Time and Work
Time and work measures how long a piece of work takes when one or more persons or machines work together, and it is a dependable scoring chapter in the ADRE Grade IV, APSC and Assam Police papers. The whole chapter turns on one idea: if a person finishes a work in n days, the work done by that person in one day is one divided by n. The method followed in the Master Book: Maths Masti (D. Hurina) is to convert every worker into a rate of work per day, that is into a fraction, and then to add or subtract these fractions as the question requires. Once the candidate begins to think in terms of one day's work instead of the total days, every question becomes a simple sum of fractions, however many workers are involved. This habit of working with the daily rate is the single most useful skill in the chapter.
Basic Terms
- Work: the task to be completed, taken as one whole unit.
- Rate of work: the part of the work done in one day.
- Time: the number of days taken to finish the work.
- Efficiency: the rate of work of a person compared with another.
- Together: the combined rate when two or more persons work at the same time.
- Leaving and joining: a person stopping or starting work in the middle of the task.
If a person finishes a work in n days, one day's work is one divided by n. This single conversion is the foundation of the whole chapter.
The Fundamental Idea and Its Forms
Time is inversely proportional to the rate of work. The more efficient the worker, the less the time. If two persons can finish a work in a days and b days alone, then working together they finish it in a b divided by a plus b days. This is the most used result of the chapter, and it comes straight from adding the two daily rates one over a and one over b.
Two Persons Working Together
When two persons work together, their rates are added. The time taken together is the reciprocal of the sum of the two daily rates. If one person works alone for some days and then the second joins, the candidate finds the work already done, subtracts it from one, and divides the remaining work by the combined rate. Keeping the total work as one unit makes the arithmetic clean.
More Than Two Workers
When three or more persons work, the method does not change: convert each person into a daily rate, add them, and take the reciprocal. Questions in which some workers join after a few days, or leave before the work is finished, are solved by multiplying each group by the number of days that group actually worked. The sum of all the work done must equal one.
Men, Women and Children
Some questions replace persons by groups, such as two men equalling three women. Here the candidate must first bring every worker to a common unit, for example express one woman as two-thirds of a man, and then apply the usual method. The comparison of efficiencies is given so that the whole group can be reduced to a single type of worker.
Work and Wages
When a wage is to be shared, it is divided in the ratio of the work done, not in the ratio of the number of days alone. The work done is the rate multiplied by the time, so the wages follow the product of efficiency and time. This is a common extension of the chapter in the examination.
Pipes and Cisterns
Filling pipes and emptying pipes are the same problem in another dress. A filling pipe has a positive rate and an emptying pipe a negative rate. The net rate is the difference of the two, and the time to fill is the reciprocal of the net rate. If the net rate is negative, the tank is emptying rather than filling.
How This Chapter Appears in the Examination
In ADRE Grade IV, APSC and Assam Police papers the questions usually ask for the time taken by two or three persons working together, the time when one joins or leaves in the middle, the time for a group of mixed workers, or the share of wages. Pipes and cisterns questions also appear, and the method is identical once the signs of the rates are noted.
Shortcut Methods for Rapid Calculation
For two workers use the product over sum formula and avoid writing the addition of fractions. Keep the total work as one so that the remaining work is easy to find. When the same number of men work for two different periods, compare the total man-days. For pipes use signed rates and add them before taking the reciprocal.
Common Mistakes to Avoid
- Adding the times instead of the rates of work.
- Forgetting to subtract the work already completed before the second worker joins.
- Using the number of days alone to divide the wages instead of the work done.
- Ignoring the sign of an emptying pipe or a worker who leaves.
- Changing the number of workers without changing the rate accordingly.
How to Prepare This Topic
- Learn to convert days into a daily rate at once.
- Memorise the two-worker product-over-sum result.
- Practise joining and leaving questions with the total work as one.
- Work on the men-women-children and wage-sharing forms.
- Finish with pipes and cisterns, and revise the method weekly.
Frequently Asked Questions
Why do we add rates and not times?
Because two workers together do more work in a day, not less. Adding the daily rates gives the combined rate, and the time is the reciprocal of that combined rate.
How do I solve a question where one worker joins later?
Find the work done by the first worker alone up to the joining day, subtract it from one, and divide the remaining work by the combined rate of all workers present.
How are wages divided in these questions?
The wage is shared in the ratio of the work done by each person, which is the rate multiplied by the time. A more efficient worker who works fewer days may still earn the same as a slower worker who works longer.
How is a pipes and cisterns problem related to time and work?
It is the same problem. A filling pipe is a positive rate and an emptying pipe a negative rate. Add the signed rates and take the reciprocal to get the time to fill the tank.
What should I do with a group of mixed workers?
First convert everyone to a common unit using the given efficiency relation, then add the daily rates as usual. The comparison exists only to reduce the group to one type of worker.