Here we will start by introducing the formula of Time and Work and then solve some examples with the help of the LCM method as mentioned in the heading. The LCM method is one of the easiest and simplest way to solve time and work questions. Let’s us now discuss this chapter where the questions are based on basic relationships between the different variables of time and work.
As we all know that in Time and Work, we have different variables like work which is noted by W, the number of persons which is denoted by P and the time which is denoted by T. So let us take the relationships between these variables.
Now we can say that the the work done is proportional to Number of persons doing the work if the time is constant.
W ∝ P (T = constant)
So when more number of persons are working together then amount of work done will be more and when less number of persons are working then the amount of work done will be less, at a given same time.
The other relationship here is between work and time, we can say that the work done is proportional to time when the number of persons are constant.
W ∝ T (P = constant)
So when the Number of persons are not changing, for less work less time is required and for more work more time is required which also means that as the work done increases the time required to do the work also increases and vice versa.
And the third relationship here is between number of persons and time, we can say that the number of persons are inversely proportional to time when the work is constant.
P ∝ 1/T (W = constant)
The variable Time can be classified into two, first one in number of days for which work is done and second one in the number of hours per day for which work is done.
Now let’s look on the important formula on which mostly more than half of the questions from Time and Work are based on.
Efficiency ∝ Work / Time
where Efficiency is the per day work for a person. Efficiency is also termed as Capacity. So if in any question you come across the word capacity that simply means that the question is talking about efficiency. As Capacity and Efficiency are both same.
Also it is necessary to remember that number of days or time required to do the work is inversely proportional to the capacity. For example, if A completes the work in 5 days then his capacity will be 1/5. Let’s now take some questions based on this chapter.
Q.1) A can complete a work in 10 days and B can complete the same work in 15 days. If they work together, in how many days the work will be completed.
Let work = 1; Capacity of A = 1/10
Capacity of B = 1/15
Combined capacity of A and B = 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6
So time taken = 6 days [ANS]
Alternative Approach : The LCM Method
Let work = 30 units (LCM of 10 & 15)
A’s Capacity = 30/10 = 3 units/day
B’s Capacity = 30/15 = 2 units/day
So (A+B) Capacity = 5 units/day
And we know,
Time taken = Work / Capacity = 30 / 5 = 6 days [ANS]
Q.) A and B together can complete a work in 24 days, B and C completes the same work in 30 days & A and C can complete it in 40 days. Find the time taken by A alone to complete the work.
Solution : We will solve this question by LCM Method, as it’s one of the easiest and fastest way to get the answer.
Let Work = 120 units (LCM of 24,30 & 40)
Capacity of (A + B) = 120/24 = 5 units
Capacity of (B + C) = 120/30 = 4 units
Capacity of (A + C) = 120/40 = 3 units
Total Capacity = 2 (A + B + C) = 12 units
Or we can say that, (A + B + C) = 6 units
Here in question since it’s asked to calculate the time taken by A alone to complete the work, we will take the capacity of A alone.
Capacity of A = 6 – Capacity of (B+C)
So, the capacity of A = 6 – 4 = 2 units.
Therefore, Time Taken by A alone to complete the work = 120/2 = 60 days [ANS]
Q.) A can do the work in 10 days and B can do the same work in 15 days. If B starts and A joins him after 5 days, then in how many days the work will be completed ?
Solution : Solving this question by LCM Method.
Let Work = 30 units (LCM of 10 and 15)
Efficiency of A = 30/10 = 3 units
Efficiency of B = 30/15 = 2 units.
As said in question that A joins B after 5 days that means B was working for the first 5 days from starting alone. Hence,
Work done by B in 5 days = 2 × 5 = 10 days
Remaining Work = Total Work – Work done by B alone,
So the left work = 30 – 10 = 20 units.
Here the remaining work is done by both A and B together,
Therefore time taken by (A+B) = 20/5 = 4 days.
And Total Time taken to complete the whole work = 5 + 4 = 9 days [ANS]
Q.) Working together A and B can complete the work in 12 days. They work together for 9 days, after which B leaves the work. If A finishes the remaining work in 5 days, then find the time taken by B to finish the work alone.
Time take by (A+B) = 12 days
Capacity of (A+B) = 1/12,
Let total work = 1
The formula we will use here,
Work done = Capacity × Time
Now according to question, A and B together work for 9 days and the remaining work was done by A alone in 5 days. So it can be represented as,
Capacity of (A+B) × 9 + Capacity of A × 5 = 1
or, 1/12 × 9 + Capacity of A × 5 = 1
Capacity of A × 5 = 1 – 3/4
So, capacity of A = 1/(4×5) = 1/20.
From this we can find the capacity of B,
Hence, Capacity of B = 1/12 – 1/20
or Capacity of B = (5-3)/60 = 2/60 = 1/30
Thus time taken by B to complete the whole work will be 30 days [ANS]
That’s all in this article, the important point which always need to be remembered while solving Time and Work problems is that capacity and efficiency are the same thing. We will be publishing more articles on Time and Work with solved examples, so stay connected with us.
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