Time and Work for the Assam Police Constable Exam
Time and work is the chapter that measures how fast a job is done, and it is a regular favourite in the SLPRB Assam Police Constable (AB & UB) paper. The chapter is built on a single idea that is easy to state and easy to forget: the rate at which a person works is the reciprocal of the time that person needs to finish the job alone. Once this idea is understood, the whole chapter opens up, because working together means adding the rates, not the times. Questions about two masons building a wall, three typists preparing a file, a group of men and women completing a project, and pipes filling a tank are all solved by the same method. The reason this chapter rewards preparation so well is that the arithmetic is usually simple and the answer depends mainly on setting the problem up correctly. The approach followed here is the one from the Master Book: Maths Masti (D. Hurina), which fixes the total work as a convenient number, usually the lowest common multiple of the given times, so that every rate comes out as a whole number.
Why Time and Work Is a Favourite in the Assam Police Paper
The topic combines fractions, ratio and unitary reasoning in a single place, which makes it ideal for an objective paper. The numbers are usually chosen so that the answer is clean, and the same three or four question patterns are repeated year after year. A candidate who learns to convert times into rates and to fix the total work at a convenient value will find that even the harder questions fall into a single line of addition or subtraction.
Terms You Should Know
- Work: the total job to be done, treated as one unit or as a convenient number of units.
- Rate of work: the part of the job finished in unit time.
- Efficiency: the rate of work of a person compared with another.
- Days and hours: the units of time in which work is measured.
- Pipes and cisterns: filling and emptying problems expressed in the same language as work.
- Work and wages: payment proportional to the work done or to the rate of work.
The most useful habit is to state clearly whether a person is doing work, which adds to the total, or undoing it, which subtracts from the total, as in the case of a leak or an emptying pipe.
The Unitary Method and the Fraction of Work Done
If a person can finish a job in ten days, then in one day the person does one tenth of the job. If another can finish it in fifteen days, that person does one fifteenth in one day. Together they do one tenth plus one fifteenth in one day, which is one sixth of the job, so they finish it in six days. This is the entire unitary method, and it works because the rate is the reciprocal of the time. Writing the work in fractions and adding them is the safest method when the numbers are small.
The LCM Method and the Assumption of Total Work
When the numbers are awkward, fix the total work as the lowest common multiple of the given times. If one worker takes ten days and another fifteen, take the total work as thirty units. Then the first worker does three units a day and the second does two units a day, and together they do five units a day, so the job of thirty units is finished in six days. The LCM method turns fractions into whole numbers and is much less prone to error, which is why it is the preferred route in the examination hall.
Two Workers Together and the Basic Formula
When two workers together finish a job in a certain time, the rate of the pair is the sum of their rates, so the reciprocal of the combined time equals the sum of the reciprocals of the individual times. From this, if the combined time and one individual time are known, the other individual time is found in one step. A worker joining a pair subtracts from the time, and a worker leaving adds to it, and the change in the rate shows the effect at once.
More Than Two Workers
With three or more workers the method does not change: add all the rates and take the reciprocal for the combined time, or fix the total work at the lowest common multiple and add the daily units. Questions in which one worker leaves partway, or joins after some days, are handled by dividing the job into periods and adding the work done in each period. The total work done in every period must add up to the whole job, and this balance gives the equation.
Men, Women and Children and the Efficiency Ratio
When the question says that two men do the work of three women, or that a boy is half as efficient as a man, it is giving an efficiency ratio. Convert all the workers into a common unit, for example into equivalent men, and then apply the same method. The number of workers multiplied by the number of days gives the total work in man-days, and if the number of workers changes, the number of days changes in inverse proportion, provided the total work stays the same.
Work and Wages
Wages are shared in the ratio of the work done, which for workers of equal efficiency is the ratio of the number of days each worked, and for workers of different efficiency is the ratio of their work, that is, efficiency multiplied by time. A question that gives the total wage and the individual times can be solved by first finding the ratio of the work done and then dividing the wage in that ratio.
Pipes and Cisterns
A filling pipe works like a worker who adds to the tank, and an emptying pipe or a leak works like a worker who takes away. The net rate is the sum of the filling rates minus the sum of the emptying rates, and the time to fill is the capacity divided by this net rate. If the net rate is negative, the tank never fills, and a question that mentions a leak is really a time and work question in disguise.
Shortcut Techniques for the Exam Hall
Fix the total work at the lowest common multiple of the given times and work in whole units. For two workers, the combined time is the product of their times divided by their sum, a result worth remembering because it appears so often. When efficiency ratios are given, convert everyone into a single type of worker before calculating. For a leak or an emptying pipe, subtract the rate instead of adding it. Always check the answer for reasonableness: two workers together must finish faster than the faster of them alone.
Common Traps and Errors
- Adding the times instead of adding the rates of work.
- Forgetting that an emptying pipe or a leak subtracts from the total rate.
- Ignoring the efficiency ratio and treating all workers as equal.
- In work and wages, dividing the money equally when the workers worked for different times.
- Mixing days and hours without converting one into the other.
- Assuming the answer is always a whole number of days, when it may be a fraction.
A Study Plan for This Chapter
- Learn to convert a time into a rate, using the reciprocal.
- Practise both the unitary method and the lowest-common-multiple method until both are easy.
- Memorise the combined-time result for two workers.
- Work on questions in which a worker joins or leaves partway.
- Cover efficiency ratio, work and wages, and pipes and cisterns as a final group.
Frequently Asked Questions
Why do we add the rates and not the times?
Because the rate is the part of the job done in one day, and when two people work together their daily contributions add up. The time is the reciprocal of the rate, so adding rates is the correct way to combine two workers.
What is the quick way to find the time when two workers work together?
For two workers whose individual times are a and b, the combined time is the product a b divided by the sum a plus b. This follows from adding the reciprocals, and it is worth remembering.
Why is the lowest common multiple method safer than fractions?
Because it replaces fractional rates with whole numbers. If the total work is taken as the lowest common multiple of the given times, each worker's rate becomes a whole number of units per day, and whole numbers are easier to add and less prone to mistakes.
How does an emptying pipe change the calculation?
An emptying pipe or a leak works against the filling pipes, so its rate is subtracted from the total filling rate. If the emptying rate is greater than the filling rate, the tank never fills.
How are wages divided among workers of different efficiency?
Wages are shared in the ratio of the work actually done, which is the efficiency of each worker multiplied by the time that worker spent. Workers of equal efficiency simply share in the ratio of the days each worked.