
- If a pipe can empty a full tank in y hours, then the part emptied in 1 hour = 1/y.
- If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where y>x) then on opening both the pipes, the net part filled in one hour = (1/x – 1/y)
- If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where x>y), then on opening both the pipes, the net part emptied in one hour = (1/y -1/x).
Pipes and Cisterns Questions and Solutions | Time and Work Formulas
Question 1: a cistern is fitted with three taps, namely P, Q, and R. P and Q can fill a cistern in 10 and 15 minutes respectively whereas R can empty it in 12 minutes. If all the three pipes are kept open, in how much time will the cistern be filled?
Assume capacity of the tank = 60 liters
The rate at which tap P fills the cistern = 6 liters/min.
The rate at which tap Q fills the cistern = 4 liters/min and
The rate at which tap R empties the cistern = 5 liters/ min
The rate at which tap P, Q, and R fill the cistern = 5 litres/min
The Time taken to fill the cistern = 12 mins
Question 2: An inlet pipe can fill in an empty cistern in 30 minutes whereas a leak in the bottom of the cistern can empty a filled tank in 40 minutes. Find the time taken to fill the cistern when both the inlet pipe and the leak are on.
Part of the cistern that is filled each minute = 1/30 − 1/40 = 1 /120.
Thus entire cistern is filled in 120 minutes.